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14/1Y. Bugeaud

Around the Littlewood conjecture in Diophantine approximation

Z. Chonoles, J. Cullinann, H. Hausman, A. M. Pacelli, S. Pegado and F. WeiArithmetic Properties of Generalized Rikuna Polynomials

A. GalateauUn théorème de zéros dans les groupes algébriques commutatifs

A. MohamedWeight reduction for cohomological mod p modular forms over imaginaryquadratic fields

J. Sijsling and J. VoightOn computing Belyi maps

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Y. BugeaudAround the Littlewood conjecture in Diophantine approximation . . . . . . . . . . . . . 5-18

Z. Chonoles, J. Cullinann, H. Hausman, A. M. Pacelli, S. Pegado and F. WeiArithmetic Properties of Generalized Rikuna Polynomials . . . . . . . . . . . . . . . . . . . . 19-33

A. GalateauUn théorème de zéros dans les groupes algébriques commutatifs . . . . . . . . . . . . . . 35-44

A. MohamedWeight reduction for cohomological mod p modular forms over imaginaryquadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45-71

J. Sijsling and J. VoightOn computing Belyi maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-131

AROUND THE LITTLEWOOD CONJECTURE IN DIOPHANTINEAPPROXIMATION

by

Yann Bugeaud

Abstract. — The Littlewood conjecture in Diophantine approximation claims thatinfq≥1

q · qα · qβ = 0

holds for all real numbers α and β, where · denotes the distance to the nearest integer. Itsp-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that

infq≥1

q · qα · |q|p = 0

holds for every real number α and every prime number p, where | · |p denotes the p-adic absolutevalue normalized by |p|p = p−1. We survey the known results on these conjectures and highlightrecent developments.

Résumé. — En approximation diophantienne, la conjecture de Littlewood stipule que tous lesnombres réels α et β vérifient

infq≥1

q · qα · qβ = 0,

où · désigne la distance à l’entier le plus proche. Son analogue p-adique, formulé par deMathan et Teulié en 2004, affirme que l’égalité

infq≥1

q · qα · |q|p = 0

est valable pour tout nombre réel α et tout nombre premier p, où | · |p est la valeur absoluep-adique normalisée par |p|p = p−1. Nous donnons un survol des résultats connus sur cesconjectures en insistant sur les développements récents.

A famous open problem in simultaneous Diophantine approximation, called the Littlewoodconjecture, claims that, for every given pair (α,β) of real numbers, we have

infq≥1

q · qα · qβ = 0,

where · denotes the distance to the nearest integer. According to Montgomery [27], thefirst occurrence of the Littlewood conjecture appeared in 1942 in a paper by Spencer [34], astudent of Littlewood.

2010 Mathematics Subject Classification. — 11J04, 11J13, 11J61.Key words and phrases. — Simultaneous approximation, Littlewood conjecture.

6 Around the Littlewood conjecture in Diophantine approximation

Since 2000, there has been much activity on and around the Littlewood conjecture, includingthe formulation by de Mathan and Teulié [26] of a closely related open problem, called themixed Littlewood conjecture. The purpose of the present survey is to highlight recent resultsand developments on these questions. We make the choice to state more than twenty theoremsand to give only a single proof.Section 1 is devoted to the Littlewood conjecture itself, while the mixed and the p-adic (aspecial case of the mixed) Littlewood conjectures are addressed in Section 2. The readerwill observe that the state-of-the-art regarding the Littlewood and the p-adic Littlewoodconjectures is essentially the same. The proof of one result from [5] is given in Section 3.We conclude in Section 4 by mentioning recent results and open questions on inhomogeneousvariations of the Littlewood and the mixed Littlewood conjectures.The number of papers just appeared, being submitted or in preparation shows that there iscurrently a lot of activities on this topic.Throughout, we assume that the reader is familiar with the theory of continued fractions and‘almost all’ (or ‘almost every’) always refers to the Lebesgue measure.

1. The Littlewood conjecture

Let α and β be real numbers. Clearly,

infq≥1

q · qα · qβ = 0 (1.1)

always holds when 1,α,β are linearly dependent over the rationals or when α or β hasunbounded partial quotients in its continued fraction expansion. Thus, we may assume thatα and β belong to the set Bad of badly approximable real numbers, where

Bad = α ∈ R : infq≥1

q · qα > 0.

The set Bad is the set of real numbers whose sequence of partial quotients is bounded. It haszero Lebesgue measure and full Hausdorff dimension (that is, Hausdorff dimension one).In 1955 Cassels and Swinnerton-Dyer [14] made the first significant contribution on theLittlewood conjecture in showing that (1.1) holds when α and β belong to the same cubicfield. Note that it is still not known whether or not cubic real numbers belong to Bad.Pollingon and Velani [32] showed in 2000 that, for every badly approximable real number α,there exist uncountably many badly approximable real numbers β such that a strong formof (1.1) holds for the pair (α,β).

Theorem 1. — For every real number α in Bad, there exists a subset G(α) of Bad with

full Hausdorff dimension such that, for any β in G(α), there exist arbitrarily large integers qsatisfying

q · (log q) · qα · qβ ≤ 1.

For an alternative proof of a slightly weaker form of Theorem 1, together with some additionalinteresting results, the reader is referred to [22]; see also Theorem 29 in Section 4.Einsiedler, Katok, and Lindenstrauss [16] (see also [35]) established that the set of exceptionsto the Littlewood conjecture is very small.

Publications mathématiques de Besançon - 2014/1

Y. Bugeaud 7

Theorem 2. — The set of pairs (α,β) of real numbers such that

infq≥1

q · qα · qβ > 0

has Hausdorff dimension zero. Furthermore, it is contained in a countable union of compact

sets of box dimension zero.

Furthermore, Lindenstrauss [24] stressed that one may deduce from the techniques of [16] anexplicit, sufficient criterion for a real number α in order that the Littlewood conjecture holdsfor every pair (α,β), where β is an arbitrary real number.To present his result (and also subsequent results given in Section 2), we adopt a point ofview from combinatorics on words. We look at the continued fraction expansion of a givenreal number α as an infinite word. For an infinite word w = w1w2 . . . and an integer n ≥ 1,we denote by p(n,w) the number of distinct blocks of n consecutive letters occurring in w,that is,

p(n,w) := Cardw+1 . . . w+n : ≥ 0.The function n → p(n,w) is called the complexity function of w. For an irrational real numberα = [a0; a1, a2, . . .], we set

p(n,α) := p(n, a1a2 . . .), n ≥ 1.

Clearly, for positive integers n, n we have

p(n+ n,α) ≤ p(n,α) · p(n,α).

This inequality implies that the sequence (log p(n,α))n≥1 is subadditive, thus, the sequence((log p(n,α)/n)n≥1 converges.

Definition 3. — The entropy of a real number α is the quantity

E(α) = limn→+∞

log p(n,α)

n.

It is an easy exercise to show that the set of real numbers α such that E(α) = 0 has Hausdorffdimension zero.With the above notation, the result alluded to below Theorem 2 and stated as Theorem 5 in[24] can be formulated as follows.

Theorem 4. — If the real number α satisfies E(α) > 0, then, for every real number β, we

have

infq≥1

q · qα · qβ = 0.

Theorem 1 is a metrical result and, as such, does not tell us how to associate explicitly tosome given badly approximable number α a badly approximable number β such that 1,α andβ are linearly independent over the integers and (1.1) holds for the pair (α,β). This problemhas been addressed in [1] (see [25] for a weaker previous result).

Theorem 5. — Let ϕ be a positive, non-increasing function defined on the set of positive

integers and satisfying ϕ(1) = 1, limq→+∞ ϕ(q) = 0 and limq→+∞ qϕ(q) = +∞. Given α in



Bad, there exists an uncountable subset Bϕ(α) of Bad such that, for any β in Bϕ(α), there

exist infinitely many positive integers q with

q · qα · qβ ≤ 1

q · ϕ(q) . (1.2)

In particular, the Littlewood conjecture holds for the pair (α,β) for any β in Bϕ(α). Further-

more, the set Bϕ(α) can be effectively constructed.

The proof of Theorem 5 rests on the theory of continued fractions. For given α and ϕ,we construct inductively the sequence of partial quotients of a suitable real number β suchthat (1.2) holds for the pair (α,β).Going back to metrical results, the following theorem of Gallagher [19] shows that (1.1) canbe improved for almost all pairs (α,β) of real numbers.

Theorem 6. — Let ψ : Z≥1 → R be a non-negative decreasing function. Then, for almost

every pair (α,β) of real numbers, the inequality

qα · qβ ≤ ψ(q)

has infinitely (resp. finitely) many integer solutions q ≥ 1 if the series

q≥1 ψ(q) log q diverges

(resp. converges). In particular, for almost every pair (α,β) of real numbers, we have

infq≥2

q · (log q)2 · qα · qβ = 0.

Since we are, at present, not able to confirm nor to deny the Littlewood conjecture, we maysearch for pairs (α,β) of real numbers for which there exists a slowly growing function ϕ suchthat

lim infq→+∞

q · ϕ(q) · qα · qβ > 0. (1.3)

In view of Theorem 6, a first non-trivial step is to show the existence of pairs (α,β) forwhich (1.3) holds with the function q → ϕ(q) = (log q)2. This has been done in 2011 in [12],by means of a method introduced by Peres and Schlag [31]. This result has been subsequentlyconsiderably improved by Badziahin [3], who used an intricate Cantor-type construction toestablish the following theorem.

Theorem 7. — For every real number α in Bad, the set of real numbers β such that

infq≥3

q · log q · log log q · qα · qβ > 0

has full Hausdorff dimension. In particular, the set of pairs (α,β) of real numbers satisfying

infq≥3

q · log q · log log q · qα · qβ > 0

has full Hausdorff dimension in R2.

It remains an open problem to show the existence of pairs (α,β) of real numbers for whichinequality (1.3) holds with the function ϕ : q → log q.


Y. Bugeaud 9

2. The mixed and the p-adic Littlewood conjectures

In 2004, de Mathan and Teulié [26] proposed a mixed Littlewood conjecture which can bestated as follows. Let D = (dk)k≥1 be a sequence of integers greater than or equal to 2. Sete0 = 1 and, for n ≥ 1,

en =

1≤k≤n

dk.

For an integer q, setwD(q) = supn ≥ 0 : q ∈ enZ

and|q|D = 1/ewD(q) = inf1/en : q ∈ enZ.

When D is the constant sequence equal to p, where p is a prime number, then | · |D is theusual p-adic value | · |p normalized by |p|p = p−1. In analogy with the Littlewood conjecture,de Mathan and Teulié formulated the following conjecture.

Conjecture 8. — (Mixed Littlewood Conjecture) For every real number α and every sequence

D as above, we have

infq≥1

q · qα · |q|D = 0. (2.1)

Obviously, (2.1) holds if α is rational or has unbounded partial quotients. Thus, we onlyconsider the case when α is an element of the set Bad defined in Section 1.By Lemme 3.1 of [26], if (α,D) does not satisfy (2.1), then there exists a real number M suchthat all the partial quotients of the real numbers enα, n ≥ 0, are less than M . Here andbelow, · denotes the fractional part function.De Mathan and Teulié proved that (2.1) and even the stronger statement

lim infq→+∞

q · log q · qα · |q|D < +∞

holds for every real quadratic number α, provided that the sequence D is bounded; see [5, 23]for alternative proofs when D is the constant sequence equal to a prime number, a particularcase which deserves to be highlighted.

Conjecture 9. — (p-adic Littlewood Conjecture) For every real number α and every prime

number p, we have

infq≥1

q · qα · |q|p = 0. (2.2)

Einsiedler and Kleinbock [17] showed that a slightly weaker form of the p-adic Littlewoodconjecture, namely Theorem 11 below, can easily be deduced from the following theorem ofFurstenberg [18].

Theorem 10. — Let r and s be multiplicatively independent integers. Then, for every ir-

rational number α, the set of real numbers αrmsn, where m and n run through the set of

non-negative integers, is dense in [0, 1].

An alternative proof of Theorem 10 was given by Boshernitzan [8] and is reproduced in themonograph [13].



Theorem 11. — Let p1 and p2 be distinct prime numbers. Then,

infq≥1

q · qα · |q|p1 · |q|p2 = 0

holds for every real number α.

Bourgain, Lindenstrauss, Michel and Venkatesh [9] established a quantitative version ofTheorem 11.

Theorem 12. — Let p1 and p2 be distinct prime numbers. There exists a positive real number

c such that, for any real number α, we have

infq≥16

q · (log log log q)c · qα · |q|p1 · |q|p2 = 0.

Harrap and Haynes [21] managed recently to extend Theorem 11. We quote below theirCorollary 1. For an integer a ≥ 2 and for D being the infinite sequence a, a, . . ., we write | · |ainstead of | · |D.

Theorem 13. — Let a ≥ 2 be an integer and D be a bounded sequence of integers coprime

to a and greater than or equal to 2. Then,

infq≥1

q · qα · |q|a · |q|D = 0

holds for every real number α.

The proof of Theorem 13 is a nice combination of ideas from [9, 15] and lower bounds forlinear forms in logarithms of algebraic numbers (Baker’s theory).Einsiedler and Kleinbock [17] established that the set of possible exceptions to the p-adicLittlewood conjecture is very small from the metric point of view.

Theorem 14. — Let p be a prime number. The set of real numbers α such that

infq≥1

q · qα · |q|p > 0

has Hausdorff dimension zero.

Theorem 14 is the analogue of Theorem 2. Einsiedler and Kleinbock also explained how tomodify their proof to get an analogous result when D is the constant sequence equal to aninteger a ≥ 2 (not necessarily prime).The analogue of Theorem 4 was very recently proved in [5].

Theorem 15. — If the real number α satisfies E(α) > 0, then for every prime number p we

have

infq≥1

q · qα · |q|p = 0.

Theorem 15 asserts that the complexity function of the continued fraction expansion of everypotential counterexample to the p-adic Littlewood conjecture cannot grow exponentially fast.We present now various explicit examples of real numbers α in Bad for which (2.2) andeven (2.1) hold. First, we need some classical results and definitions from combinatorics onwords.


Y. Bugeaud 11

A well-known result of Morse and Hedlund [28, 29] asserts that p(n,w) ≥ n + 1 for n ≥ 1,unless w is ultimately periodic (in which case there exists a constant C such that p(n,w) ≤ Cfor n ≥ 1). Infinite words w satisfying p(n,w) = n+1 for every n ≥ 1 do exist and are calledSturmian words. We start with a classical definition (see e.g. [2]).

Definition 16. — An infinite word w is recurrent if every finite block occurring in w occurs

infinitely often.

Classical examples of recurrent infinite words include periodic words, Sturmian words, theThue–Morse word, etc.

Theorem 17. — Let (ak)k≥1 be a sequence of positive integers. If there exists an integer

m ≥ 0 such that the infinite word am+1am+2 . . . is recurrent, then, for every sequence D of

integers greater than or equal to 2, the real number α := [0; a1, a2, . . .] satisfies

infq≥1

q · qα · |q|D = 0.

The proof of Theorem 17, given in Section 3, is elementary, in the sense that it uses only thetheory of continued fractions.As a particular case, Theorem 17 asserts that (2.1) holds for every quadratic number α andevery (bounded or unbounded) sequence D of integers greater than or equal to 2.As shown in [5], Theorem 17 implies a non-trivial lower bound for the complexity function ofthe continued fraction expansion of a putative counter-example to (2.1).

Corollary 18. — Let α be a real number such that

limn→+∞

p(n,α)− n < +∞.

Then, for every sequence D of integers greater than or equal to 2, we have

infq≥1

q · qα · |q|D = 0.

The next corollary of Theorem 17 deals with a special family of infinite recurrent words. Afinite word w1 . . . wn is called a palindrome if wn+1−h = wh for h = 1, . . . , n.

Corollary 19. — Let (ak)k≥1 be a sequence of positive integers. If there exists an increasing

sequence (nj)j≥1 of positive integers such that a1 . . . anj is a palindrome for j ≥ 1, then, for

every sequence D of integers greater than or equal to 2, the real number α := [0; a1, a2, . . .]satisfies

infq≥1

q · qα · |q|D = 0.

To derive Corollary 19 from Theorem 17, it is sufficient to note that, if a1 . . . an and a1 . . . an

are palindromes with n > 2n, then

an−n+1 . . . an = an . . . a1 = a1 . . . an.

The corollary then follows from Theorem 17 applied with m = 0.The next result asserts that the mixed Littlewood conjecture holds for every prime numberp and every real number α whose sequence of partial quotients contains arbitrarily longconcatenations of a given finite block.



Theorem 20. — Let α = [a0; a1, a2, . . .] be a real number. Let T ≥ 1 be an integer and

b1, . . . , bT be positive integers. If there exist two sequences (mk)k≥1 and (hk)k≥1 of positive

integers with (hk)k≥1 being unbounded and

amk+j+nT = bj , for every j = 1, . . . , T and every n = 0, . . . , hk − 1,

then, for every prime number p, we have

infq≥1

q · qα · |q|p = 0.

The following consequence of Theorem 20 deserves to be pointed out. Let α be a realnumber having exactly m distinct partial quotients in its continued fraction expansion. IfE(α) = logm, then for every prime number p we have

infq≥1

q · ||qα|| · |q|p = 0.

Clearly, this has been superseded by Theorem 15.The assumption of Theorem 20 can be restated as follows. To an irrational real numberα := [a0; a1, a2, . . .] we associate the set

Adh(α) := [0; am, am+1, . . .] : m ≥ 1,which is the closure of the set composed of the iterates of α under the Gauss transformation.Then, Theorem 20 asserts that the mixed Littlewood conjecture holds for every irrational realnumber α such that Adh(α) contains a quadratic number.In the case of the p-adic Littlewood conjecture, Badziahin [4] established a common extensionto Corollary 18 and Theorem 20.

Theorem 21. — Let α be an irrational real number. If the set Adh(α) contains a real number

αsatisfying

limn→+∞

p(n,α)− n < +∞,

then

infq≥1

q · ||qα|| · |q|p = 0

holds for every prime number p.

Badziahin’s paper [4] contains further new results, which show that the continued fractionexpansion, viewed as an infinite word, of a putative counterexample to the p-adic Littlewoodconjecture must satisfy various strong combinatorial properties.Metric considerations in the same spirit as in Gallagher’s paper [19] can be found in [11, 7].We state below Theorem 1 from [11].

Theorem 22. — Let p1, . . . , pk be distinct prime numbers and let ψ : Z≥1 → R be a non-

negative decreasing function. Then, for almost every real number α the inequality

qα · |q|p1 · · · |q|pk ≤ ψ(q)

has infinitely (resp. finitely) many integer solutions q if the series

q≥1

(log q)kψ(q)


Y. Bugeaud 13

diverges (resp. converges). In particular, for almost all real numbers α, we have

infq≥2

q · (log q)2 · qα · |q|p = 0,

for every prime number p.

It is proved in [12] that the set of real numbers α in Bad such that, for every prime numberp, we have

infq≥2

q · (log q)2 · qα · |q|p > 0

has full Hausdorff dimension. This was considerably improved by Badziahin and Velani [6],by means of a subtle Cantor-type construction.

Theorem 23. — For every sequence D of integers greater than or equal to 2, the set of real

numbers α such that

infq≥3

q · log q · log log q · qα · |q|D > 0

has full Hausdorff dimension. Moreover, if D denotes the sequence (22n)n≥1, then the set of

real numbers α such that

infq≥16

q · log log q · log log log q · qα · |q|D > 0

has full Hausdorff dimension.

Theorem 23 was proved shortly before Theorem 7.

3. Proof of Theorem 17

Without any loss of generality, we consider real numbers in (0, 1). We associate to every realirrational number α := [0; a1, a2, . . .] the infinite word a := a1a2 . . . formed by the sequenceof its partial quotients. Set

p−1 = q0 = 1, p0 = q−1 = 0,

andpnqn

= [0; a1, . . . , an], for n ≥ 1.

By the theory of continued fractions, we know thatqnqn−1

= [an; an−1, . . . , a1].

This is one of the key tools of our proof.For simplicity, we establish Theorem 17 only in the case m = 0.Assume that the infinite word a1a2 . . . is recurrent. Then, there exists an increasing sequenceof positive integers (nj)j≥1 such that

a1a2 . . . anj is a suffix of a1a2 . . . anj+1 , for j ≥ 1.Said differently, there are finite words V1, V2, . . . such that

a1a2 . . . anj+1 = Vja1a2 . . . anj , for j ≥ 1.Actually, these properties are equivalent.



Let ≥ 2 be an integer. Let k ≥ 2 + 1 be an integer. By Dirichlet’s Schubfachprinzip, thereexist integers i, j with 1 ≤ i < j ≤ k such that

qni ≡ qnj (mod ), qni−1 ≡ qnj−1 (mod )

and j is minimal with this property.Setting

Q := |qniqnj−1 − qni−1qnj |,we observe that

divides Q (3.1)

and thatqni−1

qni

= [0; ani , ani−1, . . . , a1]

is a convergent ofqnj−1

qnj

= [0; anj , anj−1, . . . , a1].

Consequently, we get0 < Q = qniqnj

qnj−1

qnj

− qni−1

qni

≤ qniqnjq−2ni

= q−1ni

qnj .

Since||Qα|| ≤ 2qniq

−1nj

,

we finally obtainQ · ||Qα|| ≤ 2. (3.2)

It then follows from (3.1) and (3.2) that

Q · ||Qα|| · |Q| ≤ 2−1,

where |Q| is equal to −a if a divides Q but a+1 does not. Since is arbitrary, this provesTheorem 17 when m = 0.Exactly the same idea works for m ≥ 1 and there is no extra difficulty, just a little more careis needed in the various estimates.

4. Inhomogeneous approximation

The Littlewood conjecture and its p-adic analogue can be extended in a natural way toinhomogeneous approximation.

Problem 24. — Let α,β be real numbers such that 1,α,β are linearly independent over the

rationals. Is it true that, for all real numbers α0,β0, we have

lim infq→+∞

q · qα− α0 · qβ − β0 = 0?

The assumption that 1,α,β are linearly independent over the rationals is clearly necessary.Shapira [33] established that the answer to Problem 24 is positive for almost all pairs (α,β),including all pairs (α,β) of cubic real numbers in a same cubic field.


Y. Bugeaud 15

Theorem 25. — Almost every pair (α,β) of real numbers satisfies

lim infq→+∞

q · qα− α0 · qβ − β0 = 0, (4.1)

for all real numbers α0,β0. Moreover, if 1,α,β forms a basis of a real cubic field, then (4.1)

holds for all real numbers α0,β0.

Gorodnik and Vishe [20] established recently a quantitative version of Theorem 25.

Theorem 26. — There exists a positive constant c such that almost every pair (α,β) of real

numbers satisfies

lim infq→+∞

q · (log log log log log q)c · qα− α0 · qβ − β0 = 0, (4.2)

for all real numbers α0,β0. Moreover, if 1,α,β forms a basis of a real cubic field, then (4.2)

holds for all real numbers α0,β0.

We highlight the following problem, which can be viewed as the p-adic analogue of Problem 24.

Problem 27. — Let p be a prime number. Let α be an irrational real number. Is it true that,

for every integer q0 and every irrational α0, we have

lim infq→+∞

q · qα− α0 · |q − q0|p = 0? (4.3)

Examples of real numbers α for which (4.3) holds with α0 = 0 are given in [10]. For metricalresults related to Problem 27, see [22].Gorodnik and Vishe [20] proved the p-adic analogue of their Theorem 26.

Theorem 28. — Let p be a prime number. There exists a positive constant c such that almost

every real number α satisfies

lim infq→+∞

q · (log log log log log q)c · qα− α0 · |q − q0|p = 0, (4.4)

for every real number α0 and every integer q0. Moreover, every quadratic real number αsatisfies (4.4) for every real number α0 and every integer q0.

Haynes, Jensen and Kristensen [22] have obtained several metrical results related to theinhomogeneous Littlewood conjecture and its p-adic analogue. One of their results is thefollowing theorem.

Theorem 29. — Let ε be a positive real number. Let (αi)i≥1 be a countable sequence of badly

approximable numbers. There exists a subset G of Bad with full Hausdorff dimension such

that, for every β in G, every i ≥ 1 and every real number β0, there exist arbitrarily large

integers q satisfying

q · (log q)1/2−ε · qαi · qβ − β0 ≤ 1.

In view of Theorem 11, we may ask whether, for some integer d ≥ 3, we have

infq≥1

q · qα1 · · · qαd = 0,

for all badly approximable real numbers α1, . . . ,αd. Except the following result of Peck [30],nothing more is known on this question than on the Littlewood conjecture.



Theorem 30. — Let d ≥ 2 be an integer and 1,α1, . . . ,αd be a basis of a real number field

of degree d+ 1. Then, we have

lim infq→+∞

q · (log q) · qα1 · · · qαd < +∞,

thus, in particular,

infq≥1

q · qα1 · · · qαd = 0.

We do not know whether the algebraic numbers α1, . . . ,αd in the statement of Theorem 30 arebadly approximable. Theorem 30 extends and improves the result of Cassels and Swinnerton-Dyer [14] mentioned in Section 1. The second statements of Theorem 25 and of Theorem 26can be viewed as inhomogeneous analogues of Theorem 30 when d = 2. This motivates thefollowing open problem.

Problem 31. — Let d ≥ 3 be an integer and 1,α1, . . . ,αd be a basis of a real number field of

degree d+ 1. Is it true that

lim infq→+∞

q · qα1 − α1 · · · qαd − α

d = 0

holds for all real numbers α1, . . . ,α

d?

Acknowledgements. I am pleased to thank Dmitry Badziahin and Alan Haynes for their verycareful reading of a preliminary version of this text.

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25 mai 2014

Yann Bugeaud, Mathématiques, Université de Strasbourg, 7, rue René Descartes, F-67084 Strasbourg,France • E-mail : [email protected]


ARITHMETIC PROPERTIES OF GENERALIZED RIKUNAPOLYNOMIALS

by

Z. Chonoles, J. Cullinan, H. Hausman, A.M. Pacelli, S. Pegado & F. Wei

Abstract. — Fix an integer ≥ 3. Rikuna introduced a polynomial r(x, t) defined overa function field K(t) whose Galois group is cyclic of order , where K satisfies some mildhypotheses. In this paper we define the family of generalized Rikuna polynomials rn(x, t)n≥1

of degree n. The rn(x, t) are constructed iteratively from the r(x, t). We compute the Galoisgroups of the rn(x, t) for odd over an arbitrary base field and give applications to arithmeticdynamical systems.

Résumé. — Soit ≥ 3 un nombre entier fixé. Rikuna a défini un polynôme r(x, t) sur uncorps de fonctions K(t) dont le groupe de Galois est cyclique d’ordre , où K satisfait à certaineshypothèses pas très restrictives. Dans cet article, nous définissons la famille des polynômes deRikuna généralisés rn(x, t)n≥1 de degré n. Les rn(x, t) sont construits de manière itérative àpartir de r(x, t). Nous calculons les groupes de Galois des rn(x, t) pour impair sur un corps debase arbitraire et donnons des applications aux systèmes dynamiques arithmétiques.

1. Introduction

In [11], Rikuna introduced a one-parameter family of polynomials with wide-ranging applica-tions to arithmetic. In particular, let > 2 be a fixed positive integer (not necessarily prime)and K a field of characteristic coprime to that does not contain a primitive -th root ofunity. Fix an algebraic closure K of K and fix a primitive -th root of unity ζ ∈ K. Assumefurther that ζ

+ := ζ + ζ

−1 ∈ K. Following [11], define the polynomials p(x), q(x) ∈ K[x] by

p(x) :=ζ−1 (x− ζ) − ζ(x− ζ

−1 )

ζ−1 − ζ

q(x) :=(x− ζ) − (x− ζ

−1 )

ζ−1 − ζ

.

Let t be an indeterminate over K and define the degree Rikuna polynomial by r(x, t) =p(x)− t q(x) ∈ K(t)[x].

2010 Mathematics Subject Classification. — 11R32, 11S20.Key words and phrases. — postcritically finite, Galois group, cyclotomic field.

20 Arithmetic Properties of Generalized Rikuna Polynomials

Rikuna’s polynomials have been very well-studied in a number of different guises. For example,when = 3, r(x, s/3) is the “simplest cubic” polynomial of Shanks [13], which has deeparithmetic implications (see, for example, [5], [8], and [12]). Rikuna originally introducedr(x, t) as a method for creating cyclic Galois extensions of a base field that are not given byKummer or Artin-Schreier theory (hence the requirement that ζ ∈ K). He proved in [11]that for all ≥ 3, r(x, t) is irreducible over K(t) and has Galois group Z/ over K(t). It wasthen shown in [6] that when is odd r(x, t) is generic in the sense that every Z/-extensionof L with L ⊃ K is obtained as a specialization of r(x, t). When is even, r(x, t) need not begeneric and in [6] an algebraic characterization of the non-genericity is given.When K is a finite field of characteristic coprime to , the Rikuna polynomials have beenused for certain class number constructions. Using the notation above, in [3] they considerϕ(x) = p(x)/q(x) and use a recursive construction to give explicit families of function fieldswith certain class number indivisibility properties.Before Rikuna defined the polynomial r(x, t) Shen and Washington introduced in [14] and[15] an interesting family Pn(x, t) of polynomials of prime-power degree. There, they take

to be prime and the base field to be K = Q(ζ+ ). Then write

(x− ζ)n = an(x)− ζbn(x),

with an, bn ∈ K[x]. Write Pn(x, t) = an(x)− (t/n) bn(x) with t ∈ OK . The authors call thisthe

n-tic polynomial, and it coincides with our “generalized Rikuna polynomial” below (onemust replace “t” by “nt”, and we consider general t ∈ K). They then determine the splittingfields of the Pn(x, t), their ramification properties, and the structure of the units. Moreover,they apply Faltings’ theorem over K to certain superelliptic curves to show that the numberof reducible specializations of their polynomials is finite. In the special case = 3, the authorsfurther show in [15] that Pn(x, t) is irreducible for all t ∈ Z.In this paper we take a different approach to the polynomials of Rikuna and Shen-Washingtonby interpreting them in the context of iterated self-maps of P1. To define them, we followclosely the conventions of [16, Ex. 4.9]: if F,G ∈ K[X,Y ] are homogenous polynomials ofdegree d ≥ 2 with no common factors, then they give rise to a rational self-map ϕ = [F,G] ofP1. Let F0(X,Y ) = X and G0(X,Y ) = Y and inductively define

Fn+1 = Fn(F (X,Y ), G(X,Y )) and Gn+1 = Gn(F (X,Y ), G(X,Y )).

Then Fn and Gn have no common factors and the n-th iterate ϕ(n) of ϕ is given in homogenouscoordinates by ϕ

(n) = [Fn, Gn].We now apply this machinery to our setup. Let P,Q ∈ K[X,Y ] be the homogeneous formsof p, q above, respectively. Then P and Q are homogenous of degree and − 1, respectively,and have no common factors (one can show that Res(P,Q) = (ζ − ζ

−1 )(−1)

Y2). Define

ϕ : P1→ P1 by the pair [P,Q] and the n-th iterate ϕ

(n) by [Pn, Qn], as above. Then ϕ(n) is

a rational self-map of P1 of degree n. In affine coordinates we may write pn(x) = Pn(x, 1)

and qn(x) = Qn(x, 1) so that

ϕ(n)(x) :=

pn(x)

qn(x).

We define the sequence of generalized Rikuna polynomials rn(x, t) ∈ K(t)n≥1 by

rn(x, t) = pn(x)− t qn(x),


Z. Chonoles, J. Cullinann, H. Hausman, A. M. Pacelli, S. Pegado and F. Wei 21

where r1(x, t) = r(x, t). These polynomials are similar to the ones in [14] and [15], but weonly require that be odd, not prime, nor do we put any restriction on K except that ζ+ ∈ K

and ζ ∈ K, and we work over the general function field K(t); namely we do not restrict tointegral specializations in the special case K = Q(ζ+ ). Our main result is the following:

Theorem 1. — Fix an odd integer > 2, let K be a field of characteristic coprime to , and

fix an algebraic closure K of K, as well as a primitive -th root of unity ζ ∈ K. Suppose

further that K does not contain ζ but does contain ζ + ζ−1 . Let Kn be the splitting field of

rn(x, t) over K(t). Then

Gal(Kn/K(t)) Z/n Z/(n/bn),

where bn is the number of roots of unity in K(t)(ζ) of order dividing n.

We prove Theorem 1 in Section 3.2 by passing to an auxiliary tower of fields Ln, whereLn = Kn(ζ), and determining the Galois group of this tower. Specifically, we show thatGal(Ln/K(t)) has the presentation

Gal(Ln/K(t)) = ρn, γn | ρ2n−vn

n = γnn = id, ρnγn = γ

−(−1)vn−1

n ρn,

where vn is the -adic valuation of bn in the statement of the Theorem 1. In particular, wedetermine Gal(Kn/K(t)) as an explicit quotient of Gal(Ln/K(t)) of index 2. This resultbuilds on that of [15] on the Galois groups and shows that the Galois group of rn(x, t) isrelatively small (but non-abelian); recall [10, thm. 1] that in characteristic 0, the nth iterateof the generic monic polynomial of degree k is isomorphic to the nth wreath power of Sk. Thusour Galois groups have order 2n/bn, compared to the maximal size !(

n−1)/(−1). Because wedo not assume that K is a number field, our iterative construction has potential applicationsto class number indivisibility problems in positive characteristic function fields, just as thebase function ϕ(x) was used in [3].The rn(x, t) also have nice applications to arithmetic dynamics. Given a rational self-mapF of P1, one can consider the tower defined as the compositum of the splitting fields of theiterates F

(n) of F . For example, when F is the Lattès map associated to an endomorphismΦ of an elliptic curve E, it is a classical result that the tower of splitting fields is finitely-ramified. However, the splitting fields of K(t)/(F − t) may be viewed in the context of theKummerian fields K(F (−n)

E(K)), and the arithmetic properties of these fields are much lesswell-understood.In the final section of the paper we introduce some of the dynamical properties of the rn(x, t)in the spirit of [1] and [2]. In particular, we show that ϕ(x) is postcritically finite and give asimple formula for the discriminant of rn(x, t). We also raise some questions for future studysurrounding the arithmetic of the towers they define.

2. Splitting Fields I – Preliminaries

Fix a positive integer ≥ 3. In this section we prove some general results on splitting fieldsassociated to the rn(x, t). We do not distinguish between even and odd until the nextsection. We fix once and for all a coherent system ζnn≥1 of primitive

n-th roots of unity;that is, ζn is a primitive

n-th root of unity and the -power map sends the level n elementto the level n − 1 element. Recall that ζ ∈ K. We first work out some of the key minimal



polynomials involved in our analysis. Given a field k with algebraic closure k, we defineIrr(π, k)(x) to be the minimal polynomial for π ∈ k over k.Let µ1 be the group of roots of unity of K(ζ). For each n ≥ 1, let Gn ⊂ µ1 be the finite(cyclic) subgroup consisting of elements of order dividing

n, and let bn = #Gn. Thus, Gn isgenerated by a primitive bn-th root of unity; we fix a generator and write Gn = ζbn. Thus,for all n ≥ 1, ζbn ∈ K(ζ) and bn|

n. But since ζ ∈ Gn for all n, we have that |bn for alln ≥ 1.Let µ2 be the group of roots of unity of K(ζ2). For each n ≥ 1, let Hn ⊂ µ2 be the finite(cyclic) subgroup consisting of elements of order dividing

n, and let cn = #Hn. A generatorof Hn is a primitive cn-th root of unity, so Hn contains all cn-th roots of unity, in particularζcn . Thus, for each n ≥ 1, ζcn ∈ K(ζ2) and cn|

n. Since ζ ∈ Hn for all n ≥ 1, we have that|cn for each n ≥ 1.

The following lemma, whose proof can be found in [7, p. 297], will be needed below.

Lemma 1. — Let k be a field, m ≥ 2 an integer, and a ∈ k×. Assume that for any prime p

with p | m we have a /∈ kp, and if 4 | m, that a ∈ 4k4. Then x

m − a is irreducible in k[x].

Next we determine three minimal polynomials that will be used extensively in the rest of thepaper.

Lemma 2. — For all integers > 2, we have Irr(ζ,K(t))(x) = x2− ζ

+ x+1. Moreover, if

is odd, or if is even and 4|bn, then Irr(ζn ,K(t)(ζ))(x) = xn/bn − ζbn . Finally, if is even

and 4 bn, then Irr(ζn ,K(t)(ζ2))(x) = xn/cn − ζcn .

Proof. — Since ζ+ ∈ K and ζ /∈ K, assertion (1) follows. For (2), first note that ζn is a root

of the (monic) polynomial xn/bn−ζbn ∈ K(t)(ζ)[x]. Next, for any d ∈ Z≥0, if ζdbn ∈ K(t)(ζ),then ζbn is a d-th power in K(t)(ζ); and if ζbn = z

d for some z ∈ K(t)(ζ), then z is a primitivedbn-th root of unity, so that all dbn-th roots of unity are in K(t)(ζ), including ζdbn . Thus,ζbn is a d-th power in K(t)(ζ) if and only if ζdbn ∈ K(t)(ζ). Therefore, if ζbn is a p-th powerin K(t)(ζ) for some prime p dividing n

bn, then ζpbn ∈ K(t)(ζ). Because p |

n

bn, we have that

pbn | n, so that ζpbn ∈ Gm; but the order of ζpbn is pbn > bn = #Gn, contradiction. Thus,ζbn is not a p-th power in K(t)(ζ) for any prime p dividing n

bn, and hence x

n/bn − ζbn isirreducible.If is not odd, we may have 4 | (n/bn), in which case Lemma 1 requires that we also showζbn = −4z4 for all z ∈ K(t)(ζ). We do this when 4 | bn. Suppose that 4|bn, that 4 | n/bn,and that ζbn = −4z4 for some z ∈ K(t)(ζ). Then (2z2)2 = −ζbn , so that 2z2 = ζ4ζ2bn , orζ34ζ2bn . Clearly 2z2 ∈ K(t)(ζ), and since 4|bn and ζbn ∈ K(t)(ζ), we have ζ4 ∈ K(t)(ζ),

whence ζ2bn ∈ K(t)(ζ). But since 4 | n/bn we have 4bn | n and thus 2bn | n, so thatζ2bn ∈ Gn, contradicting the fact that the order of ζ2bn is 2bn > bn = #Gn. Thus xn/bn − ζbnis irreducible when is even and 4|bn.For (3), note that c1 | and |c1 so that c1 = . When n = 1 it is clear that the minimalpolynomial for ζ over K(t)(ζ2) is x− ζ, as claimed. Now consider n ≥ 2. Since is even wehave that 4|n; thus if ζ4 were an element of K(t)(ζ), we would have ζ4 ∈ Gn and thus 4 | bn.Therefore, our hypothesis implies ζ4 /∈ K(t)(ζ). However, because is even, we do have that



4 | 2, and thus ζ4 ∈ K(t)(ζ2). The arguments from (2) now go through exactly as before,with K(t)(ζ) replaced by K(t)(ζ2).

Lemma 3. — For each n ≥ 2 the degree of the Galois extension K(t)(ζn)/K(t) is

[K(t)(ζn) : K(t)] =

2n/bn if is odd or is even with 4 | bn

4n/cn if is even and 4 bn.

Remark. Note that [K(t)(ζ) : K(t)] = deg Irr(ζ,K(t))(x) = 2.

Proof. — Suppose that is odd or that is even and 4 | bn. Since K(t)(ζ) ⊂ K(T )(ζn), wehave

[K(t)(ζn) : K(t)] = [K(t)(ζn) : K(t)(ζ)][K(t)(ζ) : K(t)]

= deg Irr(ζn ,K(t)(ζ))(x) · 2 = 2n/bn.

On the other hand, if is even and 4 bn, then K(t)(ζ2) ⊂ K(t)(ζn) (recall n ≥ 2). Thus

[K(t)(ζn) : K(t)] = [K(t)(ζn) : K(t)(ζ2)][K(t)(ζ2) : K(t)(ζ)][K(t)(ζ) : K(t)]

= deg Irr(ζn ,K(t)(ζ2))(x) · [K(t)(ζ2) : K(t)(ζ)] · 2

= 2n/cn · [K(t)(ζ2) : K(t)(ζ)].

Note that ζ2 is a root of x2−ζ ∈ K(t)(ζ)[x], so that [K(t)(ζ2) : K(t)(ζ)] ≤ 2. The proof ofLemma 2 shows that ζ4 /∈ K(t)(ζ), but that ζ4 ∈ K(t)(ζ2), so that [K(t)(ζ2) : K(t)(ζ)] > 1.Therefore the degree of the field extension [K(t)(ζ2) : K(t)(ζ)] = 2, which completes theproof of the lemma.

For each n ≥ 1 we know that | bn and that bn | n, so given the prime factorization of :

= 2e0pe11 · · · pemm ,

the prime factorization of bn has the form bn = 2a0pa11 · · · pamm for some ei ≤ ai ≤ nei. Alsonote that because | bn, we have K(t)(ζ) ⊂ K(t)(ζbn). Moreover, since ζbn ∈ K(t)(ζ) wehave the equality of fields K(t)(ζbn) = K(t)(ζ).

3. Splitting Fields II – Odd

In this section and for the rest of the paper we specialize to the case of odd . We workout the splitting fields of the rn(x, t) as well as describe an auxiliary tower of fields. We areultimately interested in the Galois groups of the rn(x, t) and this auxiliary tower will aid indetermining those groups.

Lemma 4. — Suppose is odd and recall that ζbn is quadratic over K(t) for each n ≥ 1.Then for each n ≥ 1 the conjugate of ζbn over K(t) is ζ

−1bn

. Hence ζ+bn

∈ K(t).

Proof. — Let ψ be the non-trivial automorphism of K(t)(ζ)/K(t) so that ψ(ζ) = ζ−1 . Let

ζabn

= ψ(ζbn) be the conjugate of ζbn over K(t). Because ψ2 is trivial, it must be the case that

a2 ≡ 1 (mod bn), whence a

2 ≡ 1 (mod paii ) for all i, where the pi are the prime divisors of



as above. By assumption, the pi are all odd and it then follows that a ≡ ±1 (mod paii ) for all

i. Now,ζ−1 = ψ(ζ) = ψ(ζbn/bn

) = (ζabn)bn/ = ζ

a ,

so that a ≡ −1 (mod ). Thus, a ≡ −1 (mod peii ) for all i. Together with a ≡ ±1 (mod p

aii ),

we must have that a ≡ −1 (mod paii ) for all i. By the Chinese Remainder Theorem, we have

that a ≡ −1 (mod bn) so that ψ(ζbn) = ζ−1bn

.

Lemma 5. — If is odd, then for each n ≥ 1, Irr(ζn ,K(t))(x) = x2n/bn − ζ

+bnxn/bn + 1.

Proof. — Since ζ+bn

∈ K(t), we have that x2n/bn − ζ

+bnxn/bn + 1 ∈ K(t)[x]. This polynomial

is monic and ζn is a root. Thus

Irr(ζn ,K(t))(x) | (x2n/bn − ζ

+bnxn/bn + 1).

However, deg Irr(ζn ,K(t))(x) = [K(t)(ζn) : K(t)] = [K(t)(ζn) : K(t)(ζ)][K(t)(ζ) :K(t)] = 2n/bn, by (1) and (2) of this Lemma. Since x

2n/bn − ζ+bnxn/bn + 1 is monic, it

must be the minimal polynomial.

Corollary 1. — The extensions K(t)(ζn)/K(t) are Galois with degree 2n/bn.

Define the rational function

α(x) =ζ − x

ζ−1 − x

∈ K(t)(ζ)(x).

Lemma 6. — We have the equality of fields K(t)(ζ) = K(t)(α(t)).

Proof. — It suffices to show that ζ ∈ K(t)(α(t)). But since ζ+ ∈ K, we can write

ζ =ζ+α(t)− t(α(t)− 1)

α(t) + 1.

Thus, ζ ∈ K(t)(α(t)), as desired.

Define the polynomial

A(x) = x2−

2 +

(ζ+ )2 − 4

t2 − ζ+ t+ 1

x+ 1 ∈ K(t)[x].

Lemma 7. — The minimal polynomial for α(t) over K(t) is A(x).

Proof. — Note that α(t)±1 are the roots of A(x). Since [K(t)(α(t)) : K(t)] = [K(t)(ζ) :K(t)] = 2, and A(x) is monic, the lemma follows.

Next, we characterize the roots of rn(x, t). Let

α(t) ∈ K(t) be an th root of α(t) and foreach positive integer d, fix a compatible system of dth roots d

α(t) ∈ K(t) of α(t) in the

sense thatdα(t)

= d−1

α(t).

Let Kn be the splitting field of rn(x, t) over K(t). Because of the cumbersome notationinvolving the surds, we will set the following notation for the remainder of the paper. Set

βn(t) :=nα(t)



so that βn(t)n≥1 forms a compatible system of -power roots as well.

Lemma 8. — For all n ≥ 1, the minimal polynomial for βn(t) over K(t)(ζn) is xn − α(t).

Proof. — By Lemma 6, α(t) ∈ K(t)(ζn) for all n ≥ 1. It is also clear that xn −α(t) is monicand has βn(t) as a root. Note that K(t)(ζn) = K(ζn)(t), so that any element of K(t)(ζn)is of the form f

g for relatively prime f, g ∈ K(ζn)[t]. Also note that α(t) = ζ−tζ−1 −t

is in lowest

terms, i.e. gcd(ζ − t, ζ−1 − t) = 1; any non-constant h ∈ K(ζn)[t] dividing both ζ − t and

ζ−1 − t would divide (ζ − t)− (ζ−1

− t) = ζ − ζ−1 ∈ K(ζn), a contradiction. If α(t) is not

an -th power in K(t)(ζn), i.e. α(t) /∈ (K(t)(ζn)), then xn−α(t) is irreducible, by Lemma 1.

Suppose that α(t) = (fg ) for some f

g ∈ K(t)(ζn). Then f = ζ − t and g

= ζ−1 − t, up

to multiplication by a unit. But if deg(f) = 0 then deg(f ) = 0, and if deg(f) ≥ 1 thendeg(f ) ≥ , while deg(ζ − t) = 1 (likewise with g). Thus α(t) is not an -th power inK(t)(ζn), so that x

n − α(t) is irreducible.

Proposition 3.1. — Fix an integer n ≥ 1. Then the roots of rn(x, t) are given by

θ(n)c =

ζ − ζcnβn(t)

1− ζζcnβn(t)

,

for all integers 0 ≤ c ≤ n − 1.

Proof. — When n = 1, z ∈ K(t) is a root of r(x, t) if and only if φ(z) = t, which is true ifand only if

ζ−1(z − ζ) − ζ(z − ζ

−1) = t((z − ζ) − (z − ζ−1)) ⇐⇒

z − ζ

z − ζ−1

=ζ − t

ζ−1 − t

= α(t)

⇐⇒z − ζ

z − ζ−1

= ζN β1(t)

for some 0 ≤ N ≤ − 1. Rearranging and reindexing, the roots of r(x, t) are given by

ζ − ζcβ1(t)

1− ζζcβ1(t)

: 0 ≤ c ≤ − 1

,

as claimed.By induction, φ(n+1)(z) = t if and only if φ(z) = θ

(n)c for some 0 ≤ c ≤

n − 1. For each valueof c, φ(z) = θ

(n)c if and only if

z =ζ − ζ

d β1(θ

(n)c )

1− ζζd β(θ

(n)c

for some 0 ≤ d ≤ − 1. Note that α(θ(n)c ) = ζζcnβn(t), so we may rewrite z as

z =ζ − ζ

dn+c+n−1

n+1 βn+1(t)

1− ζζdn+c+n−1

n+1 βn+1(t).



Because

ζdn+c+n−1

n+1 : 0 ≤ d ≤ − 1, 0 ≤ c ≤ n−1

= ζen+1 : 0 ≤ e ≤

m+1− 1,

and because, for each n, the θ(n)c are distinct, the the roots of rn+1(x, t) are precisely as

claimed. This proves the proposition.

For each n ≥ 0, we define Kn to be the splitting field of rn(x, t) (we define p0 = x and q0 = 1so that r0(x, t) = x− t and K0 = K(t)). Proposition 3.1 shows that for each n, the fields Kn

are Galois over K(t) since they are the splitting fields of separable polynomials. Moreover,for each n, we have Kn ⊂ Kn+1 because the roots of φ(n)(x) are the images under φ of theroots of φ(n+1)(x).

3.1. The auxiliary tower Ln. — For each n ≥ 0 we define the field Ln to be Ln =K(t)(ζn ,βn(t)). Thus,

L0 = K(t)(ζ1,1

α(t)) = K(t)(α(t)) = K(t)(ζ).

In Appendix A, we give a field diagram of the Kn and the Ln towers.

Lemma 9. — For each n ≥ 0, Kn ⊂ Ln.

Proof. — Since K0 = K(t) ⊂ K(t)(ζ) = L0, it suffices to consider n > 0. But for each c

with 0 ≤ c ≤ n − 1, the field Ln contains the elements

θ(n)c =

ζ − ζcnβn(t)

1− ζζcnβn(t)

,

which generate Kn/K(t). This proves the lemma.

For ease of notation in the subsequent sections, we define the positive integer vn by vn = ν(bn),where ν : Z −→ Z≥0 ∪ ∞ is the -adic valuation.

Lemma 10. — For each n ≥ 0, the extension Ln/K(t) is Galois with degree 22n−vn .

Proof. — Because K(t) ⊂ K(t)(ζn) ⊂ Ln, it follows that

[Ln : K(t)] = [Ln : K(t)(ζn)][K(t)(ζn) : K(t)] = n· 2n/bn = 22n−vn .

By Lemma 7, the minimal polynomial for α(t) over K(t) is A(x). Let F be the splittingfield of B(x) = A(x

n). Note that z ∈ K(t) is a root of B if and only if zn is a root of A,

i.e. zn = α(t)±1. Thus, the roots of B are precisely ζ

cnβn(t)

±1, for 0 ≤ c ≤ n − 1. Because

F contains both βn(t) and ζnβn(t), it contains ζn . Thus Ln ⊂ F . On the other hand, sinceLn contains ζn and βn(t), it contains ζ

cnβn(t)

±1. Thus F = Ln and it follows that Ln isGalois over K(t).



3.2. Galois groups. — In this section we determine the Galois groups of the field extensionsKn/K(t) keeping the convention that be odd. We begin with an explicit description of theGalois groups of the field extensions Ln/K(t).

Proposition 3.2. — For all n ≥ 0, the Galois groups Gal(Ln/K(t)) are generated by the

automorphisms ρn and γn, which are determined by

ρn(ζn) = ζ(−1)

bn−1

n γn(ζn) = ζn

ρn(βn(t)) = βn(t)−1

γn(βn(t)) = ζnβn(t).

Moreover, they satisfy the relations

ρ2n−vn

n = γnn = id, ρnγn = γ

−(−1)vn−1

n ρn.

Proof. — The extensions K(t)(ζn)/K(t) and Ln/K(t) are Galois, and by Lemma 8 we have

[Ln : K(t)(ζn)] = n.

Thus, each automorphism of K(t)(ζn)/K(t) extends to n automorphisms of Ln/K(t). It is

easy to check that the mapping

ρn : ζn → ζ(−1)

(vn−1)

n

is an element of Gal(K(t)(ζn)/K(t)). Since is odd, the congruence

(− 1)vn−1

≡ (−1)vn−1

≡ −1 (mod )

holds, whence

ρn(ζ) = ρn(ζn−1

n ) = ρn(ζn)n−1

=

ζ(−1)

(vn−1)

n

n−1

= ζ(−1)

vn−1

= ζ−1 .

Therefore, ρn must act on α(t) as follows:

ρn(α(t)) = ρn

ζ − t

ζ−1 − t

=

ζ−1 − t

ζ − t= α(t)−1

.

It follows that any extension of ρn to an element of Gal(Ln/K(t)) must send βn(t) to ζdnβn(t),

for some 0 ≤ d ≤ n − 1. An extension of any automorphism is determined by its action on

βn(t) by definition of the field Ln. We have thus identified all n extensions of ρn. The ρn

defined in the statement of the Proposition is indeed an automorphism of Ln/K(t) becauseit is one of the extensions of ρn. By Lemma 8 it is also clear that ζnβn(t) is a conjugate ofβn(t) over K(t)(ζn), whence the map γn defined in the statement of the Proposition is anautomorphism of L/K(t). It is clear that the order of γn is

n and the order of ρn is 2n−vn

because ρdn = id if and only if

ζn = ρdn(ζn) = ζ

(−1)

vn−1d

n , and

βn(t) = ρdn(βn(t)) =

βn(t)

−1d

,



which is the case if and only if

1 ≡

(− 1)

vn−1d

= (− 1)dvn−1

(mod n),

and d is even. But this is true if and only if 2n−vn | d, and because 2n−vn is even, it is theleast d ≥ 1 such that ρ

dn = id.

Because an automorphism of Ln/K(t) is determined by its action on ζn and βn(t), it suffices to

check the relation ρnγn = γ−(−1)

vn−1

n ρn on these two elements. This is a routine computationwhich we omit. Thus, the subgroup of Gal(Ln/K(t)) generated by ρn and γn consists of the(possibly non-distinct) automorphisms ρ

xn, γ

yn, where 0 ≤ x ≤ 2n−vn and 0 ≤ y ≤

n − 1. Itis also not difficult to show that these automorphisms are all distinct.

Theorem 2. — For all n ≥ 0 the Galois group Gal(Kn/K(t)) is generated by σn := ρn|Kn

and τn := γn|Kn . They satisfy the relations

σn−vn

n = τnn = id, σnτn = τ

−(−1)vn−1

n σn.

Granting the theorem for a moment, the description of Gal(Kn/K(t)) in terms of generatorsand relations lends itself to a description as a semidirect product:

Gal(Kn/K(t)) Z/nZφn Z/(n−vn)Z,

where φn : Z/(n−vn)Z −→ Aut(Z/nZ) is given by φn(1) = (1 → (−1)( − 1)vn−1

). Notethat φn is injective, i.e. (1 → (−1)(− 1)

vn−1) is always an automorphism of Z/nZ of order

n−vn because the order of −(− 1)

vn−1 modulo n is

n−vn .

Proof of Theorem 2. — Basic Galois theory tells us that the quotient mapGal(Ln/K(t)) −→ Gal(Kn/K(t))

is given by restriction; thus Gal(Kn/K(t)) is generated by σn and τn. Because σn and τn arethe restrictions of ρn and γn, respectively, they satisfy the same relation. Since

ρn−vn

n (ζn) = ζ

(−1)

vn−1n−vn

n = ζ(−1)

n−1

n = ζ−1n , and

ρn−vn

n (βn(t)) = (βn(t))(−1)

n−vn

= βn(t)−1

,

it follows that ρn(ζ) = ζ−1 .

Since Kn is the splitting field of rn(x, t) over K(t), it is generated over K(t) by

θ(n)c =

ζ − ζcnβn(t)

1− ζζcnβn(t)

,

for each 0 ≤ c ≤ n − 1. Note that for each c we have

ρn−vn

n (θ(n)c ) =ρn−vn

n (ζ)− ρn−vn

n (ζcnβn(t))

1− ρn−vnn

ζζ

cnβn(t)

=ζ−1 − ζ

−cn βn(t)−1

1− ζ−1 ζ

−cn βn(t)−1

=ζcnβn(t)− ζ

ζζcnβn(t)− 1

= θ(n)c .

Thus, ρn−vn

n restricts to the identity on Kn so that ρn−vn

n ∈ Gal(Ln/Kn). By Proposition 3.2the order of ρn−vn

n in Gal(Ln/K(t)) is 2, hence 2 divides #Gal(Ln/Kn) = [Ln : Kn]. Together



with Lemma 10 we can conclude that 2 [Kn : K(t)] and so it follows that ζ /∈ Kn. Sinceθ(n)c ∈ Kn ⊂ Kn(ζ) for all 0 ≤ c ≤

n − 1, it must be the case that

θ(n)c − ζ

−1 =

ζ − ζ−1

1− ζζcnβn(t)

∈ Kn(ζ).

But since ζ−ζ−1 ∈ Kn(ζ), we have that ζcnβn(t) ∈ Kn(ζn) for all 0 ≤ c ≤

n−1. Thereforeboth βn(t) and ζnβn(t) are elements of Kn(ζ), whence ζn ∈ Kn(ζ). Altogether this resultsin the inclusion of fields:

Kn ⊂ Ln ⊂ Kn(ζ).

Since 2 divides [Ln : Kn] and [Kn(ζ) : Kn] = 2, we have that #Gal(Ln/Kn) = 2. We canalso conclude that Ln = Kn(ζ) and that #Gal(Kn/K(t)) =

2n−vn .In light of these observations on the Galois groups, we see there is exactly one non-trivialautomorphism of Ln/K(t) that restricts to the identity automorphism on Kn/K(t). We haveseen that ρn−vn

n has this property, so it must be the only one. Because the order of σn is theleast d ≥ 1 such that σd

n = id, the order of σn must be n−vn . Finally, since γ

yn does not equal

ρn−vn

n for any y, γyn restricts to the identity on Kn if and only if it is the identity on Ln. Thusthe order of τn is

n. This completes the proof of the theorem.

4. Dynamical Properties

In this final section we adopt the notational conventions and context of [1]. Let K be anumber field and fix a rational self-map ϕ of P1 defined over K; in coordinates, we maytake ϕ(x) = g(x)/h(x) with g(x), h(x) coprime elements of OK [x]. Then for n ≥ 1, thesplitting fields of the iterates ϕ

(n)(x) − t give rise to tower of splitting fields of the previousiterates. More precisely, we let Fn be the Galois closure of the field K(t)/(ϕn(x) − t) andFϕ the compositum over all n of the splitting fields: Fϕ = ∪nFn. Fix a compatible systemof specialization maps σn : OFn −→ K and set Kn,t0 to be the Galois closure over K of thespecialized extension. In this way the compositum Kϕ,t0 can be viewed as a specialization ofthe tower Fϕ,t0 .It is quite difficult to determine the exact primes ramifying in a tower, and the ones which areknown tend to use auxilliary information. For example, given an elliptic curve E/Q withoutcomplex multiplication and a rational prime , the iterates of the Lattès map ϕ give rise totowers ramified above and the primes of bad reduction for E. A similar example concernsthe Chebyshev polynomial ψ2(x) = x

2 − 2. The tower of splitting fields of the iterates ofψ2 is ramified only above 2. Moreover the Chebyshev polynomials ψd arise as the image ofprojection-to-x for the d-power map on the algebraic group S

1. At the same time, in bothexamples the associated Galois groups are smaller than one would expect from a “random”tower (the Galois group of a Lattès tower is an open subgroup of GL2(Z), while the Galoisgroup of the Chebyshev tower is isomorphic to the additive group Z2 of 2-adic integers). Bothof these examples come from the specialization t = 0; other specializations would potentiallygive rise to towers whose Galois groups are less well-understood (though in the case of Lattèsmaps, if t were the x-coordinate (in Weierstrass form) of a point of infinite order, then morecan be said in terms of arboreal representations; see [4] for more details).



The common characteristics of these two examples are “small” Galois group, finite ramification,and that ϕ is postcritically finite; moreover, both arise via endomorphisms of algebraic groups.The rn(x, t) share some of these characteristics. To give a little more detail, with all notationas above let

Rϕ := θ ∈ K : (hg − gh)(θ) = 0 and Bϕ := ϕ(θ) : θ ∈ Rϕ

be the sets of ramification points and branch points of ϕ, respectively. In particular, Rϕ

consists of the roots of hg− gh counted without multiplicity. A rational function ϕ is said to

be postcritically finite if the forward orbit of the critical points under all iterations is a finiteset. In other words, if

Bϕ(n) = Bϕ ∪ ϕ(Bϕ) ∪ · · · ∪ ϕ(n−1)(Bϕ)

is the set of branch points of ϕ(n), then ϕ is postcritically finite if ∪nBϕ(n) is a finite set.We can apply this setup to the rn(x, t). Fix > 2 and set ϕ(x) = p(x)/q(x), where p(x) andq(x) are as in the Introduction. Then the following lemma is a simple computation.

Lemma 11. — Let > 2 be an integer and K a field of characteristic coprime to . Suppose

that ζ+ ∈ K, where ζ is a primitive th root of unity. Then ϕ(x) = p(x)/q(x) ∈ K(x) is

postcritically finite.

Proof. — The critical points of ϕ are ζ and ζ−1 , each of which is fixed by ϕ.

In the context where K is a number field, the fact that the ϕ are postcritically finite meansthe function field towers are finitely ramified. Moreover, the discriminant formulæ of [1, 2]imply that for all t ∈ K, the specialized towers at t are finitely ramified as well. Indeed,applying the results of [1, 2] to the present setup, we obtain:

disc rn(x, t) = ±n(n)(ζ − ζ

−1 )(

n−2)(n−1)(t2 − (ζ + ζ−1 )t+ 1)

n−1.

This bounds the number of primes ramifying at any level of the tower and moreover showsexaclty what the potential ramified primes are. In particular, the primes of OK dividing areramified in the tower, while those dividing t

2 − ζ+ t+ 1 may be ramified.

In the special case where is a prime number and K = Q(ζ+ ), we have the factorization = l(−1)/2 as ideals of OK . It would be interesting to determine specializations at whichonly a few primes in addition to l ramify in the tower above K. If we set = 3, however,and take for example t = 0, 1 so that 3 is the only ramified prime, then the tower aboveQ specializes to the abelian cyclotomic-3 tower. A delicate problem would be to determinean explicit relationship between the ramified primes and the index of the specialized Galoisgroup inside the geometric Galois group. Finally, it would be interesting to determine, alongthe lines of the Lattès and Chebyshev towers, whether there is an alternative “geometric”description of the rn(x, t), which would make the analogy more complete.

Acknowledgements. We would like to express our gratitude to Farshid Hajir and to thereferee. The impetus for this paper developed from numerous conversations with FarshidHajir, while the referee’s comments and suggestions greatly improved the exposition of thepaper.



References

[1] W. Aitken, F. Hajir, C. Maire. Finitely ramified iterated extensions. Int. Math. Res. Not. 2005,no. 14, 855-880.

[2] J. Cullinan, F. Hajir. Ramification in iterated towers for rational functions. Manuscripta Math.137 (2012), no. 3-4, 273-286.

[3] M. Daub, J. Lang, M. Merling, A. Pacelli, N. Pitiwan, M. Rosen. Function Fields with ClassNumber Indivisible by a Prime . Acta Arith. 150 (2011), no. 4, 339-359.

[4] R. Jones, J. Rouse. Iterated endomorphisms of abelian algebraic groups. Proc. Lond. Math. Soc.100 (2010), no. 3, 763-794.

[5] Y. Kishi. A family of cyclic cubic polynomials whose roots are systems of fundamental units.J. Number Theory 102 (2003), no. 1, 90-106.

[6] T. Komatsu. Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummertheory. Manuscripta Math. 114 (2004) 265-279.

[7] S. Lang, Algebra, Graduate Texts in Mathematics 211. Springer-Verlag, New York, 2002.[8] E. Lehmer. Connection between Gaussian periods and cyclic units. Math. Comp. 50 (1988), no.

182, 535-541.[9] J. Neukirch, Algebraic Number Theory, Springer-Verlag, Berlin, 1999.

[10] R.W.K. Odoni. The Galois theory of iterates and composites of polynomials. Proc. London.Math. Soc. 51 (1985), no. 3, 385-414.

[11] Y. Rikuna. On simple families of cyclic polynomials. Proc. Amer. Math. Soc. 130 (2002), no. 8,2215-2218

[12] R. Schoof, L. Washington. Quintic polynomials and real cyclotomic fields with large class num-bers. Math. Comp. 50 (1988), no. 182, 543-556.

[13] D. Shanks. The simplest cubic fields. Math. Comp. 28 (1974), 1137-1157[14] Y.Y. Shen, L.C. Washington. A family of real 2n-tic fields. Trans. Amer. Math. Soc. 345 (1994),

no. 1, 413-434.[15] Y.Y. Shen, L.C. Washington. A family of real pn-tic fields. Canad. J. Math. 47 (1995), no. 3,

655-672.[16] J. Silverman, The arithmetic of dynamical systems. Graduate Texts in Mathematics, 241.

Springer, New York, 2007.

Appendix A

Field Diagrams

The following diagram shows the relationship between the towers Kn and Ln. Defineb ∈ N ∪ ∞ to be

b = supm ∈ N : K(ζm) = K(ζ).

For example, if K = Q then b = 1; if K = R then b = ∞; and if K = Q(ζ2 + ζ−12 ) then

K(ζ) = K(ζ2) but K(ζ) K(ζ3), so b = 2. In terms of the towers, when n ≥ b, thereare + 1 intermediate fields between Kn and Kn+1. Finally, we make the convention thata single line denotes that all infinite primes are inert, while a double line indicates that allinfinite primes split completely.



Lm+1 = K0(ζm+1 , m+1α(T ))

2

Km+1

Lm( m+1α(T ))

Lm(ζ−1

m+1m+1

α(T ))

··· Lm(ζm+1 )

Km(θ(m+1)0 )

Km(θ(m+1)−1 )

··· Km(ζm+1 + ζ−1

m+1 )

Lm = K0(ζm , m

α(T ))

2

Km Lb+1 = K0(ζb+1 ,b+1

α(T ))

2

...

Kb+1

...

Lb(b+1

α(T ))

Lb(ζ−1

b+1b+1

α(T ))

··· Lb(ζb+1 )

Kb(θ(b+1)0 )

Kb(θ(b+1)−1 )

··· Kb(ζb+1 + ζ−1

b+1 )

Lb = K0(ζ,b

α(T ))

2

Kb

Lb−1 = K0(ζ,b−1

α(T ))

...2

Kb−1

...

L1 = K0(ζ,

α(T ))

2

K1

L0 = K0(ζ1,α(T )) = K0(ζ)

2

K0 = K(T )

Lm

m

2

Km

2m−v K0(ζm )

2m−v

K0

6 mai 2013



Z. Chonoles, Department of Mathematics, The University of Chicago, 5734 S. University Avenue Chicago,IL 60637, USA • E-mail : [email protected]

J. Cullinan, Department of Mathematics, Bard College, Annandale-On-Hudson, NY 12504, USAE-mail : [email protected]

H. Hausman, Department of Mathematics, Williams College, Williamstown, MA 01267, USAE-mail : [email protected]

A.M. Pacelli, Department of Mathematics, Williams College, Williamstown, MA 01267, USAE-mail : [email protected]

S. Pegado, Department of Mathematics, Williams College, Williamstown, MA 01267, USAE-mail : [email protected]

F. Wei, Department of Mathematics, Harvard University, One Oxford Street, Cambridge MA 02138, USAE-mail : [email protected]


UN THÉORÈME DE ZÉROS DANS LES GROUPES ALGÉBRIQUESCOMMUTATIFS

par

Aurélien Galateau

Résumé. — Dans ces notes, on présente un théorème de zéros, dû à Amoroso et David, quigénéralise le résultat principal de [Phi96] et constitue une version avec multiplicités, dans lecadre élargi des groupes algébriques commutatifs, du lemme de zéros de [AD03]. Cet énoncés’avère utile dans certaines approches diophantiennes du problème de Bogomolov effectif sur lesvariétés abéliennes (cf. [Gal10]).

Abstract. — In these notes, we give a new zero theorem, due to Amoroso and David, whichgeneralises the main result of [Phi96]. This is a version with multiplicity, in the general settingof commutative algebraic groups, of the zero lemma proven in [AD03]. This new result turnsout to be useful in a recent diophantine approach of the effective Bogomolov problem on abelianvarieties (cf. [Gal10]).

1. Introduction

L’objet de ces notes est d’énoncer et de prouver une généralisation du théorème de zéros

annoncé par Amoroso et David dans [AD03]. Ce théorème s’inscrit dans la lignée des lemmes

de zéros classiques de Philippon, qu’il généralise (voir [Phi86], théorème 2.1 et [Phi96]

théorèmes 1 et 2).

On fixe pour cet article un corps k algébriquement clos et un groupe algébrique commutatif

G de dimension g ≥ 1 défini sur k. On sait qu’il existe un plongement (quasi-projectif), qu’on

fixe maintenant, de G dans un espace projectif Pn sur k. Soit V une sous-variété de G. Un

lemme de zéros permet classiquement de comparer le degré d’un polynôme homogène P non

trivial, nul sur une réunion de translatés de V , au degré d’une “variété obstructrice” contenant

un translaté de V . Le théorème principal de [Phi96] permet ainsi de minorer le degré d’un

polynôme P nul sur le translaté de V par un ensemble de la forme Σ0 + · · ·+Σd, où d est la

codimension de V et les (Σi)1≤i≤d sont des ensembles finis de points pondérés par des ordres

d’annulation.

Classification mathématique par sujets (2010). — 11J95, 14C17, 14L40.Mots clefs. — géométrie diophantienne, groupes algébriques commutatifs, intersection géométrique.

36 Un théorème de zéros dans les groupes algébriques commutatifs

Sous l’hypothèse que les supports des Σi sont des réunions d’un nombre fini de sous-groupes

(Hi,j)1≤j≤si de G, on obtient ici une inégalité plus précise concernant le degré de sous-variétés

obstructrices. En reprenant les idées développées par Philippon dans [Phi96] (notamment le

formalisme des dessous d’escalier), on donne un énoncé précis suivant la multiplicité. La

première partie rassemble des préliminaires sur les dérivations, les dessous d’escalier et la

multiplicité de Samuel. Dans la seconde, on énonce le théorème de zéros puis on le démontre,

en considérant des suites de gros ensembles algébriques emboîtés obtenus en intersectant des

translatés de l’hypersurface définie par P . On donne enfin une information supplémentaire

sur les variétés obstructrices dont l’existence est donnée par le théorème de zéros.

Rappelons que ce raffinement dans les lemmes de zéros a été introduit par Amoroso et

David pour démontrer une minoration fine du minimum essentiel des sous-variétés du tore

multiplicatif. Un analogue partiel de leur minoration a été démontré dans le cadre des variétés

abélienne en utilisant le résultat principal de cet article (voir [Gal10]).

2. Définitions et résultats préliminaires

Avant d’énoncer le théorème, il est utile de rappeler un certain nombre de définitions. On note

g l’idéal de définition (premier, homogène) de l’adhérence de Zariski G de G dans Pn et :

A := k[X0, . . . , Xn]/g.

On dira qu’une sous-variété V de G est définie dans G par un premier p de A si V = G∩Z(p).On dira aussi qu’elle est incomplètement définie dans G par des formes de degré au plus L si

V est une composante irréductible d’un fermé G∩Z(I), où I est un idéal de A engendré par

des polynômes de degré au plus L (voir [Phi86], définitions 3.5).

Soit U :=

k

i=1[ni]Xi et V :=

k

i=1[mi]Xi deux cycles algébriques sur G (quitte à rajouter

des zéros, on peut supposer que la somme porte sur les mêmes composantes). On définit la

réunion de U et de V , notée U ∪ V , comme étant le cycle :

k

i=1

[maxni,mi]Xi.

Le degré du cycle

k

i=1[ni]Xi est par définition :

k

i=1 nideg(Xi).

Soit x ∈ G défini par un idéal m de A. On pose Am le localisé homogène de A en m et Am

son complété pour la topologie m-adique. On sait (cf. [Bou83], 8, §5) que Am est isomorphe

à B := k[[T1, . . . , Tg]] = k[[T]]. De plus, la donnée d’un tel isomorphisme Φ0 en x = 0 (défini

par un idéal m0) détermine par translation un isomorphisme Φx pour tout x ∈ G.

On fixe x ∈ G. L’isomorphisme Φx permet de munir l’espace des opérateurs différentiels

Homk(Am,k) de la base (∂κ

Φx)κ∈Ng , où : ∂κ

Φx:=

1κ!

∂|κ|

∂Tκ , avec :

|κ| :=g

i=1

κi et : κ! := κ1! · · ·κg!

Pour 1 ≤ i ≤ g, on note i := (0, . . . , 0, 1, 0, . . . , 1), le 1 étant sur la i-ème coordonnée. Les

(∂iΦx

)1≤i≤g forment alors une base du tangent Homk(m/m2,k) de G en x.


A. Galateau 37

Si F ∈ A est une forme homogène de degré D non nulle en x, pour toute forme homogène

P ∈ A de degré D, on note Px,F l’image deP

Fdans Am.

Définition 1. — Soit P ∈ A une forme homogène de degré D et T ≥ 1 un entier. On dit

que P s’annule à un ordre au moins T en x si pour toute forme F ∈ A, de degré D, non nulle

en x, on a :

∀κ ∈ Ng, |κ| ≤ T − 1 : ∂

κ

ΦxPx,F = 0.

2.1. Dessous d’escalier et ensembles pondérés. — On reprend ici le formalisme des

dessous d’escaliers développé par Philippon dans [Phi96], 2. Ce formalisme permet classique-

ment d’intégrer la multiplicité dans un lemme de zéros, en travaillant avec des ensembles de

points de G pondérés par des sous-ensembles de Ng.

Soit d ≥ 1 et 1 ≤ i1 < · · · < id ≤ g. On appelle d-face d’indice i := (i1, . . . , id) le sous-ensemble

Fi := N · i1 + · · ·+ N · id de Ng.

On dit qu’un ensemble E ⊂ Ng est un escalier si pour tout κ ∈ E, on a : κ + Ng ⊂ E.

Le complémentaire dans Ng d’un escalier est appelé dessous d’escalier. Si W est un dessous

d’escalier, on note aussi IW l’idéal de B engendré par les monômes (Tβ)β∈Ng\W .

On appelle ensemble pondéré un sous-ensemble Σ de Ng × G tel que pour tout x ∈ G,

l’ensemble Wx,Σ := κ ∈ Ng, t.q. (κ, x) ∈ Σ est un dessous d’escalier (on notera encore

Wx ce dessous d’escalier lorsqu’il n’y aura pas d’ambigüité sur Σ). Le support d’un ensemble

pondéré Σ, noté Supp(Σ), est sa projection sur G. La somme de deux ensembles pondérés Σ

et Σ, notée Σ+ Σ

, est l’ensemble pondéré :

(κ+ κ, x+ x

), (κ, x) ∈ Σ, (κ

, x

) ∈ Σ

.On identifie aussi un sous-ensemble X de G à l’ensemble pondéré 0×X. Si P ∈ A est une

forme de degré D et Σ est un ensemble pondéré, on dit que P est nulle sur Σ si :

∀(κ, x) ∈ Σ, ∀F de degré D non nulle en x : ∂κ

ΦxPx,F = 0.

Si Fi est une d-face, et W un dessous d’escalier, on note Ci(W ) l’enveloppe convexe de

(Ng \ Fi) ∩ W dans Rd+. Si V est une sous-variété de G de codimension d, associée à un

idéal premier p, et si W est un dessous d’escalier, on note :

mW (V ) := d! maxi,x

Vol

Rd

+ \ Ci(W )

,

où le maximum porte sur les indices i associés à une d-face Fi, et les points x ∈ V (définis

par un idéal m) tels que les dérivations ∂i1Φx

, . . . , ∂idΦx

se projettent sur une base de l’espace

Homk(p/p ∩m2,k), qui est le cotangent de V en x.

On peut comparer naturellement la multiplicité mW (V ) à la multiplicité de Samuel, définie

sur les idéaux de la façon suivante (voir [Bou83], chapitre 8, §7.1) :

Définition 2. — Soit A un anneau local, noethérien, de dimension d et p son idéal maximal.

En notant ψ la longueur sur les modules, on pose :

ep(A) := d! liml→∞

ψ(A/plA)

ld.

On a alors le lemme (lemme 7 de [Phi96]) :



Lemme 1. — Soit I un idéal homogène de A, p un idéal minimal associé à I, définissant

une sous-variété V de G et W un dessous d’escalier fini de Ng. Si ∆W

Φ0(I) ⊂ p, on a :

eIAp(Ap) ≥ mW (V ).

2.2. Ensembles pondérés et idéaux. — On aura besoin, pour prouver le théorème de

zéros, de faire agir des ensembles pondérés sur des idéaux homogènes, via des opérateurs

différentiels qu’on définit dans ce paragraphe.

On suppose désormais qu’il existe un entier c ≥ 1 tel qu’on peut trouver un entier c ≥ 1, un

recouvrement ouvert (Uα)α et des formes bi-homogènes (Fα)α définies sur k et de degré (c, c)

représentant l’addition :

s : G × G → G.

Toutes ces données sont fixées pour la suite de l’article. On sait d’après [Lan87] qu’un tel

c existe, et qu’on peut prendre (c, c) = (2, 2) si l’adhérence de Zariski G est projectivement

normale et équivariante.

Soit x ∈ G et soit (Fα,0, . . . , Fα,n) une famille de polynômes bi-homogènes représentant

l’addition au voisinage de (0, x) ∈ G × G. On note :

s∗: A −→ A⊗k Am,

le morphisme induit sur les algèbres. Soit P ∈ A, et κ ∈ Ng. On pose alors ∆κ

ΦxP le coefficient

dans A du monôme Tκ

de (1 ⊗ Φx) s∗(P ) ∈ A ⊗k B. Ce polynôme dépend du choix de la

famille (Fα,i)0≤i≤n, mais pas son hypersurface d’annulation. En particulier, P s’annule sur un

ensemble pondéré Σ si et seulement si :

∀(x,κ) ∈ Σ, on a : ∆κ

ΦxP ∈ m0.

Les relations suivantes sont immédiates :

∆κ

Φx(P +Q) = ∆

κ

ΦxP +∆

κ

ΦxQ,

∆κ

Φx(PQ) =

κ+κ=κ

∆κ

ΦxP ·∆κ

Φx

Q,

pour tout x ∈ G, (P,Q) ∈ A, κ ∈ Ng.

On définit maintenant :

Définition 3. — Soit I un idéal homogène de A et W un dessous d’escalier de Ng. On pose :

∆W

Φx(I) :=

∆

κ

Φx(P ), P ∈ I,κ ∈ W

,

et si Σ est un ensemble pondéré :

∆Σ(I) :=

∆

Wx,Σ

Φx(I), x ∈ Supp(Σ)

.

On dit que I s’annule sur Σ si ∆Σ(I) ⊂ m0.

On a alors les propriétés :

Proposition 2. — Soit W et W deux dessous d’escalier de Ng, (x, x

) ∈ G2

, I et J deux

idéaux homogènes de A.


A. Galateau 39

– (1) On a : ∆W

Φx(I+ J) = ∆

W

Φx(I) +∆

W

Φx(J). Et si I ⊂ J : ∆W

Φx(I) ⊂ ∆

W

Φx(J).

– (2) On a : ∆0Φ0(I) = I et ∆0

Φx(I) = τ

∗x(I), où τ

∗x désigne la translation par x sur l’algèbre A.

– (3) L’idéal ∆W

Φxne dépend pas du choix du recouvrement ouvert et des formes bi-homogènes

représentant l’addition sur ce recouvrement. De plus :

∆W

Φx∆W

Φx

(I) = ∆W+W

Φx+x(I).

Démonstration. — Pour les points (1), (2), et la première assertion de (3), voir la preuve de la

Proposition 3 de [Phi96], où l’hypothèse que les dessous d’escalier sont finis n’est pas utilisée.

L’égalité du point (3) est démontrée dans cette même référence si l’un des dessous d’escalier

est trivial, et plus généralement dans [DH00] (4.5, (iv)).

3. Le théorème de zéros et sa démonstration

3.1. Le théorème principal. — On peut maintenant énoncer le principal résultat de cet

article.

Théorème 3. — Soit V une sous-variété algébrique de G de codimension d ≥ 1 et soient

s0, . . . , sd des entiers strictement positifs. Pour 0 ≤ i ≤ d, soit Σi un ensemble pondéré tel

que, pour tout x ∈ Supp(Σi), Wx,Σi est fini et ne dépend pas de x. On suppose que :

Σi = Γi,1 ∪ · · · ∪ Γi,si ,

pour des ensembles pondérés Γi,j (1 ≤ j ≤ si) à support contenant l’origine de G. Pour tout

1 ≤ j ≤ si, fixons un sous-groupe Hi,j de StabSupp(Γi,j)

. Soit enfin P ∈ A non trivial,

s’annulant sur l’ensemble pondéré :

V + Σ0 + · · ·+ Σd.

Il existe alors :

– Des entiers 1 ≤ k ≤ k ≤ d ;

– Des indices j0, . . . , jk−1 tels que, ∀ 0 ≤ i ≤ k − 1 : 1 ≤ ji ≤ si ;

– Des sous-variétés algébriques Zj de G, pour 1 ≤ j ≤ sk , propres, irréductibles, de codimen-

sion k, telles que :

deg

x∈H0,j0+···+Hk−1,jk−1

1≤j≤sk

y∈Γk,j

[mWy(x+ y + Zj)]x+ y + Zj

≤ cgdeg(G) · Lk

.

De plus, chaque variété Zj, pour 1 ≤ j ≤ sk , est incomplètement définie dans G par des

formes de degré au plus cL et contient au moins une composante isolée du fermé :

Supp

V +H1,j1 + · · ·+Hk−1,jk−1

+ Σk + · · ·+ Σd

.

Remarques.

(i) Ce théorème implique le théorème 2 de [Phi96]. Il suffit de prendre ici tous les si égaux à 1

et les sous-groupes Hi,1 égaux à 0 (pour 0 ≤ i ≤ d). Notre théorème affirme alors l’existence

d’une unique variété obstructrice qui vérifie la formule donnée dans le théorème 2 de [Phi96]



(la variété obstructrice étant notée V dans cette référence). C’est notamment pour cela qu’on

n’a pas directement écrit les Supp(Σi) comme des réunions de sous-groupes.

(ii) Ce théorème implique aussi le théorème 4.2 de [AD03], en considérant la compactification

standard de Gnm, en prenant c = 1 et tous les supports des Γi,j égaux à des sous-groupes Hi,j

de Gnm. Outre la généralité du groupe algébrique G, la principale nouveauté de ce théorème,

en comparaison du théorème 4.2 de [AD03], est la prise en compte des multiplicités.

3.2. Démonstration du théorème 3. — Passons à la preuve du théorème proprement

dite. L’idée, classique, consiste à considérer de gros ensembles algébriques construits en in-

tersectant des translatés de l’hypersurface définie par P , avec multiplicité. On construit ainsi

une suites de fermés emboîtés, et on s’assure que deux fermés successifs ont même dimension.

On peut alors comparer les degrés de ces deux fermés (en même dimension), ce qui achève de

prouver le théorème 3.

Soit maintenant V une sous-variété de G de codimension d, des ensembles pondérés Σ0, . . . ,Σd

tels que dans le théorème, et un polynôme homogène P de degré L, non trivial, et qui s’annule

sur V + Σ0 + · · ·+ Σd.

Pour pouvoir travailler avec des ensembles algébriques, et en particulier ne retenir que les

composantes irréductibles pertinentes de tels ensembles, on aura besoin de la définition sui-

vante :

Définition 4. — Soit Z1 et Z2 deux fermés algébriques inclus dans G. On appelle trace de

Z1 relativement à Z2, notée tr(Z1, Z2), la réunion des composantes isolées de Z1 contenant

au moins une composante isolée de Z2.

3.2.1. Construction des fermés emboîtés. — Commençons par définir, pour 1 ≤ j ≤ s0 :

I0,j := ∆Γ0,j (P ),

X0,j := Z(I0,j),

Y0,j := tr

X0,j , Supp(V +H0,j + Σ1 + · · ·+ Σd)

.

On pose ensuite j0 comme étant le plus petit entier 1 ≤ j ≤ s0 tel que la dimension de Y0,j

soit minimale. On pose ensuite :

I0 := I0,j0 , X0 := X0,j0 , et Y0 := Y0,j0 .

De la même façon, on définit, par récurrence sur 1 ≤ i ≤ d, les ensembles suivants, où j varie

entre 1 et si :

Ii,j := ∆Γi,j (Ii−1),

Xi,j := Z(Ii,j),

Yi,j := tr

Xi,j , Supp(V +H0,j0 + · · ·+Hi−1,ji−1 +Hi,j + Σi+1 + · · ·+ Σd)

.

On choisit aussi l’entier ji comme étant le plus petit entier 1 ≤ j ≤ si pour lequel la dimension

de Yi,j est minimale, et l’on pose :

Ii := Ii,ji , Xi := Xi,ji , et Yi := Yi,ji .


A. Galateau 41

Fait 1. — La suite de fermés ainsi construite vérifie les propriétés suivantes :

– (1) Pour tout 0 ≤ i ≤ d, on a : Yi = ∅.– (2) Pour tout 0 ≤ i ≤ d− 1, on a l’inclusion : Yi+1 ⊂ Yi.

– (3) On a l’inégalité : codim(Yd) ≤ d.

Démonstration. — Pour démontrer la propriété (1), on observe par une récurrence immédiate

que, pour 0 ≤ i ≤ d et 1 ≤ j ≤ si :

Ii,j = ∆Γ0,j0+···+Γi−1,ji−1+Γi,j

(P ).

Comme P s’annule sur V + Σ0 + · · ·+ Σd, on a donc l’inclusion :

Supp(V + Σi+1 + · · ·+ Σd) ⊂ Xi,j .

Par définition de Yi,j , on en déduit :

Yi,j ⊃ tr(Xi,j , Supp(V + Σi+1 + · · ·+ Σd)) = ∅.

En particulier, Yi est non vide pour tout 0 ≤ i ≤ d.

Pour démontrer le second point, fixons 0 ≤ i ≤ d−1 et remarquons que pour tout 1 ≤ j ≤ si+1,

Γi+1,j contient l’origine de G. On en déduit les inclusions : Ii ⊂ Ii+1,j , puis pour j = ji :

Xi+1 ⊂ Xi. On remarque enfin que la réunion de translatés de V intervenant dans la définition

des Yi,j est décroissante avec i, et on obtient : Yi+1 ⊂ Yi.

Enfin, le point (3) se déduit immédiatement du point (1) : si la variété Yd est non vide, alors

elle contient un translaté de V , et est de codimension inférieure à d.

Fait 2. — Il existe un indice 1 ≤ k ≤ d tel que : dim(Yk−1) = dim(Yk).

Démonstration. — On remarque que : (P ) ⊂ I0, donc que : Y0 ⊂ X0,j0 ⊂ Z((P )). Les Yi

forment donc une suite de d + 1 fermés emboîtés de codimension comprise entre 1 et d, et il

existe deux fermés successifs ayant même dimension.

L’entier k du théorème est donc fixé, ainsi que les entiers j0, . . . , jk−1. On pose aussi

k := codim(Yk).

On doit maintenant déterminer les variétés Zj (1 ≤ j ≤ sk) dont l’existence est affirmée par

le théorème.

3.2.2. Choix des variétés obstructrices. — Le fermé Yk étant de dimension minimale parmi

les Yk,j , et chacun de ces fermés étant inclus dans Yk−1, on en déduit l’égalité, pour tout

1 ≤ j ≤ sk :

dim(Yk,j) = dim(Yk−1),

et il existe une composante isolée commune à ces deux fermés. Pour tout 1 ≤ j ≤ sk , on

choisit une telle variété (de codimension k), et on note Zj son intersection avec G.

On est déjà en mesure de démontrer le dernier point du théorème. Fixons 1 ≤ j ≤ sk . L’idéal

Ik−1 est engendré par des formes de degré au plus cL et la variété Zj est une composante



isolée de G∩Xk−1, donc est incomplètement définie dans G par des formes de degré au plus

cL. De plus, Zj contient, par construction, une composante isolée du fermé :

Supp

V +H1,j1 + · · ·+Hk−1,jk−1

+ Σk + · · ·+ Σd

.

3.2.3. Comparaison des degrés. — Il reste maintenant à établir l’inégalité du théorème. La

prise en compte des multiplicités rend cette comparaison délicate. On procède en deux étapes.

La première consiste à montrer que le fermé Yk−1 contient de nombreux translatés des Zj .

La seconde est de comparer le degré du cycle naturellement formé à partir de cette réunion

(en introduisant les multiplicités liées aux dessous d’escaliers des Σi) à celui de la réunions

des composantes de codimension k du fermé Yk−1.

On commence par remarquer que pour tout 1 ≤ j ≤ sk et pour tout y ∈ Supp(Γk,j), la

variété y + Zj est incluse dans Yk−1. En effet, par définition de Ik,j , on a :

∆0Φy

(Ik−1) ⊂ Ik,j .

Comme Ik,j est nul sur Zj par construction, il suit que Ik−1 est nul sur y + Zj (par la

proposition 2, (2)). Soit maintenant une composante isolée Vj de :

Supp

V +H1,j1 + · · ·+Hk−1,jk−1

+ Σk + · · ·+ Σd

,

contenue dans Zj (dont l’existence a été démontrée à la fin du paragraphe 3.2.2). Comme :

y + Vj ⊂ y + Zj et y ∈ Supp(Γk,j),

on a alors :

y + Zj ⊂ tr

Xk−1, Supp(V +H0,j0 + · · ·+Hk−1,jk−1

+Hk,j + Σk+1 + · · ·+ Σd + Γk,j)

= tr

Xk−1, Supp(V +H0,j0 + · · ·+Hk−1,jk−1

+ Σk + Σk+1 + · · ·+ Σd)

= Yk−1,

la dernière égalité provenant du fait que Hk,j stabilise Supp(Γk,j). De plus, ces deux fermés

ayant même dimension, la variété y + Zj est en fait une composante isolée de Yk−1.

L’étape suivante consiste à montrer que H0,j0 + · · ·+Hk−1,jk−1stabilise Yk−1. Pour 0 ≤ i ≤

k − 1, soit donc xi ∈ Hi,ji et notons x := x0 + · · ·+ xk−1. Par la proposition 2, (2) et (3), on

a :

τ∗x(Ik−1) = ∆

0Φx

∆Γ0,j0+···+Γi−1,ji−1 (P ),

et puisque pour tout 0 ≤ i ≤ k − 1, xi ∈ Stab

Supp(Γi,ji)

, on a :

τ∗x(Ik−1) = Ik−1.

Le fermé Xk−1 est donc stabilisé par H0,j0 + · · ·+Hk−1,jk−1, et il en va de même pour Yk−1

vue sa définition.

On en déduit que la réunion :

x∈H0,j0+···+Hk−1,jk−1

1≤j≤sk

y∈Γk,j

(x+ y + Zj)(1)


A. Galateau 43

est formée de composantes isolées de Yk−1. Il s’agit maintenant, en utilisant cette information,

de majorer le degré du cycle considéré dans le théorème 3.

Soit donc 1 ≤ j ≤ sk , x ∈ H0,j0 + · · · + Hk−1,jk−1, et y ∈ Supp(Γk,j). Notons p l’idéal

premier définissant x+ y + Zj . On a l’inclusion suivante entre idéaux :

∆Wy

Φ0(Ik−1) ⊂ τ

∗−x−y(Ik,j) ⊂ p.

La première inclusion suit la définition de l’idéal Ik,j , en remarquant que x stabilise Xk−1, et

la seconde est directe vue la construction de Zj . Toutes les conditions sont maintenant réunies

pour appliquer le lemme 1 qui donne :

eIk−1Ap(Ap) ≥ mWy(x+ y + Zj).(2)

On peut désormais majorer le degré du cycle suivant :

Z :=

x∈H0,j0+···+Hk−1,jk−1

1≤j≤sk

y∈Γk,j

[mWy(x+ y + Zj)]x+ y + Zj

,

On a :

deg(Z) ≤

p∈Ass(Ik−1),rg(p)=k

eIk−1Ap(Ap) · deg(p)

≤ deg(G) · (cL)k.

L’inégalité (2) et les définitions rappelées en 2 sur les cycles algébriques établissent en effet

la première inégalité. La seconde est le lemme 5 de [Phi96], la réunion (1) étant incluse dans

Xk−1. Le théorème 3 est donc entièrement démontré.

3.3. Une remarque sur les variétés obstructrices. — Pour conclure ce travail, donnons

une dernière précision sur les variétés obstructrices. On peut en fait, au moins dans un cas

standard, montrer que les formes qui les définissent ont une multiplicité. Précisons ce qu’on

entend par là :

Définition 5. — Soit V une sous-variété de G et soient T, L deux entiers positifs. On dit que

V est incomplètement définie dans G par des formes de degré ≤ L, avec multiplicité ≥ T +1,

s’il existe des éléments (q1, . . . , ql) de A, tous de degrés ≤ L, définissant un idéal I, tels que

V soit une composante irréductible de G ∩ Z(∆WTΦ0

(I)), où W est le dessous d’escalier :

WT := κ ∈ Ng, |κ| ≤ T.

On reprend les hypothèses du théorème 3, et on suppose en plus que Σk (où k est donné par

le théorème) est de la forme :

Σk = κ ∈ Ng, |κ| ≤ Tk× Supp(Σk),

pour un entier positif Tk . Dans ce cas :

Proposition 4. — Les variétés Zj dont l’existence est donnée par le théorème 3 sont incom-

plètement définies dans G par des formes de degré ≤ cL avec multiplicité ≥ Tk .



Démonstration. — Dans la preuve du théorème 3, on a construit les Zj comme composantes

de G ∩ Z(Ik−1), où l’idéal Ik−1 est engendré par des formes de degré au plus cL. De

plus, par construction, pour 1 ≤ j ≤ sk , la variété Zj est une composante irréductible

de G ∩Xk,j = G ∩ Z(Ik,j). En particulier, si pj désigne l’idéal de définition de Zj :

∆WTkΦ0

(Ik−1) ⊂ pj ,

et Zj est bien une composante irréductible de G ∩ Z(∆WTkΦ0

(Ik−1)).

Références

[AD03] F. Amoroso et S. David : Minoration de la hauteur normalisée dans un tore. J. Inst. Math.Jussieu, 2(3):335–381, 2003.

[Bou83] N. Bourbaki : Algèbre commutative. Masson, Paris, 1983.

[DH00] S. David et M. Hindry : Minoration de la hauteur de Néron-Tate sur les variétés abéliennes

de type C.M. J. Reine Angew. Math., 529:1–74, 2000.

[Gal10] A. Galateau : Le problème de Bogomolov effectif sur les variétés abéliennes. Algebra andNumber Theory, 4:547–598, 2010.

[Lan87] H. Lange : Families of translations of commutative algebraic groups. Journal of algebra,

109:260–265, 1987.

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22/04/2014

Aurélien Galateau, Laboratoire de Mathématiques de Besançon, Facultés des sciences et techniques, CNRS,UMR 6623, 16 route de Gray, 25030 Besançon, France • E-mail : [email protected]


WEIGHT REDUCTION FOR COHOMOLOGICAL MOD p

MODULAR FORMS OVER IMAGINARY QUADRATIC FIELDS

by

Adam Mohamed

Abstract. — Let F be an imaginary quadratic field and O its ring of integers. Let n ⊂ O be

a non-zero ideal and let p > 5 be a rational inert prime in F and coprime with n. Let V be

an irreducible finite dimensional representation of Fp[GL2(Fp2)]. We establish that a system

of Hecke eigenvalues appearing in the cohomology with coefficients in V already lives in the

cohomology with coefficients in Fp ⊗ dete for some e ≥ 0; except possibly in some few cases.

Resume. — Soient F un corps quadratique imaginaire et O son anneau d’entiers. Soient

n ⊂ O un ideal non nul et p > 5 un nombre premier inerte dans F copremier avec n. SoitV une representation irreductible de dimension finie de Fp[GL2(Fp2)]. Nous etablissons qu’un

systeme de valeurs propres de Hecke appartenant au groupe de cohomologie coefficients dans Vappartient aussi au groupe de cohomologie coefficients dans Fp ⊗ dete pour e ≥ 0 a l’exception,

eventuellement, de quelques cas.

1. Introduction

Let F be an imaginary quadratic field with O as its ring of integers. The class number ofF is denoted as h. Let Γ be a congruence subgroup of GL2(O). Let σ be the non-trivial

element of Gal(F/Q). We consider the representations of GL2(O) defined as Va,br,s (O) =

Symr(O2) ⊗ deta ⊗ (Syms(O2))σ ⊗ (detb)σ where a, b, r, s are positive integers. For an O-

algebra A, we define Va,br,s (A) := V

a,br,s (O) ⊗O A. A cohomological modular form of level Γ

and weight V a,br,s (A) over F is a class in H1(Γ, V a,b

r,s (A)). As in the classical setting, the space

H1(Γ, V a,br,s (A)) can be endowed with a structure of Hecke module. The Hecke algebra acting

on H1(Γ, V a,br,s (A)) is commutative and has its elements indexed over the integral ideals of

F. So, one can consider eigenclasses or eigenforms which are eigenvectors for all the Heckeoperators Ta. Hence to such an eigenform corresponds a system of Hecke eigenvalues.Integral systems of eigenvalues when reduced modulo a prime p are believed to be relatedto mod p representations of Galois groups as conjectured by Ash et al. in [4]. One instance

2010 Mathematics Subject Classification. — 11F75, 11F67, 11F25, 11F41.

Key words and phrases. — Modular forms modulo p, imaginary quadratic fields, Hecke operators, Serre

weight.

46 Weight reduction for cohomological mod p modular forms over imaginary quadratic fields

of this correspondence being the theorem of Deligne constructing l-adic representations ofthe absolute Galois group of Q, GQ := Gal(Q/Q), via systems of Hecke eigenvalues arisingfrom modular forms over Q. Let N be a positive integer and Γ0(N) a congruence subgroupof SL2(Z). Take V to be the SL2(Z)-module given as V := Sym

k−2(Z2) = Z[X,Y ]k−2, thespace of homogeneous polynomials of degree k − 2 over Z in two variables and with k even.The converse of Deligne’s theorem, Serre’s modularity conjecture, which is now a theorem ofKhare and Wintenberger, has been formulated in the language of group cohomology in [5]and the standard conjecture in there relates mod p Galois representations of GQ to systemsof Hecke eigenvalues on H1(Γ0(N), V ⊗Z Fp).Next let N and n be positive integers. In [2], it was shown that a system of Hecke eigenvaluesoccurring in the cohomology of Γ1(N) with coefficients in some GLn(Fp)-module also occursin the cohomology with coefficients in some irreducible GLn(Fp)-module. This fact has someinteresting features. In fact it allows one to obtain a cohomological avatar of the so-calledHasse invariant, see [9]. That is, one can produce congruences between weight two and higherweight modular forms using cohomological methods.As for the case of an imaginary quadratic field F of class number one, then when p splits in F

and is coprime with n, in [12], it is established that a Hecke system of eigenvalues occurringin the first cohomology with non-trivial coefficients can be realized in the first cohomologywith trivial coefficients. This should also hold when the class number of F is greater than one.Let p be a rational prime coprime to n and inert in F. Let E be a finite dimensionalrepresentation of GL2(Fp2) over Fp. Let Γ be a congruence subgroup of GL2(O). Then acohomological mod p modular form of level Γ and weight E is defined to be a class in H1(Γ,E).As in the classical setting there is a Hecke algebra action on the space H1(Γ,E) and onecan consider systems of Hecke eigenvalues for the space H1(Γ,E). Our aim will be to saysomething more precise about systems of Hecke eigenvalues in this setting. We will provethat a system of Hecke eigenvalues living in ⊕h

i=1H1(Γ1,[bi](n),M) where M is an irreducible

Fp[GL2(Fp2)]-module also occurs in ⊕h

i=1H1(Γ1,[bi](pn),Fp⊗det

e) for some e ≥ 0 depending onM ; except possibly for some cases. See Theorem 4.11 for the precise statement. Here Γ1,[bi](n)are some congruence subgroups defined in Section 3. The strategy for proving Theorem 4.11was initiated by Ash and Stevens in [2], and it was also adapted in [12] where a reduction toweight 2 statement is proved.There is an application of Theorem 4.11 related to Serre type questions about mod p Galoisrepresentations of the absolute Galois group of F. When we are dealing with cohomologicalmodular forms mod p with trivial coefficients Fp, we shall say that we are in weight two. LetGF := Gal(F/F ) and let be given

ρ : GF → GL2(Fp)

an irreducible mod p Galois representation of conductor n. Let Tr denotes the trace of amatrix. Then the following questions arise:

(a) Does there exist a cohomological Hecke eigenform of some weight V and level n witheigenvalues ψ(Tλ) such that Tr(ρ(Frobλ)) = ψ(Tλ) for all unramified prime ideals λ pn?

(b) Does there exist a cohomological Hecke eigenform of weight 2 (V = Fp ⊗ dete for some

e ≥ 0) and level pn with Tr(ρ(Frobλ)) = ψ(Tλ) for all unramified prime ideals λ pn?

Publications mathematiques de Besancon - 2014/1

A. Mohamed 47

As a consequence of Theorem 4.11, we shall see that the two questions above are equivalent.See Proposition 4.12 for the precise statement. Proposition 4.12 proves that when investigatingSerre type questions as above, it is enough to work in weight two. For example, in [11], somecomputational investigations of Serre’s conjecture over imaginary quadratic fields were carriedout and the principle illustrated by Proposition 4.12 was assumed to hold.Here is our outline. We shall first recall Hecke theory in our context. This is the contentof Section 2. In Section 3, we shall compare some modules. The main result is proved inSection 4.

Acknowledgement. — The present article is extracted from the author dissertation which wassupervised by Gabor Wiese. The author thanks him for his time and teaching. This workstarted as one of the FP6 European Research Training Networks “Galois Theory and ExplicitMethods” project (GTEM; MRTN-CT-2006-035495), I acknowledge their financial support.

2. Hecke operators

We set some of the necessary notation and recall briefly how Hecke operators are defined inour setting. As in the classical setting, we define Hecke operators via Hecke correspondenceson hyperbolic 3-manifolds. This section is mainly notational as what we shall recall is verywell explained in for example the works of Hida or Shimura.So, some of the notation are as follows. Let F be an imaginary quadratic field of class numberh ≥ 1. Denote by O its ring of integer and let n be an ideal of O. The class group of F isdenoted by Cl and we fix a rational prime p inert in F and p = pO. We also assume that pis coprime with n. Let O be the profinite completion of O : O =

q =0Oq. We will denote

the adeles of F by A, and Af , A∞ stand for the finite part and the infinite part of A. Wewrite G := GL2, so that, G(A), G(F ), G(Af ) are the usual linear algebraic groups of 2 × 2matrices with entries in A, F,Af , respectively. Let H3 := G(C)/C∗U2

∼= C× R>0, the threedimensional equivalent of the classical Poincare upper half plane H2 = G(R)/R∗O2. Here U2

is the unitary subgroup of G(C). Let K be an open compact subgroup of G(O) such that thedeterminant homomorphism

det : K → O∗

is surjective. We define the following homogeneous space

YK := G(F )\(H3 ×G(Af )/K)

= G(F )\(G(C)/C∗U2 ×G(Af )/K)

= G(F )\G(A)/K.U2.C∗.

By the determinant map we have

YK F∗\A∗

/O∗C∗ ∼= F

∗\A∗

f/O∗ ∼= Cl.

2.1. Hecke correspondences and Hecke operators. — Let σ be the generator ofGal(F/Q). Let

VO = Va,b

r,s (O) = Symr(O2)⊗ deta ⊗ (Syms(O2))σ ⊗ (detb)σ



be an O[G(O)]-module endowed with the discrete topology. We define Va,br,s (Fp) := VO ⊗O Fp.

This space is also endowed with the discrete topology.Let X = G(A)/U2C

∗ ∼= H3×G(Af ). Under the assumption that K acts freely on X×Va,br,s (Fp),

one has a topological cover

π1 : G(F )\(X × Va,b

r,s (Fp))/K → G(F )\X/K ∼= YK .

We consider the locally constant sheaf VFp

on YK given by the sections of π1 : for an opensubset U of YK , we have

VFp(U) = s : U → G(F )\(X × V

a,b

r,s (Fp))/K; π1 s = id.

We take g ∈ Mat2(O) =0 (by which we mean the 2× 2 matrices with entries in O and non-zerodeterminant) be such that all its local factors gq at almost all the finite places q including those

dividing pn are ( 1 00 1 ) and otherwise gq are of the form

πq 00 1

or

π2q 00 1

with πq a uniformizer

of Oq.

Remark 2.1. — Often one takes g ∈ Mat2(O) with the component at only one finite placeq away from pn gq being of the form

πq 00 1

and all the remaining components are the identity

matrices.

We introduce K

g−1 = K ∩ g−1

Kg and Kg = gKg

−1 ∩K. The group isomorphism

K

g−1∼= K

g; λ → gλg

−1

induces the isomorphism g∗ : YK

g−1∼= YK

g; y → gy, that allows us to form the Hecke

correspondence diagram

YKg−1

g∗

−−−−→ YKgsg

sg

YK YK ,

where sg and sg are the natural projections.The Hecke operator Tg acting on the Fp-vector spaces Hi(YK ,V

Fp) is defined by the following

diagram:

Hi(YKg−1

, s−1g V

Fp)

conj∗g−−−−→ Hi(YK

g, s

−1g V

Fp)

res

cor

Hi(YK ,VFp) Hi(YK ,V

Fp).

So we have Tg = cor conj∗g res. Here conj∗g is the isomorphism induced by the conjugation

map, cor, res are the corestriction and the restriction maps. It is also known that Tg isindependent of the choice of the uniformizers πq but in fact depends only on the double cosetKgK.

2.2. Explicit formulas for the Hecke action. — Let n be a non-zero ideal of O. For ourpurposes, we choose the following representatives of the class group Cl of F. By the Chebotarevdensity theorem, we can choose representatives of the class group [b1] = [O], [b2], · · · , [bh],


A. Mohamed 49

where for i > 1, the bi are prime ideals coprime with pn. Thus we denote the classgroup as Cl = [b1], · · · , [bh]. Let πbi be a uniformizer of the local ring Obi . We definet1 := (1, · · · , 1, 1, 1, · · · , 1, · · · ), and for i > 1, ti := (1, · · · , 1,πbi , 1, · · · , 1, · · · ) ∈ A

∗

f, i.e, ti

is the idele having 1 at all places expect at the place bi where we have πbi . Via the grouphomomorphism

A∗

f→ Cl

(· · ·xq · · · ) → [

q =∞

qvq(xq)],

where vq is the normalized valuation of Oq, we see that ti corresponds to bi. We define

gi :=ti 00 1

, i.e, (gi)q =

(ti)q 00 1

. Similarly gi corresponds to the class [bi] via the determinant

map.From the strong approximation theorem, the topological space YK decomposes into the disjointunion of its connected components as:

YK = h

i=1Γ[bi]\H3,

where Γ[bi] := G(F ) ∩ giKg−1i

. This is an arithmetic subgroup of G(F ). We next recall thedefinition of neatness for subgroups of G(Af ).Let us take K to be neat so that the groups Γ[bi] are torsion free. To achieve this, if K = K1(n),the open compact subgroup of level n defined below, where the positive generator of n ∩ Z isgreater than 3, then Γ[bi] are torsion free. This is Lemma 2.3.1 from [15].From a general comparison theorem it is known that an isomorphism Hr(Γ[bi]\H3,VFp

) =

Hr(Γ[bi], Va,br,s (Fp)) holds, see [7] for details. Hence we can write

Hr(YK ,VFp) = ⊕h

i=1Hr(Γ[bi]\H3,VFp

) = ⊕h

i=1Hr(Γ[bi], V

a,b

r,s (Fp)).

Let us further specialize the open compact subgroup K. We define the open compact subgroupof level n

K1(n) =

a b

c d

∈

q∞

G(Oq) : c, d− 1 ∈ nO.

This is an open compact subgroup which surjects on O∗ by the determinant map. Thecorresponding congruence subgroups G(F ) ∩ giK1(n)g

−1i

are denoted as Γ1,[bi](n). As already

alluded to, the Hecke operators Tg do not act componentwise on the Fp-vector space

⊕h

i=1Hr(Γ1,[bi], V

a,br,s (Fp)). By this we mean that in general Tg permutes the components

when acting on an element from ⊕h

i=1Hr(Γ1,[bi], V

a,br,s (Fp)) as we will soon see.

2.2.1. Prelude to the formulas. — Let q be an integral ideal away from pn. We consider thefollowing subset of Mat2(O). Define

∆q1(n) =

a b

c d

∈ Mat2(O) : (ad− bc)O = qO,

a b

c d

≡ ( ∗ ∗

0 1 ) (mod n).The open compact subgroup K1(n) acts on ∆q

1(n) via multiplication: for g ∈ K1(n) andδ ∈ ∆q

1(n) we have gδ ∈ ∆q1(n). We have that ∆q

1(n)K1(n) = K1(n)∆q1(n) = ∆q

1(n). Forδ ∈ ∆q

1(n) we define the subgroup

K

1,δ(n) = δK1(n)δ−1 ∩K1(n)



of K1(n). The subsets ∆q1(n) act on any left Fp[GL2(Fp2)]-module via reduction modulo p.

There is the following fact that is worth mentioning.

Lemma 2.2. — Let δ ∈ ∆q1(n). Then there is a bijection between the coset space

K1(n)/K

1,δ(n) and the orbit space K1(n)δK1(n)/K1(n) given as

K1(n)/K

1,δ(n) → K1(n)δK1(n)/K1(n)

λK

1,δ(n) → λδK1(n).

Proof. — There is a surjective map K1(n) → K1(n)δK1(n)/K1(n) which sends λK

1,δ(n) toλδK1(n). Two distinct elements λ and λ

map to the same orbit if and only of they lie in thesame class modulo K

1,δ(n).

For δ ∈ ∆q1(n), there are finitely many γj ∈ ∆q

1(n) such that the double coset K1(n)δK1(n)decomposes as

K1(n)δK1(n) = jγjK1(n).

Let g ∈ Mat2(O) be such that its components at a finite number of finite places q away from

pn are of the formπq 00 1

or

π2q 00 1

where πq is a uniformizer of Oq and are the identity

otherwise. When we denote c = (det(g)) the ideal corresponding to g, then g ∈ ∆c1(n).

Lemma 2.3. — Let g ∈ ∆c1(n) as above. Let gi corresponding to [bi] and K1(n) as above.

Then, for each i there exist a unique index ji, 1 ≤ ji ≤ h, matrices ki =ui 00 1

∈ giK1(n)g

−1i

and βi := gjigg−1i

ki =yi 00 1

∈ G(F ) such that K1(n)gK1(n) = K1(n)g

−1ji

βigiK1(n).

Proof. — For each i let ji be the unique index such that the ideal (det(gjigg−1i

)) is principal.

Set then αi := gjigg−1i

=

det(αi) 00 1

. The ideal (det(αi)) being principal means that

det(αi) = xiyi with yi ∈ F∗ and xi ∈ O∗

. Set ui = x−1i

and define ki =ui 00 1

∈ K1(n). Then

ki ∈ giK1(n)g−1i

and βi := αiki =yi 00 1

∈ G(F ). Hence for each i there exists a matrix

βi ∈ gji∆c1(n)K1(n)g

−1i

∩G(F ) = gji∆c1(n)g

−1i

∩G(F )

such that K1(n)gK1(n) = K1(n)g−1ji

βigiK1(n). Indeed, g−1ji

βigi = g−1ji

αikigi = gg−1i

kigi, and

we observe that we have g−1i

kigi ∈ K1(n).

For 1 ≤ i ≤ h, let ji and βi as given in the above lemma. Let fi := (det(βi)) = bjib−1i

c. Define

Λfi1,[bi]

(n) := gji∆c1(n)g

−1i

∩G(F ). Explicitly this is the set

a b

c d

∈ G(F ) : a ∈ bjib

−1i

, b ∈ bji , c ∈ b−1i

, d− 1 ∈ nO; (ad− bc)O = fi.

We set j := ji. Let α ∈ Λfi1,[bi]

(n) (we have in mind βi). We consider the following double coset

Γ1,[bj ](n)αΓ1,[bi](n). This double coset defines a Hecke operator Tα mapping

Hr(Γ1,[bi](n), Va,b

r,s (Fp)) to Hr(Γ1,[bj ](n), Va,b

r,s (Fp))

as follows. Firstly one needs to introduce the following subgroups

1. Γ,α−1

1,[bi](n) := Γ1,[bi](n) ∩ α

−1Γ1,[bj ](n)α

2. Γ,α

1,[bj ](n) := αΓ

1,[bi](n)α−1 = αΓ1,[bi](n)α

−1 ∩ Γ1,[bj ](n).


A. Mohamed 51

The operator Tα is defined as the composition of the following maps:

Hr(Γ,α−1

1,[bi](n), V a,b

r,s (Fp))conjα−−−−→ Hr(Γ,α

1,[bj ](n), V a,b

r,s (Fp))res

cor

Hr(Γ1,[bi](n), Va,br,s (Fp)) Hr(Γ1,[bj ](n), V

a,br,s (Fp)).

Here res is the restriction map, conjα is the isomorphism induced by the compatible maps:

Γ,α

1,[bj ](n) ∼= Γ,α−1

1,[bi](n)

ω → α−1

ωα

and

Va,b

r,s (Fp) → Va,b

r,s (Fp)

v → α.v.

Here cor is the corestriction homomorphism. We explicitly describe Tα in degree zero andone. In degree zero Tα is given as

H0(Γ,α−1

1,[bi](n), V a,b

r,s (Fp))v →αv−−−−→ H0(Γ,α

1,[bj ](n), V a,b

r,s (Fp))v →v

v →

h hv

H0(Γ1,[bi](n), Va,br,s (Fp)) H0(Γ1,[bj ](n), V

a,br,s (Fp)).

where the sum is over a set of cosets representatives of Γ1,[bj ](n)/Γ,α

1,[bj ](n). Hence one obtains

that:

Tα : H0(Γ1,[bi](n), Va,b

r,s (Fp)) → H0(Γ1,[bj ](n), Va,b

r,s (Fp))

v →

λ∈Γ1,[bj ](n)/Γ,α

1,[bj ](n)

(λα).v.

It is worthwhile observing that the decomposition Γ1,[bj ](n) = rλrΓ,α

1,[bj ](n) is equivalent to

the decomposition of the double cosets Γ1,[bj ](n)αΓ1,[bi](n) = rλrαΓ1,[bi](n).

2.2.2. Formula on degree one. — We now give the formula of Tα on degree one cohomology. Tothis end, we need to recall the formulas describing the isomorphism conjα and the corestrictionin terms of non-homogeneous cocycles. The conjugation isomorphism is described by theformula

conjα : H1(Γ,α−1

1,[bi](n), V a,b

r,s (Fp)) → H1(Γ,α

1,[bj ](n), V a,b

r,s (Fp))

c → (ω → α.c(α−1ωα)).

For the corestriction homomorphism, let Γ1,[bj ](n) = nγnΓ,α

1,[bj ](n). For ω ∈ Γ1,[bj ](n), let sn

be the unique index such that γ−1n ωγsn ∈ Γ1,[bj ](n). Then the corestriction homomorphism is



given as

cor : H1(Γ,α

1,[bj ](n), V a,b

r,s (Fp)) → H1(Γ1,[bj ](n), Va,b

r,s (Fp))

c → (ω →

n

γn.c(γ−1n ωγsn)).

The formula of Tα on degree one cohomology is thus

Tα : H1(Γ1,[bi](n), Va,b

r,s (Fp)) → H1(Γ1,[bj ](n), Va,b

r,s (Fp))

c → (ω →

n

γnα.c((γnα)−1

ωγsnα)).

Indeed with the given formulas we have

(cor(conjα(c)))(w) =

γn∈Γ1,[bj ](n)/Γ

1,[bj ](n)

γn.(conjα(c)(γ−1n wγsn))

=

γn∈Γ1,[bj ](n)/Γ

1,[bj ](n)

γnα.c(α−1

γ−1n wγsnα).

Let λi be another set of representatives of Γ1,[bj ](n)/Γ,α

1,[bj ](n), and σi ∈ Γ,α

1,[bj ]such that

λi = γiσi. With this we have

(cor(c))(w) = γiσ.c(σ−1i

γ−1i

wγjiσi).

Because taking conjugation by an element from Γ,α

1,[bj ](n) gives cohomologous cocycle, we

deduce that the corestriction map does not depend on the choice of representatives ofΓ1,[bj ](n)/Γ

,α

1,[bj ](n). This means that Tα does not depend on the choice of set of representatives

and so only depends on the double coset Γ1,[bj ]αΓ1,[bi] since we know that

Γ1,[bj ](n) = nγnΓ,α

1,[bj ]⇐⇒ Γ1,[bj ](n)αΓ1,[bi](n) = nγnαΓ1,[bi](n).

2.2.3. Action of Tg on ⊕h

i=1Hr(Γ1,[bi](n), V

a,br,s (Fp)). — Now that we have recalled the formulas

of the Hecke operators on group cohomology, let us say how Hecke operators act on theFp-vector spaces ⊕h


a,br,s (Fp)). Let g be as in Lemma 2.3 and consider βi

and ji provided by the lemma loc. cit. Let Tβi the Hecke operator corresponding to the

double coset Γ1,[bji ](n)βiΓ1,[bi](n). Then Tβi sends an element from Hr(Γ1,[bi](n), V

a,br,s (Fp))

to Hr(Γ1,[bji ](n), V a,b

r,s (Fp)). It was proved by Shimura, see [14], that for (c1, · · · , ch) ∈⊕h


a,br,s (Fp)), the Hecke action of Tg is

Tg.(c1, · · · , ch) = (d1, · · · .dh),where dji = Tβi .ci.

Remark 2.4. — In the idyllic situation where the ideal (det(g)) is principal, then, the

Hecke operator Tg does not permute the summands in ⊕h

l=1Hr(Γ1,[bl](n), V

a,br,s (Fp)). Indeed

(det(gjigg−1i

)) = (det(g)), so ji = i in Lemma 2.3. Therefore Tg.(c1, · · · , ch) = (d1, · · · , dh)where di = Tβi .ci.


A. Mohamed 53

Remark 2.5. — Let g be as in Lemma 2.3. Let us denote the ideal (det(g)) as c. Then Tg

maps the Fp-vector spaces ⊕r

l=1Hr(Γ1,[bl](n), V

a,br,s (Fp)) to ⊕r

l=1Hr(Γ1,[c−1bl](n), V

a,br,s (Fp)). To

see this, one needs to just recall that Tg maps

⊕r


a,b

r,s (Fp)) to ⊕r

i=1 Hr(Γ1,[bji ]

(n), V a,b

r,s (Fp))

where ji is such that (det(gjigg−1i

)) is principal. In terms of ideals this means that [c−1bi] =[bji ].

Remark 2.6. — (Diamond action) There is an action of Diamond operators inducing an

action of the group (O/n)∗ on ⊕h

i=1H1(Γ1,[bi](n), V

a,br,s (Fp)). Let χ : (O/n)∗ → F

∗

p be acharacter. As a representation of the abelian group (O/n)∗, then when p (O/n)∗, the

space ⊕h

i=1H1(Γ1,[bi](n), V

a,br,s (Fp) decomposes as a direct sum of χ-eigenspaces. So by denoting

the spaces ⊕h

i=1H1(Γ1,[bi](n), V

a,br,s (Fp) as MV

a,br,s (Fp)

(n) and a χ-eigenspace as MV

a,br,s (Fp)

(n,χ),

then we haveM

Va,br,s (Fp)

(n) = ⊕χMVa,br,s (Fp)

(n,χ).

Let us turn next to the definition of the Hecke algebra.

2.2.4. Hecke algebra. — We start by defining first what we call mod p cohomological modularforms over F. Recall that we have denoted pO as p and we are assuming that p is inert in F.

The residue field is then Fp2 . The congruence subgroups Γ1,[bi](n) act on Va,br,s (Fp) via reduction

modulo p.

Definition 2.7. — A cohomological mod p modular form of weight V a,br,s (Fp) and level n over

F is a class in⊕h

i=1H1(Γ1,[bi](n), V

a,b

r,s (Fp)).

We have denoted this Fp-vector spaces as MV

a,br,s (Fp)

(n).

We next define the Hecke algebra of interest for our purposes.

Definition 2.8 (Hecke algebra). — 1. The abstract Hecke algebra H is the polynomialalgebra Z[Tq, Sq| q pnmaximal ideal ⊂ O].

2. The Hecke algebra H(MV

a,br,s (Fp)

(n)) acting on MV

a,br,s (Fp)

(n) is the homomorphic image

of: H → EndFp(M

Va,br,s (Fp)

(n)));Tq, Sq → Tq, Sq.

As we said an eigenform for all the Hecke operator Tq for q away from pn gives rise to a systemof Hecke eigenvalues. Here is a formal definition of a system of Hecke eigenvalues with valuesin Fp.

Definition 2.9. — A system of Hecke eigenvalues with values in Fp is a ring homomorphismψ : H → Fp. We say that it occurs in M

Va,br,s (Fp)

(n) if there is a non-zero f ∈ MV

a,br,s (Fp)

(n)

such that Tf = ψ(T )f for all T ∈ H.

In the next section we shall relate the induced modules IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp) to some irreducible

Fp[GL2(O)]-modules of the form Va,br,s (Fp).



3. The relevant induced modules

We recall that by assumption we have fixed a rational inert prime p and p = pO does notdivide an integral ideal n which was also fixed. Here we will be concerned with the induced

modules IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp). We shall derive a more explicit decomposition of the latter. Let

G = GL2(Fp2) and S = SL2(Fp2).Define the following congruence subgroups of GL2(F ) :

Γ11,[bi]

(n) := giK1(n)g−1i

∩ SL2(F ).

Becauseti 00 1

a b

c d

t−1i 00 1

=

a tib

t−1i c d

, one obtains that

Γ11,[bi]

(n) =

a b

c d

∈ SL2(F ) : a− 1, d− 1 ∈ n; b ∈ bi, c ∈ b−1

in.

In particular with our assumptions one has that

Γ11,[b1]

(n) = Γ1(n) :=

a b

c d

∈ SL2(O) : a− 1, d− 1, c ≡ 0 (mod n)

.

Furthermore, let

Γ(n) =

a b

c d

∈ SL2(O) : a− 1, d− 1, b, c ≡ 0 (mod n)

.

Lemma 3.1. — Let S = SL2(Fp2). Then, we have an exact sequence

1 → Γ(p) ∩ Γ1(n) → Γ1(n) → S → 1

where the third arrow is reduction modulo p.

Proof. — It is clear that Γ(p)∩ Γ1(n) is the kernel of the reduction modulo p of Γ1(n). So, weare left to see the surjectivity of the third arrow. To this end let a, b, c, d ∈ O with ad− bc ≡ 1(mod p). We need to find α,β, γ, δ ∈ O such that αδ − βγ = 1 with the congruences:

α ≡ a (mod p)

α ≡ 1 (mod n)

β ≡ b (mod p)

γ ≡ c (mod p)

γ ≡ 0 (mod n)

δ ≡ d (mod p)

δ ≡ 1 (mod n).

It is readily seen that if 0 = c ∈ n and is coprime with p then the Chinese Remainder Theorempermits to conclude. Indeed, set γ = c, there exist α, δ ∈ O with α ≡ a (mod p), α ≡ 1(mod γ), δ ≡ d (mod p), δ ≡ 1 (mod γ). This gives αδ ≡ 1 (mod γ), and so there existsβ ∈ O such that αδ − βγ = 1 and β ≡ b (mod p). So we need to see that we can alwaysreduce to this case. To this end as n is coprime with p, we can find n ∈ n, k ∈ p, and r, s ∈ Osuch that nr − ks = 1. The image of the matrix ( r 0

n n) belongs to S and can be lifted by the

previous arguments. Then ( r 0n n

)a b

c d

=

ra rb

n(c+a) n(d+b)

is a matrix in S whose bottom line

has entries in n. Then if n(c+ a) = 0 we are done, otherwise we just have to multiply fromthe right by ( 1 0

n 1 ) for the condition to hold.


A. Mohamed 55

Corollary 3.2. — For i ≥ 1, the congruence subgroup Γ11,[bi]

(n) surjects onto S via reductionmodulo p.

Proof. — From Lemma 3.1, we have that Γ1(n) surjects onto S. So let M =a b

c d

∈ S and

α β

γ δ

∈ Γ1(n) be a lift of M. For each i > 1 take λi ∈ bi such that λi ≡ 1 (mod p) (this is

possible since bi is coprime with p). Then the matrix

α λiβ

λ−1i γ δ

belongs to Γ1

1,[bi](n) and its

reduction is M.

From the fact that Γ11,[bi]

(n) ⊂ Γ1,[bi](n), we deduce that the reduction modulo p of Γ1,[bi](n)

contains S. Now suppose we are given two subgroups H1, H2 of G = GL2(Fp2) containing S

and such that their images by the determinant map are the same: det(H1) = det(H2) < F∗

p2.

The fact det(H1 ∩H2) = det(H1) ∩ det(H2) implies the following commutative diagram withexact rows:

1 −−−−→ S −−−−→ H1 ∩H2det−−−−→ det(H1) −−−−→ 1

1 −−−−→ S −−−−→ H1det−−−−→ det(H1) −−−−→ 1.

Therefore one has H1 = H2, and we have established that any subgroup H of G containing S

is uniquely determined by the image of the determinant map Hdet−−→ F

∗

p2. From this fact we

derive that Γ1,[bi](n) reduces to

T1(n) :=g ∈ G : det(g) ∈ Im(O∗ reduction−−−−−−→ F

∗

p2).

We also derive that Γ1,[bi](pn) reduces to

T1(pn) :=

a b0 1

∈ G : a ∈ Im(O∗ reduction−−−−−−→ F

∗

p2).

In summary, reduction mod p gives the following bijection:

Γ1,[bi](pn)\Γ1,[bi](n) → T1(pn)\T1(n).

Define U =

a b0 1

∈ G

. Next we have the following bijection

T1(pn)\T1(n) ←→ U\GT1(pn)g → Ug.

Indeed the map is surjective and two elements from T1(n) are sent to the same class modulo U

if and only they belong to the same class modulo T1(pn) because we have U ∩ T1(n) = T1(pn).Composing these two bijections, we obtain the bijection

Γ1,[bi](pn)\Γ1,[bi](n) ←→ U\G.

3.1. Induced modules. — Let H be a group and J < H a subgroup of finite index. For aleft J-module M the induced module, and a twisted induced module are defined as follows.

Definition 3.3. — 1. IndHJ(M) = f : H → M : f(gh) = gf(h) ∀ g ∈ J, h ∈ H.



2. Given a character χ : J → F∗

p, we define a twisted induced module as

IndHJ (Fχ

p ) = f : H → Fp : f(gh) = χ(g)f(h) ∀ g ∈ J, h ∈ H.

Recall how a left action of H on IndHJ(M) can be defined: for g ∈ H and f ∈ IndH

J(M) we

have (g.f)(h) := f(hg).

Let B =

a b0 e

∈ G

be the Borel subgroup of G and define the character χ of B by

χ : B → F∗

p2a b0 e

→ e.

For an integer d, we also set χd(.) = (χ(.))d. The homomorphism χ induces a group isomorphism

U\B ∼= F∗

p2.

From this isomorphism we obtain the following isomorphism of B-modules

IndBU(Fp) ∼= Ind

F∗p2

1(Fp).

The isomorphism is defined as follows:

Φ : IndF∗p2

1(Fp) → IndBU(Fp)

f →

a b0 e

→ f(e)

.

The representation IndF∗p2

1(Fp) is the regular representation of F∗

p2. This is a (p2−1)-dimensional

representation of an abelian group of order prime to p and hence it admits a decompositioninto a direct sum of one-dimensional representations of F∗

p2. By a slight abuse of notation,

the summands are the F∗

p2-modules F

χd

p , where for x ∈ F∗

p2, y ∈ Fp, we have x.y := x

dy with

0 ≤ d ≤ p2 − 2.

Proposition 3.4. — For all i, there is the following isomorphism of left Γ1,[bi](n)-modulesand left Γ1

1,[bi](n)-modules respectively:

1. IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp) ∼= ⊕p

2−2d=0 IndG

B(F

χd

p );

2. IndΓ11,[bi]

(n)

Γ11,[bi]

(pn)(Fp) ∼= ⊕p

2−2d=0 IndS

B∩S(F

χd

p ).

Proof. — Because of the bijection Γ1,[bi](pn)\Γ1,[bi](n) ←→ U\G given by reducing modulo p,

the transitivity of Ind, and the observation above, we have the following identifications of leftΓ1,[bi](n)-modules:

IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp) ∼= IndG

U(Fp)

∼= IndGB(IndB

U(Fp))

∼= IndGB(⊕p

2−2d=0 (F

χd

p ))

∼= ⊕p2−2

d=0 IndGB(F

χd

p ).


A. Mohamed 57

For the second item, one uses the bijection Γ11,[bi]

(pn)\Γ11,[bi]

(n) ←→ (U ∩ S)\S.

We shall need a more explicit version of IndGB(F

χd

p ). For 0 ≤ d ≤ p2−2, we define the following

G-module which we denote by Ud(Fp) :

Ud(Fp) = f : F2p2

→ Fp : f(xa, xb) = xdf(a, b) ∀x ∈ F

∗

p2.

Next we define the following homomorphism

ϕ : Ud(Fp) → IndGB(F

χd

p )

F → (( a bc e

) → F (c, e)).

We shall show that it is an isomorphism of G-modules. It is well defined since

ϕ(F )(( x y

0 z) ( a b

c e)) = F (zc, ze) = z

dF (c, e) = χ

d(( x y

0 z))ϕ(F )(( a b

c e)).

It is also easy to see that ϕ is an G-homomorphism. In order to conclude that ϕ is anisomorphism, one can define the inverse ψ of ϕ as follows. We first note that for c, e ∈ Fp2 notboth zero we can find a, b ∈ Fp2 such that ae− bc = 0. Hence an element (c, e) = (0, 0) gives

rise to a matrix ( a bc e

) in G. Another choice of a, b with ae − b

c = 0 amounts to multiply

( a bc e

) from the left by a matrix of the form ( 1 ∗0 1 ) which acts trivially on F

χd

p . This implies thatthe map

ψ : IndGB(F

χd

p ) → Ud(Fp)

f → ((c, e) → f(( a bc e

)),

is well defined, that is, to mean that any choice of a, b ∈ Fp2 with ae − bc = 0 will do.

Furthermore it is easy to verify that it is an G-homomorphism and it is the inverse of ϕ.

In the next remark, there is another proof of the isomorphism of G-modules : IndGB(F

χd

p ) ∼=Ud(Fp).

Remark 3.5. — We start with the identification of Fp-vector spaces

Fp[X,Y ]/(Xp2 −X,Y

p2 − Y ) ∼= f : F2

p2→ Fp

where P (X,Y ) maps to the function (a, b) → P (a, b). To see this we observe that the spaces onboth sides have dimensions p4 as Fp-vector spaces. So, we just have to prove injectivity. To thisend for any x ∈ Fp2 if the polynomial fx(Y ) = P (x, Y ) ∈ Fp[Y ] vanishes for all y ∈ Fp2 , then

this means that Y and Yp2−1−1 divide fx(Y ) for all x ∈ Fp2 . Because x is arbitrarily chosen we

deduce that Y p2 −Y divides P (X,Y ). As the role of X and Y are symmetric, one obtains that

P (X,Y ) lies in the ideal (Xp2 −X,Y

p2 − Y ). In fact this is an isomorphism of Fp[G]-modules.

Let W(p,Fp) := f ∈ Fp[X,Y ]/(Xp2 − X,Y

p2 − Y ) : f((0, 0)) = 0. This module can be

identified with IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp) as Γ1,[bi](n)-module, see [16] for details. Then Ud(Fp) is the

subspace of homogeneous polynomial classes of degree d in Fp[X,Y ]/(Xp2 −X,Y

p2 − Y ) with

f((0, 0)) = 0. And as a graded Γ1,[bi](n)-module W(p,Fp) decomposes as follows:

W(p,Fp) = ⊕p2−2

d=0 Ud(Fp).



The isomorphism in Proposition 3.4 will permit us to obtain a better understanding of thenon-semisimple G-module Ud(Fp). We shall turn to this among other things.

4. Irreducible G-modules

We keep the same notation as in the previous sections. We will prove here the main results.For an irreducible Fp[G]-module W, this is done by embedding a cohomology group withcoefficients in W into another cohomology group with trivial coefficients roughly speaking.First of all, we shall see how the irreducible Fp[G]-modules can be embedded in a twist ofUd(Fp). Let τ be the non-trivial automorphism of Fp2 . For 0 ≤ r, s ≤ p− 1, 0 ≤ l, t,≤ p− 1,recall that when l and t are not both equal to p− 1, the representations

Vl,t

r,s(Fp) := Symr(F2p)⊗Fp2

detl ⊗Fp2

Syms(F2p)

τ ⊗Fp2(dett)τ ,

exhaust all the irreducible Fp[G]-modules. Here, we identify Symr(F2p) with the homogeneous

polynomials in the variables X,Y over Fp of degree r which we denote by Fp[X,Y ]r. A

matrixa b

c d

∈ G acts from the left on V

l,tr,s(Fp) as follows: on the first factor

a b

c d

.X

iY

j :=(aX + bY )i(cX + dY )j followed by multiplication by (ad − bc)l and on the second factorwe apply first τ on

a b

c d

and proceed as for the first factor followed by multiplication by

(ad− bc)pt, e.g,a b

c d

.X

iY

j ⊗XiY

j:= (ad− bc)l+pt(aX + bY )i(cX + dY )j ⊗ (apX + b

pY )i

(cpX + d

pY )j

.

For e ≥ 0, we write Ue

d(Fp) to mean the Fp[G]-module Ud(Fp) with the natural action of G

followed by multiplication by dete, e.g, Ue

d(Fp) = Ud(Fp)⊗Fp2

dete.

Lemma 4.1. — We have the following embedding of left Fp[G]-modules

Ψ : V q,t

r,s (Fp) → Uq+pt

r+ps(Fp)

f ⊗ g → ((a, b) → f(a, b)g(ap, bp)).

Proof. — In polynomial terms we can write Ψ(f(X,Y ) ⊗ g(X,Y )) = f(X,Y )g(Xp, Y

p).By definition of Ψ we have Ψ(

fi ⊗ gi) =

Ψ(fi ⊗ gi). Now let M =

a b

c d

∈ G, we

need to check that Ψ(M.f ⊗ Mτ.g) = M.Ψ(f ⊗ g). For f(X,Y ) =

n+m=r

an,mXnY

m,

g(X,Y ) =

l+k=sbl,kX

lY

k, then M.f = (ad−bc)q

n+m=r

an,m(aX+bY )n(cX+dY )m and

Mτ.g = (ad− bc)pt

l+k=s

bl,k(apX + bpY )l(cpX + d

pY )k. We set α = (ad− bc)q+pt

. Henceby denoting Ψ(M.(f ⊗ g)) as (∗), we have

(∗) = α

n+m=r

l+k=s

an,m(aX + bY )n(cX + dY )mbl,k(apX

p + bpY

p)l(cpXp + dpY

p)k

= α

n+m=r

l+k=s

an,m(aX + bY )n(cX + dY )mbl,k(aX + bY )pl(cX + dY )pk

= M.Ψ(f ⊗ g).


A. Mohamed 59

One would then like to have a more concrete description of the cokernel of Ψ. In other words,one has to compute the Jordan-Holder series of Ue

r+ps(Fp).

Remark 4.2. — In the special case s = 0, the semi-simplification of Uer(k) for k a finite field

can be obtained by immediate generalization of the case k = Fp which is treated for instancein [16]. But as it seems that this method does not apply when s > 0, we will naturally followthe Brauer character theory approach which gives the semisimplification of Ue

d(k) in complete

generality.

For our purpose we shall next see that

(Ur+ps(Fp))ss = V

0,0r,s (Fp)⊕ V

r,s

p−r−1,p−1−s(Fp)⊕ V

0,s+1r−1,p−2−s

(Fp)⊕ Vr+1,0p−r−2,s−1(Fp).

From this we deduce the semisimplification of Uer+ps(Fp) by twisting.

4.1. The constituents of Uer+ps(Fp). — Let k be a finite field and G = GL2(k), andB its

Borel subgroup of upper triangular matrices. For a character φ of B with values in Fp, we

consider IndGB(Fφd

p ) where 0 ≤ d ≤ k − 2. The semisimplication of IndGB(Fφd

p ) is computed

in [8] via Brauer character theory. Given two homomorphisms χ1, χ2 : k∗ → Q∗(or Q

∗

p), oneobtains a character of B induced by χ1, χ2 as

a b0 e

→ χ1(a)χ2(e).

Furthermore for V = Q ( or Qp) let I(χ1,χ2) := IndGB(Vχ1,χ2) where

IndGB(Vχ1,χ2) = f : G → V : f(

a b0 e

g) = χ1(a)χ2(e)f(g) ∀

a b0 e

∈ B, g ∈ G.

This is a (q + 1)-dimensional representation of G where q = k. It is known as a principalseries representation of G. Next let E be the set of embeddings k → Fp. Then the completelist of irreducible Fp-representations of G is given by:

R−→m,

−→n = ⊗τ∈E(Sym

nτ−1(k2)))τ ⊗ (detmτ )τ ⊗ Fp;

for integers 0 ≤ mτ ≤ p− 1 and 1 ≤ nτ ≤ p associated with each τ ∈ E, and some nτ is lessthan p− 1. Here one makes the convention Sym−1(k2) = 0, the null module. Before we gofurther, note that in our notation we have

Vl,t

r,s(Fp) = R(l,t),(r+1,s+1)

as irreducible G-modules.To obtain the semisimplification of an Fp-representation of G, the approach is via Brauercharacter theory. One starts with a Qp-representation W of G, and reduction modulo the

maximal ideal of Zp yields an Fp-representation of G. More precisely for such a W, we knowthat there exists a Zp-lattice L inside W invariant under the action of G. Then reduction of Lmodulo the maximal ideal of Zp gives rise to an Fp-representation whose Brauer character is therestriction of the character of W to the p-regular classes of G. In this way the semisimplificationthus obtained is independent of the lattice L.

Any group homomorphism ϕ : k∗ → Q∗

p can be written as ϕ =

ττaτ with 0 ≤ aτ ≤ p− 1

and τ the Teichmuller lift of τ. Then the reduction of ϕ is ϕ =

ττaτ . By a twist it suffices

to consider the irreducible representation of the form I(1,χ). Then it was shown in [8] that



Proposition 4.3 (Diamond). — Let M = I(1,

ττaτ ) with 0 ≤ aτ ≤ p−1 for each τ ∈ E.

Let Frob be the Frobenius in E. Then the semisimplification of the reduction M of M isM ∼= ⊕J⊂SMJ with MJ = R−→

mJ ,−→n J

, where

mJ,τ =

0 if τ ∈ J

aτ + δJ(τ) otherwise;nJ,τ =

aτ + δJ(τ) if τ ∈ J

p− aτ − δJ(τ) otherwise;

with δJ the characteristic function of J(p) = τ Frob : τ ∈ J.

We shall next specialize to our setting. We take G = G and B = B. Let ψ : Fp2 → Fp be a

fixed injection. We also denote by ψ the character obtained by restriction to F∗

p2. Now, let

ψ : F∗

p2→ Q

∗

p be the Teichmuller lift of ψ. The homomorphism ψ induces the character of B :

B → Q∗

p; ( x y

0 z) → ψ(z).

Via ψ, Qp is endowed with a structure of B-module which we denote Qψ

p : for h ∈ B, x ∈ Qp

we have h.x = ψ(h)x. The Qp-representation

I(1,ψ) = IndGB(Q

ψ

p ) = f : G → Qp : f(hg) = ψ(h)f(g) ∀h ∈ B, g ∈ G

of G has reduction the Fp-representation IndGB(F

ψ

p ) where by abuse of notation ψ is the

character of B induced by ψ :B → F

∗

p; ( x y

0 z) → ψ(z).

For 0 ≤ d ≤ p2 − 2, we consider the Fp-representations of G : IndG

B(F

ψd

p ). We write d = r+ ps

with 0 ≤ r, s ≤ p− 1, E = id, τ so that ψd= id

rτs. From Proposition 4.3, we have

(Ur+ps(Fp))ss = V

0,0r,s (Fp)⊕ V

r,s

p−r−1,p−1−s(Fp)⊕ V

0,s+1r−1,p−2−s

(Fp)⊕ Vr+1,0p−r−2,s−1(Fp);

where we have used the identification of Ud(Fp) with IndGB(F

ψd

p ) as Fp[G]-modules from page 57.

We define the representation Wl,tr,s by the exact sequence

0 → Vl,t

r,s(Fp) → Ul+pt

r+ps(Fp) → W

l,t

r,s → 0.

Thus the semisimplification of W l,tr,s is

(W l,t

r,s)ss = V

r+l,s+t

p−r−1,p−s−1(Fp)⊕ Vl,s+1+t

r−1,p−s−2(Fp)⊕ Vr+l+1,tp−r−2,s−1(Fp).

4.2. Some invariants. — Let Γ1,[bi](n) be the congruence subgroups of G(F ) defined in

Section 2. We view Fp as a trivial left G-module. We need to remind us once more how Heckeoperators act on the degree zero group cohomology. Let g ∈ ∆q

1(n) where ∆q1(n) is the subset of

Mat2(O) =0 defined in Section 2. From Lemma 2.3, we have that for each i, 1 ≤ i ≤ h, there area unique index ji and a matrix βi ∈ Λc

1,[bi](n) such that K1(n)gK1(n) = K1(n)gjiβig

−1i

K1(n)with gi the matrices corresponding to the ideal classes as defined in Subsection 2.2.Let M be a finite dimensional left Fp[G]-module. We have seen that the Hecke opera-tor corresponding to the double coset K1(n)gK1(n) which we have denoted as Tg sends(m1, · · · ,mh) ∈ ⊕h

i=1H0(Γ1,[bi](n),M) to (n1, · · · , nh) with nji = Tβi .mi. Here Tβi is the


A. Mohamed 61

Hecke operator corresponding to the double coset Γ1,[bji ](n)βiΓ1,[bi](n). Explicitly one defines

Γ,βi

1,[bji ](n) = βiΓ1,[bi](n)β

−1i

∩ Γ1,[bji ](n), then we have

Tβi : H0(Γ1,[bi](n),M) → H0(Γ1,[bji ]

(n),M)

m →

λ∈Γ1,[bji](n)/Γ

,βi1,[bji

](n)

(λβi).m.

This being said here are the Γ11,[bi]

(n) and Γ1,[bi](n)-invariants for Ue

d(Fp), V

l,tr,s(Fp), (W

l,tr,s)ss

and Wl,tr,s.

Lemma 4.4. — Let d and n be integers greater than or equal to zero. Then one has

1. for all n ≥ 0, one has

⊕h

i=1H0(Γ1

1,[bi](n),Un

d(Fp)) =

⊕h

i=1Fp if d ≡ 0 (mod p2 − 1)

0 otherwise

as Fp-vector spaces.

2. ⊕h

i=1H0(Γ1,[bi](n),U

n

d(Fp)) =

⊕h

i=1Fp if d ≡ 0 (mod p2 − 1) and (O∗)n = 1

0 otherwise


3. the Hecke operator Tg acts on (m1, · · · ,mh) ∈ ⊕h

i=1H0(Γ1,[bi](n),U

n

d(Fp)) = ⊕h

i=1Fp,

where d ≡ 0 (mod p2 − 1) and (O∗)n = 1, by sending mi to nji with

nji = [Γ1,[bji ](n) : βiΓ1,[bi](n)β

−1i

∩ Γ1,[bji ](n)]mi.

Proof. — As for the first item, because Γ11,[bi]

(n) reduces modulo p to S and since we can

identify Un

d(Fp) with Ud(Fp) as S-module, we see that the invariants do not depend on the

values of n. Having this, it is suitable to view Ud(Fp) as the set of Fp-valued functions onF2p2

and homogeneous of degree d. Observe first that a non-null constant function belongs

to (Ud(Fp))Γ11,[bi]

(n)if and only if d ≡ 0 (mod p

2 − 1). Any nonzero f ∈ (Ud(Fp))Γ11,[bi]

(n)

is a constant function. Indeed let (0, 0) = (a, b), (a, b) ∈ F2p2

and suppose that f(a, b) =

x = f(a, b) = y. Now since (a, b), (a, b) = (0, 0) there are c, e, c, d

∈ Fp2 such that

( a bc e

) ,ab

cd∈ S. Then (a, b) = (a, b)

e −b

−c a

ab

cd. Therefore Γ1

1,[bi](n) acts transitively

on F2p2

− (0, 0). Indeed from Lemma 3.1, reduction modulo p is a surjective homomorphism

Γ11,[bi]

(n) S. So we have y = f(a, b) = f((a, b)γ) = γf((a, b)) = f(a, b) = x, contradicting

the hypothesis x = y. Hence f ∈ (Ud(Fp))Γ11,[bi]

(n)if and only if f is constant.

For the second item one firstly observes that a non-null constant function belongs to(Un

d(Fp))

Γ1,[bi](n) if and only if d ≡ 0 (mod p

2 − 1) and (O∗)n = 1. From here the sameproof as the one given for the first item applies.For the third item, let fx ∈ (Un

d(Fp))

Γ1,[bi](n) with f(a, b) = x for all (a, b) ∈ F

2p2

− (0, 0) and

let given Γ1,[bji ](n)βiΓ1,[bi](n) = kδkΓ1,[bi](n). Then

Tβi .fx(a, b) =

δkβi.fx(a, b) = [Γ1,[bji ](n) : βiΓ1,[bi](n)β

−1i

∩ Γ1,[bji ](n)]x.



From this what we have claimed follows.

We also have the following

Lemma 4.5. — Let 0 ≤ r, s ≤ p − 1, and let l, t be integers greater than or equal to zero.Then one has

1. ⊕h

1=iH0(Γ1

1,[bi](n), V l,t

r,s(Fp)) =

⊕h

1=iFp if r = s = 0 and for all l, t

0 otherwise


2. ⊕h

1=iH0(Γ1,[bi](n), V

l,tr,s(Fp)) =

⊕h

1=iFp if r = s = 0 and (O∗)l+pt = 10 otherwise


3. the Hecke operator Tg acts on (m1, · · · ,mh) from ⊕h

i=1H0(Γ1,[bi](n), V

l,t

0,0(Fp)) which

is equal to ⊕h

i=1Fp when (O∗)l+pt = 1, by sending mi to nji with nji = [Γ1,[bji ](n) :

βiΓ1,[bi](n)β−1i

∩ Γ1,[bji ](n)]mi.

Proof. — Firstly when r = s = 0, and for all l, t then Vl,tr,s(Fp) = Fp as S-module. By definition

we have FΓ11,[bi]

(n)p = Fp. Otherwise use the fact that V l,t

r,s(Fp) is irreducible as Γ11,[bi]

(n)-module.The second item is proved similarly. The statement about the Hecke action is verified similarlyas in the proof of Lemma 4.4.

Lemma 4.6. — Let 0 ≤ r, s ≤ p− 1, e := l + tp, e1 := e+ p(p− 1), e2 := e+ p− 1 ≥ 0 andlet f be the order of O∗

. The following isomorphisms of Fp-vectors spaces hold:

1. ⊕h

1=iH0(Γ1

1,[bi](n), (Wr,s)

ss) =

⊕h

1=iFp if

r = s = p− 1 or

r = 1, s = p− 2 or

r = p− 2, s = 10 otherwise

2. suppose that (r = 1 or s = p− 2) and (r = p− 2 or s = 1); then we have

⊕h

1=iH0(Γ1

1,[bi](n),Wr,s) =

⊕h

1=iFp if r = s = p− 10 otherwise

3. ⊕h

1=iH0(Γ1,[bi](n), (W

l,tr,s)ss) =

⊕h

1=iFp if

r = s = p− 1 and f | e orr = 1, s = p− 2 and f | e1 orr = p− 2, s = 1 and f | e2

0 otherwise.

4. suppose that (r = 1 or s = p− 2 or f e1) and (r = p− 2 or s = 1 or f e2); then wehave

⊕h

1=iH0(Γ1,[bi](n),W

l,t

r,s) =

⊕h

1=iFp if r = s = p− 1 and f | e0 otherwise

Lastly, the Hecke action on these spaces is as in the previous lemmas.


A. Mohamed 63

Proof. — We have (Wr,s)ss = V

r,s

p−r−1,p−s−1(Fp)⊕ V0,s+1r−1,p−s−2(Fp)⊕ V

r+1,0p−r−2,s−1(Fp). From the

above lemma we know that H0(Γ11,[bi]

(n), V r,s

p−r−1,p−s−1(Fp)) is non zero only when r = s = p−1.

Indeed Vr,s

p−1−r,p−s−1(Fp) ∼= Fp as S-modules if and only if r = p− 1, s = p− 1. In this case we

have (Wr,s)ss = V

r,s

p−r−1,p−s−1(Fp) ∼= Fp. Therefore we obtain that

H0(Γ11,[bi]

(n), (Wr,s)ss) = Fp.

From the same lemma V0,s+1r−1,p−s−2(Fp) has non zero invariants only when r = 1, s = p− 2. In

this case we have

(W1,p−2)ss = V

1,p−2p−2,1 (Fp)⊕ V

0,p−10,0 (Fp)⊕ V

2,0p−3,p−3(Fp).

From this one hasH0(Γ1

1,[bi](n), (W1,p−2)

ss) = Fp.

From the same lemma the invariants of V r+1,0p−r−2,s−1(Fp) are non zero if and only if r = p−2, s =

1. Similarly we obtain that

H0(Γ11,[bi]

(n), (Wp−2,1)ss) = Fp.

As for the second item, when r = s = p− 1, then (Wr,s)ss = Wr,s. Otherwise, from the fact

that H0(Γ11,[bi]

(n), (Wr,s)ss) = 0, one deduces that H0(Γ1

1,[bi](n),Wr,s).

The remaining items are proved in a similar fashion.

In the cases (r = 1, s = p− 2 and f | e1) or (r = p− 2, s = 1 and f | e2), further analysis isneeded.We shall next discuss the case r = 1, s = p − 2 and f | e1 in detail as it is symmetric to

the remaining one. We suppose in addition that p > 5. So the representation Vl,t

1,p−2(Fp) has

dimension 2(p− 1) and we identify it with its image in Ul+pt

(p−1)2(Fp). Inside Ul+pt

(p−1)2(Fp) lies the

submodule M generated by the homogeneous monomials of degree (p− 1)2. The dimension of

M is (p− 1)2 + 1 and it contains V l,t

1,p−2(Fp) as submodule. By dimensional consideration (it is here that we need to have p > 5 to avoid discussing many cases), one deduces an exactsequence of Fp[G]-modules

0 → Vl,t

1,p−2(Fp) → M → V2+l,t

p−3,p−3(Fp) → 0.

Indeed from

(W l,t

1,p−2)ss = V

l,p−1+t

0,0 (Fp)⊕ V2+l,t

p−3,p−3(Fp)⊕ V1+l,p−2+t

p−2,1 (Fp),

we know that the constituents of any submodule of W l,t

1,p−2 are among the representations

Vl,p−10,0 (Fp), V

2+l,t

p−3,p−3(Fp) and V1+l,p−2+t

p−2,1 (Fp). From the equality (p−1)2+1 = (p−2)2+2(p−1),it follows that

M/Vl,t

1,p−2(Fp) ∼= V2+l,t

p−3,p−3(Fp).

Therefore V2+l,t

p−3,p−3(Fp) is a submodule of Ul+pt

(p−1)2/Vl,t

1,p−2(Fp) = (W l,t

1,p−2)ss.

Next we can realize the module Vl,p−1+t

0,0 (Fp) as submodule of (W l,t

1,p−2)ss by sending 1 to the



class Xp(p−1)Y

p(p−1) + Vl,t

1,p−2(Fp). To see this we define

ϕ : V l,p−1+t

0,0 (Fp) → Ul+pt

(p−1)2(Fp)/Vl,t

1,p−2(Fp)

1 → Xp(p−1)

Yp(p−1) + V

l,t

1,p−2(Fp).

Then for g = ( a bc e

) ∈ G, we need to check that ϕ(1) = g.ϕ(1). We have

g.Xp(p−1)

Yp(p−1) = (ae− bc)l+pt(apXp + b

pY

p)p−1(cpXp + epY

p)p−1.

The latter polynomial is a linear combination of the monomials X2p2−2p−iY

i with p|i. For allmultiples i of p less or equal to 2p(p− 1) except p(p− 1) the monomials X2p2−2p−i

Yi belong

to Vl,t

1,p−2(Fp). Indeed let i = pk, we recall the relations Xp2= X,Y

p2= Y in Ul+pt

(p−1)2(Fp),

then we have

X2p2−2p−pk

Ypk = X

p2−2p−pk

Xp2Y

pk = X(p−1)2−pk

Ypk ∈ V

l,t

1,p−2(Fp).

Therefore g.ϕ(1) ≡ ϕ(1) (mod Vl,tr,s(Fp)). This implies that the direct sum V

l,p−10,0 (Fp) ⊕

V1+l,t

p−3,p−3(Fp) is a submodule of W l,t

1,p−2. Thus we get an exact sequence

0 → Vl,p−1+t

0,0 (Fp)⊕ V2+l,t

p−3,p−3(Fp) → (W l,t

1,p−2) → V1+l,p−2+t

p−2,1 (Fp) → 0.

Hence for (r = 1, s = p− 2 and l + pt ≡ p− 1 (mod p2 − 1)), we obtain that

H0(Γ1,[bi](n),Wl,t

r,s) = Fp.

For (r = p− 2, s = 1) and l + pt ≡ 1− p (mod p2 − 1), similar arguments yield

H0(Γ1,[bi](n),Wl,t

r,s) = Fp.

In summary we have the following

Lemma 4.7. — Let p > 5 and e := l + pt. Then we have

1. H0(Γ11,[bi]

(n),Wr,s) = Fp if

r = 1, s = p− 2 or

r = p− 2, s = 1

2. H0(Γ1,[bi](n),Wl,tr,s) = Fp if

r = 1, s = p− 2 and f | p(p− 1) + e or

r = p− 2, s = 1 and f | e+ p− 1.

4.2.1. Some indexes. — Let t be a finite place coprime with pn. The matrix g ∈ Mat2(O)which has at the t-th place the matrix

πt 00 1

where πt is a uniformizer of Ot and in all the

remaining places has the identity matrix, belongs to ∆t1(n). We shall fix in this subsection

such a g. Because π−1t 00 1

a b

c d

πt 00 1

=

a π

−1t b

πtc d

;

we deduce that

K

1,g−1(n) := g−1

K1(n)g ∩K1(n) =

a b

c d

∈ K1(n) : πt | ct

.

Consider also the subgroup

K11 (n) = α ∈ K1(n) : det(α) = 1.


A. Mohamed 65

Similarly as in Lemma 3.1, one can prove that reduction modulo t provides us with a surjectivehomomorphism

K11 (n) SL2(O/t).

From this we deduce that we have a surjective map

K1(n) P1(O/t)

a b

c d

→ (c : d).

This map is surjective because there is a surjective map SL2(O/t) P1(O/t);

a b

c d

→ (c : d)

and also a surjective map K1(n) ⊃ K11(n) SL2(O/t). Since the subgroup K

1,g−1(n) is the

subset of all elements that are mapped to (0 : 1), we deduce that we have a bijection

K

1,g−1(n)\K1(n) ←→ P1(O/t).

Therefore we obtain the index [K1(n) : K

1,g−1(n)] = N(t) + 1. Recall the definition

Γ,β

−1i

1,[bi](n) := Γ1,[bi](n) ∩ β

−1i

Γ1,[t−1bi](n)βi. Similarly as YK1(n) decomposes into disjoint union

of its connected component h

i=1Γ1,[bi](n)\H3, YK1,g−1 (n)

decomposes as follows. We have

YK1,g−1 (n)

= h

i=1Γ,β

−1i

1,[bi](n)\H3. Indeed we know that the connected components of YK

1,g−1 (n)

are Γ

1,[bi](n)\H3 where Γ

1,[bi](n) = giK

1,g−1(n)g−1i

∩ G(F ) = gig−1

K1(n)gg−1i

∩ Γ1,[bi](n).

Recall that βi = gjigg−1i

ki =yi 00 1

∈ G(F ) with ki =

ui 00 1

∈ giK1(n)g

−1i

. For

σ ∈ Γ1,[bji ](n) = gjiK1(n)g

−1ji

∩G(F ) we have

β−1i

σβi ∈ β−1i

gjiK1(n)g−1ji

βi ∩G(F )

= k−1i

gig−1

g−1ji

gjiK1(n)gjig−1ji

gg−1i

ki ∩G(F )

= gig−1

K1(n)gg−1i

∩G(F )

where we have used the facts that gi, g, ki commute and ki, k−1i

∈ giK1(n)g−1i

. This meansthat β−1

iΓ1,[bji ]

(n)βi = Γ

1,[bi](n). Therefore we deduce that

Γ,β

−1i

1,[bi](n) = Γ1,[bi](n) ∩ β

−1i

Γ1,[bji ](n)βi = Γ

1,[bi](n).

So we have the following projection map

YK1,g−1 (n)

= h

i=1Γ,β

−1i

1,[bi](n)\H3

sg

YK1(n) = h

i=1Γ1,[bi](n)\H3.

The map sg is of degree [K1(n) : K

1,g−1(n)] = N(t) + 1. The maps induced by sg on the

connected components are also of degree N(t) + 1. The discussion we just made implies thatfor βi corresponding to g, that is to mean K1(n)gK1(n) = K1(n)g

−1ji

βigiK1(n) where ji is the

unique index such that the ideal (det(gjigg−1i

)) is principal, the following holds.



Lemma 4.8. — Keeping the same assumptions as above, then for any ideal n coprime witht = (det(g)), we have

[Γ1,[bji ](n) : βiΓ1,[bi](n)β

−1i

∩ Γ1,[bji ](n)] = N(t) + 1.

Therefore, the Hecke eigenvalue corresponding to the action of Tt on the Fp-vector space

H0(Γ1,[bi](n), Vl,tr,s(Fp)) where t is a prime ideal coprime with pn is N(t) + 1. Hence eigenvalue

systems coming from the Fp-vector space ⊕h

i=1H0(Γ1,[bi](n), V

l,tr,s(Fp)) are Eisenstein because

the semisimplification of the conjecturally attached Galois representations is the direct sum ofthe cyclotomic character and the trivial character.As we shall make use of Shapiro’s isomorphism, we need to verify that it is compatible withthe Hecke action on group cohomology. From the above discussion, we deduce that we canchoose identical coset representatives for the double cosets

Γ1,[t−1bi](pn)βiΓ1,[bi](pn)/Γ1,[bi](pn) and for Γ1,[t−1bi](n)βiΓ1,[bi](n)/Γ1,[bi](n).

This will be used for the compatibility of the Hecke action with the Shapiro’s isomorphism.

4.2.2. Compatibility of Shapiro’s lemma with the Hecke action. — Recall that when Γ< Γ

are congruence subgroups and M is a Γ-module then Shapiro’s isomorphism reads as

H∗(Γ, IndΓΓ(M)) ∼= H∗(Γ,M).

It is the isomorphism induced by the restriction j : Γ→ Γ and the homomorphism

φ : IndΓΓ(M) → M

f → f(1).

Therefore in terms of cocycles we have

Sh : H∗(Γ, IndΓΓ(M)) → H∗(Γ,M)

c → φ c j.From Subsection 2.2.1, we know that for g ∈ ∆a

1(pn) with a coprime with pn, any set of orbit rep-resentatives of the orbit space K1(pn)gK1(pn)/K1(pn) belongs to ∆a

1(pn) where a = (det(g))O.

In a similar fashion any set of orbit representatives of K1(n)gK1(n)/K1(n) belongs to ∆a1(n).

We also obtain that any set of representatives of the orbit space Γ1,[bji ](n)βiΓ1,[bi](n)/Γ1,[bi](n)

belong to Λc1,[bi]

(n) when βi is from Λc1,[bi]

(n). For the forthcoming statement we need to recallsome important facts. On page 56, we saw that reduction modulo p provides us the followingisomorphism of Γ1,[bi](n)-modules:

IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp) ∼= IndG

U(Fp) ∼= Ind

Γ1,[a−1bi](n)

Γ1,[a−1bi](pn)(Fp)

where U = a b0 1

∈ G ⊂ B with B the Borel subgroup of G and Fp is endowed with the

structure of a trivial left Γ1,[bi](n)-module. The left action of the latter on IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp) is

as follows: for γ ∈ Γ1,[bi](n) and f ∈ IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp) we have (γ.f)(h) := f(hγ). By definition

IndGU(Fp) := f : G → Fp : f(uh) = uf(h) = f(h), ∀u ∈ U , h ∈ G,


A. Mohamed 67

that is the collection of all the U -left invariant maps from G to Fp. Because for each i = 1, · · · , h,any element λ ∈ Λc

1,[bi](pn) has its reduction belonging to U , we derive that any f ∈ IndG

U(Fp)

satisfiesf(λ) = f(1).

Proposition 4.9. — Let a be a prime ideal coprime with pn. Let Tg = Ta be the Heckeoperator associated with g ∈ ∆a

1(n) where a = det(g)O the ideal corresponding to g. Explicitlyg is the matrix with the identity matrix in all finite places expect at the a-place where there isthe matrix

πa 00 1

. Here πa is a uniformizer of Oa. Then the following diagram

H1(Γ1,[bi](n), IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp))

Sh−−−−→ H1(Γ1,[bi](pn),Fp)Tβi

Tβi

H1(Γ1,[a−1bi](n), IndΓ1,[a−1bi]

(n)

Γ1,[a−1bi](pn)(Fp))

Sh−−−−→ H1(Γ1,[a−1bi](pn),Fp)

is well defined and is commutative.

Proof. — For the proof of this proposition, we refer to [1, p. 42].

One other important fact that tells us that we only have to look at Serre weights, that isto mean irreducible Fp[G]-modules for the analysis of Hecke eigenclasses is the followingproposition.

Proposition 4.10. — Let n be an integral ideal such that the positive generator of n ∩ Z

is greater than 3. Consider the open compact subgroup K1(n) of level n. Let M be a finitedimensional Fp[G]-module. Let Ψ be an eigenvalue system occurring in ⊕h

i=1H1(Γ1,[bi](n),M)

and taking values in Fp. Then, there exists W, an irreducible subquotient of M such that Ψalso occurs in ⊕h

i=1H1(Γ1,[bi](n),W ).

Proof. — Let W be an irreducible submodule M. Denote the quotient M/W as N. SetK = K1(n). This is an open compact subgroup of G(O) which is neat and surjects onto O∗

via the determinant. Write the following exact sequence of locally constant sheaves on YK

associated with W,M,N respectively:

0 → W → M → N → 0.

From this one obtains the exact sequence in cohomology:

· · · → H1(YK , W ) → H1(YK , M) → H1(YK , N) → · · · .Let s be a system of Hecke eigenvalues from H1(YK , M). If the image of s is zero, then s occursin H1(YK , W ), and we are done. Otherwise it is arisen from H1(YK , N). We then replace M

by N and repeat the argument.

4.2.3. Statements and proofs of the main results. — The statement about the reduction toweight two is as follows.

Theorem 4.11. — Let F be an imaginary quadratic field of class number h. Let n be anintegral ideal in F and let p > 5 be a rational prime which is inert in F and coprime withn. Suppose that the positive generator of n ∩ Z is greater than 3. Let 0 ≤ r, s ≤ p − 1 and



0 ≤ l, t ≤ p− 1, with l, t not both equal to p− 1. Let ψ be a system of Hecke eigenvalues in⊕h

i=1H1(Γ1,[bi](n), V

l,tr,s(Fp)). Then ψ occurs in ⊕h

i=1H1(Γ1,[bi](pn),Fp ⊗ det

l+pt) except possiblywhen (r = 1, s = p− 2) or (r = p− 2, s = 1). In these potential exceptions, the obstruction iscoming from Hecke eigenvalue systems which are Eisenstein.

Proof. — The proof is divided in two parts. Firstly, we show that

⊕h

i=1H1(Γ1

1,[bi](n), Vr,s(Fp)) → ⊕h

i=1H1(Γ1

1,[bi](pn),Fp)

as Fp-vector spaces except in the exceptional cases named in the statement. Secondly fromthis, we use an inflation restriction exact sequence and obtain an embedding of Hecke modules⊕h

i=1H1(Γ1,[bi](n), V

l,tr,s(Fp)) → ⊕h

i=1H1(Γ1,[bi](pn),Fp ⊗ det

l+pt).First part:

The exact sequence0 → Vr,s(Fp) → Ur+ps(Fp) → Wr,s → 0

gives rise to the long exact sequence in cohomology

0 → ⊕h

i=1H0(Γ1

1,[bi](n), Vr,s(Fp)) → ⊕h

i=1H0(Γ1

1,[bi](n),Ur+ps(Fp)) →

→ ⊕h

i=1H0(Γ1

1,[bi](n),Wr,s) → ⊕h

i=1H1(Γ1

1,[bi](n), Vr,s(Fp)) →

→ ⊕h

i=1H1(Γ1

1,[bi](n),Ur+ps(Fp)) → ⊕h

i=1H1(Γ1

1,[bi](n),Wr,s) → · · · .

This is an exact sequence of Fp-vector spaces. If r = s = p− 1, from Lemmas 4.4, 4.5 and 4.6,we get the exact sequence of Fp-vector spaces for each i = 1, · · · , h :

0 → Fp → Fp → H1(Γ11,[bi]

(n), Vr,s(Fp)) → H1(Γ11,[bi]

(n),Ur+ps(Fp)) → · · · .This means that the third arrow is the null map and hence we have an injection

⊕h

i=1H1(Γ1

1,[bi](n), Vr,s(Fp)) → ⊕h

i=1H1(Γ1

1,[bi](n),Ur+ps(Fp)).

From Lemmas 4.4, 4.5 and 4.6, we see that when (r = 1 or s = p−2) and (r = p−2 or s = 1),we have an exact sequence of Fp-vector spaces

0 → H1(Γ11,[bi]

(n), Vr,s(Fp)) → H1(Γ11,[bi]

(n),Ur+ps(Fp)) → · · · .

Therefore in all cases this is an exact sequence of Fp-vector spaces. From Proposition 3.4, we

know that the representation Ur+ps(Fp) is a direct summand of IndΓ11,[bi]

(n)

Γ11,[bi]

(pn)(Fp). So, one has

an embedding of Fp-vector spaces

⊕h

i=1H1(Γ1

1,[bi](n),Vr,s(Fp)) → ⊕h

i=1H1(Γ1

1,[bi](n), Ind

Γ1,[bi](n)

Γ11,[bi]

(pn)(Fp)).

By Shapiro’s lemma, one concludes that we have an injection of Fp-vector spaces

α : ⊕h

i=1H1(Γ1

1,[bi](n), Vr,s(Fp)) → ⊕h

i=1H1(Γ1

1,[bi](pn)),Fp).

Lastly when (r = 1, s = p− 2) or (r = p− 2, s = 1), then from Lemmas 4.4, 4.5, 4.6and 4.7, we have the exact sequence of Fp-vector spaces

0 → Fp → H1(Γ11,[bi]

(n), Vr,s(Fp)) → H1(Γ11,[bi]

(n),Ur+ps(Fp)) → · · · .


A. Mohamed 69

Second part:

Consider the inflation-restriction exact sequence

0 → H1(Γ1,[bi](n)/Γ11,[bi]

(n), (V l,t

r,s(Fp))Γ11,[bi]

(n))

infl−−→ H1(Γ1,[bi](n), Vl,t

r,s(Fp))res−−→

res−−→ H1(Γ11,[bi]

(n), V l,t

r,s(Fp))Γ1,[bi]

(n)/Γ11,[bi]

(n) → H2(Γ1,[bi](n)/Γ11,[bi]

(n), V l,t

r,s(Fp)Γ11,[bi]

(n)).

Because of the assumption concerning p we have that

H1(Γ1,[bi](n)/Γ11,[bi]

(n), (V l,t

r,s(Fp))Γ11,[bi]

(n)) = H2(Γ1,[bi](n)/Γ

11,[bi]

(n), V l,t

r,s(Fp)Γ11,[bi]

(n)) = 0.

Then we get the isomorphism of Fp-vector spaces induced by the restriction map:

H1(Γ1,[bi](n), Vl,t

r,s(Fp))∼−→ (H1(Γ1

1,[bi](n), V 0,0

r,s (Fp))⊗Fpdet

l+pt)Γ1,[bi]

(n)/Γ11,[bi]

(n)

where we have used the isomorphism

H1(Γ11,[bi]

(n), V l,t

r,s(Fp)) H1(Γ11,[bi]

(n), V 0,0r,s (Fp))⊗Fp

detl+pt

.

Next notice that for all 1 ≤ i ≤ h, we have isomorphisms of abelian groups

Γ1,[bi](n)/Γ11,[bi]

(n) ∼= O∗ ∼= Γ1,[bi](pn)/Γ11,[bi]

(pn).

From the first part, when we are in the situation (r = 1 or s = p−2) and (r = p−2 or s = 1),then there is an embedding of Fp-vector spaces:

H1(Γ11,[bi]

(n), Vr,s(Fp)) → H1(Γ11,[bi]

(pn),Fp).

When tensoring with detl+pt

, we obtain the embedding

H1(Γ11,[bi]

(n), V l,t

r,s(Fp)) → H1(Γ11,[bi]

(pn),Fp ⊗ detl+pt).

We next take O∗-invariants and we get

(H1(Γ11,[bi]

(n), V l,t

r,s(Fp)))O∗

→ (H1(Γ11,[bi]

(pn),Fp ⊗ detl+pt))O

∗.

This and the isomorphism induced by the inflation restriction exact sequence implies that


r,s(Fp)) → (H1(Γ11,[bi]

(pn),Fp ⊗ detl+pt))O

∗.

Using once more the inflation restriction exact sequence for the right hand of this embedding,we derive that


r,s(Fp)) → H1(Γ1,[bi](pn),Fp ⊗ detl+pt).

This natural map is compatible with the Hecke action, and so this is an injection of Heckemodules.Now when the cases (r = 1, s = p− 2) or (r = p− 2, s = 1) hold, then the first part providesus with an exact sequence

0 → Fp → H1(Γ11,[bi]

(n), Vr,s(Fp)) → H1(Γ11,[bi]

(pn),Fp).

This implies that the following sequences are exact:

0 → Fp ⊗ detl+pt → H1(Γ1

1,[bi](n), V l,t

r,s(Fp)) → H1(Γ11,[bi]

(pn),Fp ⊗ detl+pt),

thus

0 → (Fp ⊗ detl+pt)O

∗ → H1(Γ1,[bi](n), Vl,t

r,s(Fp)) → H1(Γ1,[bi](pn),Fp ⊗ detl+pt).



Then, when (Fp ⊗ detl+pt)O

∗= 0, the embedding


r,s(Fp)) → H1(Γ1,[bi](pn),Fp ⊗ detl+pt)

holds. Otherwise, we know that the obstruction is coming from (Fp ⊗ detl+pt)O

∗and is hence

Eisenstein as shown by Lemma 4.8.

Systems of Hecke eigenvalues arising from Fp ⊗ dete are Eisenstein and hence conjecturally

correspond to reducible Galois representations. Because of this, the statement about Serre’sconjecture is not affected since it only concerns irreducible mod p Galois representations. Nowthe statement related to Serre type questions is as follows.

Proposition 4.12. — We keep the same conditions as in Theorem 4.11. A positive answerto Question (a) on page 46 answers positively Question (b) and the reciprocal also holds.

Proof. — The part (b) ⇒ (a) is obtained as follows. By Shapiro’s lemma the system is realizedin

⊕h

i=1H1(Γ1,[bi](n), Ind

Γ1,[bi](n)

Γ1,[bi](pn)(Fp)).

By Proposition 4.10, this system of Hecke eigenvalues already appears in

⊕h

i=1H1(Γ1,[bi](n),M)

where M is a simple module from the Jordan-Holder series of IndΓ1,[bi]

(n)

Γ1,[bi](pn)(Fp). This module

M is a Serre weight.The part (a) ⇒ (b) follows from Theorem 4.11.

References

[1] A. Mohamed, Some explicit aspects of modular forms over imaginary quadratic fields, PhD Thesis,Universitat Duisburg Essen, Campus Essen, June 2011, available from http://duepublico.uni-duisburg-essen.de/servlets/DocumentServlet?id=26232&lang=en.

[2] A. Ash and G. Stevens, Modular forms in characteristic l and special values of their L-function,Duke Math. J 53, no 3 849-868.

[3] A. Ash and G. Stevens, Cohomology of arithmetic groups and congruences between systems ofHecke eigenvalues, J. Reine Angew. Math. 365 (1986), 192220.

[4] A. Ash, D. Doud, and D. Pollack, Galois representations with conjectural connections to arithmeticcohomology, Duke Mathematical Journal, Vol. 112, No. 3, 2002.

[5] A. Ash and W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Heckeeigenclasses in the mod-p cohomology of GL(n;Z), Duke Math. J. 105 (2000), 1-24.

[6] J. S. Bygott, Modular forms and modular symbols over imaginary quadratic fields, PhD thesis,University of Exeter, 1998.

[7] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, Springer-Verlag New-York,1982.

[8] F. Diamond, A correspondence between representations of local Galois groups and Lie-type groups,Proceedings of the LMS Durham Symposium on L-functions and Galois Representations, 2004.

[9] B. Edixhoven, C. Khare, Hasse invariant and group cohomology, Documenta Math 8 (2003) 43-50.


A. Mohamed 71

[10] M. Emerton, p-Adic families of modular forms, Seminaire Bourbaki, 62eme annee, 2009-2010,No. 1013, (2009).

[11] L. M. Figueiredo, Serre’s conjecture for imaginary quadratic fields, Compositio Mathematica. 118(1999), No. 1, 103-122.

[12] M. H. Sengun and S. Turkelli, Weight Reduction for mod l Bianchi Modular forms, Journal ofNumber Theory, Volume 129, Issue 8, August 2009, Pages 2010-2019.

[13] R. Taylor, On congruences between modular forms, PhD Thesis, Princeton University 1988.

[14] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, DukeMathematical Journals, Vol.45, . 3, (1978), 637-679.

[15] E. Urban, Formes automorphes cuspidales pour GL2 sur un corps quadratique imaginaire. Valeursspeciales de fonctions L et congruences, Compositio Mathematica, tome 99, No. 3 ( 1995), 283-324.

[16] G. Wiese, On the faithfulness of parabolic cohomology as a Hecke module over a finite field, J.Reine Angew. Math. ( 2007), 79-103.

14 octobre 2013

Adam Mohamed, Universitat Duisburg-Essen, Institut fur Experimentelle Mathematik, Ellernstr 29, 45326

Essen, Germany • E-mail : [email protected]


ON COMPUTING BELYI MAPS

by

J. Sijsling & J. Voight

Abstract. — We survey methods to compute three-point branched covers of the projective line,also known as Belyı maps. These methods include a direct approach, involving the solution of asystem of polynomial equations, as well as complex analytic methods, modular forms methods,and p-adic methods. Along the way, we pose several questions and provide numerous examples.

Résumé. — Nous donnons un aperçu des méthodes actuelles pour le calcul des revêtementsde la droite projective ramifiés en au plus trois points, connus sous le nom de morphismes deBelyı. Ces méthodes comprennent une approche directe, se ramenant à la solution d’un systèmed’équations polynomiales ainsi que des méthodes analytiques complexes, de formes modulaireset p-adiques. Ce faisant, nous posons quelques questions et donnons de nombreux exemples.

Contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741. Background and applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752. Gröbner techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823. Complex analytic methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904. Modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965. p-adic methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066. Galois Belyı maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107. Field of moduli and field of definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128. Simplification and verification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189. Further topics and generalizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2010 Mathematics Subject Classification. — 11G32, 11Y40.Key words and phrases. — Belyi maps, dessins d’enfants, covers, uniformization, computational

algebra.

74 On computing Belyi maps

Introduction

Every compact Riemann surface X is an algebraic curve over C, and every meromorphicfunction on X is an algebraic function. This remarkable fact, generalized in the GAGAprinciple, links the analytic with the algebraic in a fundamental way. A natural problem isthen to link this further with arithmetic; to characterize those Riemann surfaces that can bedefined by equations over Q and to study the action of the absolute Galois group Gal(Q/Q)

on these algebraic curves. To this end, Belyı [12, 13] proved that a Riemann surface X overC can be defined over Q if and only if X admits a Belyı map, a map f : X → P1

C that isunramified away from 0, 1,∞. Grothendieck, in his Esquisse d’un Programme [62], calledthis result “deep and disconcerting”.Part of Grothendieck’s fascination with Belyı’s theorem was a consequence of the simplecombinatorial and topological characterization that follows from it. Given a Belyı mapf : X → P1

C, the preimage f−1([0, 1]) of the real interval [0, 1] can be given the structure of a

dessin (or dessin d’enfant, “child’s drawing”): a connected graph with bicolored vertices (so thetwo vertices of an edge are colored differently) equipped with a cyclic ordering of the edgesaround each vertex. Conversely, a dessin determines the corresponding Belyı map uniquelyup to isomorphism over C or Q. The idea that one can understand the complicated groupGal(Q/Q) by looking at children’s pictures casts an alluring spell indeed. As a consequence,hundreds of papers have been written on the subject, several books have appeared, and thetopic remains an active area of research with many strands.In a number of these papers, computation of particular examples plays a key role in under-standing phemonena surrounding Belyı maps; arguably, part of the richness of the subject liesin the beauty in these examples. Shabat and Voevodsky [144, 0.1.1, 0.3] say on this point:

Here we have no general theory and only give a number of examples. The complete-ness of our results decrease rapidly with growing genus; we are able to give somecomplete lists (of non-trivial experimental material) for genus 0, but for generaexceeding 3 we are able to give only some general remarks. [...] The main reasonsto publish our results in the present state is our eagerness to invite our colleaguesinto the world of the divine beauty and simplicity we have been living in since wehave been guided by the Esquisse.

In spite of this important role, no survey of computational methods for Belyı maps has yetappeared, and in our own calculations we found many techniques, shortcuts, and some tricksthat others had also (re)discovered. In this article, we collect these results in one place inthe hope that it will be useful to others working in one of the many subjects that touch thetheory of Belyı maps. We also give many examples; to our knowledge, the larger examplesare new, unless mentioned otherwise. We assume that the reader has some familiarity withalgebraic curves and with computation, but not necessarily with the theory of Belyı maps ordessins; at the same time, we hope that this paper will also be a useful and comprehensivereference, so we will also make some remarks for the experts.We take as input to our methods the simple group theoretic description of Belyı maps: thereis a bijection between permutation triples

σ = (σ0,σ1,σ∞) ∈ S3d that satisfy σ0σ1σ∞ = 1

up to simultaneous conjugation in the symmetric group Sd, and


J. Sijsling and J. Voight 75

Belyı maps f : X → P1 of degree d

up to isomorphism over Q. In this bijection, the curve X can be disconnected, such as thetrivial cover of degree d > 1; the cover X is connected if and only if (the dessin is connectedif and only if) the corresponding permutation triple σ generates a transitive subgroup of Sd,in which case we call σ transitive. If σ corresponds to f in this bijection, we say that f hasmonodromy representation σ.Given the description of a Belyı map f in the compressed form of a permutation triple, it hasproven difficult in general to determine explicitly an algebraic model for f and the curve X.As a result, many authors have written on this subject of explicit computation of Belyı maps,usually subject to certain constraints or within a certain class of examples. That this is adifficult problem is a common refrain, and the following quote by Magot and Zvonkin [106,§1] is typical:

An explicit computation of a Belyi function corresponding to a given map is reducedto a solution of a system of algebraic equations. It may turn out to be extremelydifficult. To give an idea of the level of difficulty, we mention that our attempts tocompute Belyi functions for some maps with only six edges took us several months,and the result was achieved only after using some advanced Gröbner bases softwareand numerous consultations given by its author J.C. Faugère.

The paper is organized as follows. In Section 1, we collect the basic background (includinga discussion of fields of definition), and mention some applications and generalizations. InSection 2, we discuss a direct method using Gröbner methods, augmented by the Atkin–Swinnerton-Dyer trick. We then turn to other, more practical methods. We begin in Section 3with complex analytic methods; in Section 4, we consider methods using modular forms; inSection 5, we consider p-adic methods. In Section 6, we briefly discuss alternative methods forGalois Belyı maps. In Section 7, we discuss the delicate subjects of field of moduli and field ofdefinition with an eye to its implications for computation. In Section 8, we treat simplificationand verification of Belyı maps, and finally in Section 9 we conclude by considering somefurther topics and generalizations. Along the way, we give explicit examples and pose severalquestions.The authors would like to thank Noam Elkies, Ariyan Javanpeykar, Curtis McMullen, JohnMcKay, David Roberts, Steffen Rohde, Sam Schiavone, Matthias Schütt, Marco Streng, BerndSturmfels, Mark Watkins and Bruce Westbury for their comments on this work, as well as thereferee for his or her many suggestions. The first author was supported by Marie Curie grantIEF-GA-2011-299887, and the second author was supported by an NSF CAREER Award(DMS-1151047).

1. Background and applications

The subject of explicit characterization and computation of ramified covers of Riemannsurfaces is almost as old as Riemann himself. Klein [90] and Fricke–Klein [56] calculated someexplicit Belyı maps, most notably the icosahedral Galois Belyı map P1 → P1 of degree 60 [90,I, 2, §13–14]. These appeared when constructing what we would today call modular functionsassociated with the triangle groups ∆(2, 3, 5) and ∆(2, 4, 5) (see Section 4). This in turnallowed them to find a solution to the quintic equation by using analytic functions. Around



the same time, Hurwitz [77] was the first to consider ramified covers in some generality:besides considering covers of small degree, he was the first to give the classical combinatoricaldescription of covers of the projective line minus a finite number of points, which would laterresult in Hurwitz spaces being named after him.Continuing up to the modern day, the existing literature on Belyı maps with an explicit flavoris extremely rich: surveys include Birch [18], Jones–Singerman [82, 83], Schneps [138], andWolfart [172]; textbooks include the seminal conference proceedings [139], work of Malle–Matzat [107], Serre [142], and Völklein [165], mainly with an eye toward applications toinverse Galois theory, the tome on graphs on surfaces by Lando–Zvonkin [99], and the book byGirondo–Gonzalez-Diaz [58], which interweaves the subject with an introduction to Riemannsurfaces.We begin this section by reviewing basic definitions; we conclude by mentioning applica-tions and generalizations as motivation for further study. (We postpone some subtle issuesconcerning fields of moduli and fields of definition until Section 7.)

Definitions, and equivalent categories. — Let K be a field with algebraic closure K.An (algebraic) curve X over K is a smooth proper separated scheme of finite type over K thatis pure of dimension 1.We now define precisely the main category of this paper whose objects we wish to study. ABelyı map over K is a morphism f : X → P1 of curves over K that is unramified outside0, 1,∞. Given two Belyı maps f1, f2 : X1, X2 → P1, a morphism of Belyı maps from f1 tof2 is a morphism g : X1 → X2 such that f1 = f2g. We thereby obtain a category of Belyımaps over K.A curve X that admits a Belyı map is called a Belyı curve. Belyı [12, 13] proved that acurve X over C can be defined over Q if and only if X is a Belyı curve. Consequently, inwhat follows, we may pass freely between Belyı maps over Q and over C: we will simply referto both categories as the category of Belyı maps. The absolute Galois group Gal(Q/Q) actsnaturally on (the objects and morphisms in) the category of Belyı maps (over Q); this action isfaithful, as one can see by considering the j-invariant of elliptic curves. We denote the actionby a superscript on the right, so the conjugate of a curve X over Q by an automorphismτ ∈ Gal(Q/Q) is denoted by Xτ , and that of a Belyı map f by f τ .Let f : X → P1 be a Belyı map of degree d. The ramification of f above 0, 1,∞ is recordedin its ramification type, the triple consisting of the set of ramification multiplicities above0, 1,∞, respectively. Such a ramification type is therefore given by a triple of partitions of d,or alternatively by a triple of conjugacy classes in the symmetric group Sd.Part of the beauty of subject of Belyı maps is the ability to pass seamlessly between combi-natorics, group theory, algebraic geometry, topology, and complex analysis: indeed, one candefine categories in these domains that are all equivalent. In the remainder of this subsection,we make these categories and equivalences precise; the main result is Proposition 1.2 below.To begin, we record the ramification data, or more precisely the monodromy. A permutation

triple of degree d is a triple σ = (σ0,σ1,σ∞) ∈ S3d such that σ0σ1σ∞ = 1. Let σ

= (σ0,σ

1,σ

∞)

be another such triple of degree d. Then a morphism of permutation triples from σ to σ isa map t : 1, . . . , d → 1, . . . , d such that t(σ0(x)) = σ

0(t(x)) for all x ∈ S and the samefor σ1,σ∞. In particular, two permutation triples σ,σ are isomorphic, and we write σ ∼ σ

and say they are simultaneously conjugate, if and only if they have the same degree d = d and



there exists a τ ∈ Sd such that

στ= τ−1

(σ0,σ1,σ∞)τ = (τ−1σ0τ, τ−1σ1τ, τ

−1σ∞τ) = (σ0,σ

1,σ

∞).

It is a consequence of the Riemann existence theorem that the category of Belyı maps isequivalent to the category of permutation triples. More precisely, let

(1.1) F2 = x, y, z | xyz = 1

be the free group on two generators. Given a group G, a finite G-set is a homomorphismα : G → Sym(S) on a finite set S, and a morphism between finite G-sets from α to α is a mapof sets t : S → S such that α

(g)(t(x)) = t(α(g)(x)) for all g ∈ G and x ∈ S. We see thatgiving a permutation triple is the same as giving a finite F2-set, by mapping x, y, z ∈ F2 toσ0,σ1,σ∞, and that two permutation triples are isomorphic if and only if the correspondingF2-sets are isomorphic.Returning to covers and topological considerations, we have an isomorphism

F2∼= π1(P1

\ 0, 1,∞);

the generators x, y, z chosen above can be taken to be simple counterclockwise loops around0, 1,∞. We abbreviate P1

∗ = P1 \ 0, 1,∞. The category of finite topological covers of P1∗

is equivalent to the category of finite π1(P1∗)-sets; to a cover, we associate one of its fibers,

provided with the structure of π1(P1∗)-set defined by path lifting. Therefore, a Belyı map gives

rise to a cover of P1∗ by restriction, and conversely a finite topological cover of P1

∗ can be giventhe structure of Riemann surface by lifting the complex analytic structure and thereby yieldsa map from an algebraic curve to P1 unramified away from 0, 1,∞.Let f be a Belyı map, corresponding to a permutation triple σ. The corresponding F2-setρ : F2 → Sd is called the monodromy representation of f , and its image is called the monodromy

group of f . The monodromy group, as a subgroup of Sd, is well-defined up to conjugacy andin particular up to isomorphism, and we denote it by G = Mon(f). By the correspondencesabove, the automorphism group of a Belyı map is the centralizer of its monodromy group (asa subgroup of Sd).We consider a final category, introduced by Grothendieck [62]. A dessin D is a triple (Γ, C,O)

where:(D1) Γ is a finite graph with vertex set V , edge set E, and vertex map v : E → V × V ;(D2) C : V → 0, 1 is a bicoloring of the vertices such that the two vertices of an edge are

colored differently, i.e., C(v(e)) = 0, 1 (and not a proper subset) for all edges e ∈ E; and(D3) O is a cyclic orientation of the edges around every vertex.Due to the presence of the bicoloring C, the cyclic orientation in (D3) is specified by twopermutations O0, O1 ∈ Sym(E) specifying the orderings around the edges marked with 0 and1 respectively. Note that once the bicoloring C is given, the possible orientations O = (O0, O1)

can be chosen to be any pair of permutations with the property that two edges e, e are inthe same orbit under O0 (resp. O1) if and only if the corresponding vertices marked 0 (resp.1) coincide. A morphism of dessins is a morphism of graphs ϕ : Γ → Γ

such that ϕ takes thebicoloring C to C (i.e., C

(ϕ(v)) = C(v)) and similarly the cyclic orientation O to O.The category of dessins is also equivalent to that of Belyı maps. Indeed, associated to a Belyımap f is the graph given by f−1

([0, 1]), with the bicoloring on the vertices given by f andwith the cyclic ordering induced by the orientation on the Riemann surface. Conversely, given



a dessin we can algebraize the topological covering induced by sewing on 2-cells as specifiedby the ordering O.Dessins were introduced by Grothendieck [62] to study the action of Gal(Q/Q) on Belyı mapsthrough combinatorics. So far, progress has been slow, but we mention one charming result[138]; the Galois action is already faithful on the dessins that are trees (as graphs).We summarize the equivalences obtained in the following proposition and refer to Lenstra[101] for further exposition and references.

Proposition 1.2. — The following categories are equivalent:

(i) Belyı maps;

(ii) permutation triples;

(iii) finite F2-sets; and

(iv) dessins.

In particular, the equivalence in Proposition 1.2 yields the key bijection considered in thispaper:

(1.3)

permutation triples σ = (σ0,σ1,σ∞) ∈ S3

d

/ ∼

1:1Belyı maps f : X → P1 of degree d

/ ∼=Q

where the notions of isomorphism are taken in the appropriate categories. Concretely, underthe correspondence (1.3), the cycles of the permutation σ0 (resp. σ1,σ∞) correspond to thepoints of X above 0 (resp. 1,∞) and the length of the cycle corresponds to the ramificationindex of the corresponding point under the morphism f . Note in particular that becausethe first set of equivalence classes in (1.3) is evidently finite, there are only finitely manyQ-isomorphism classes of curves X with a Belyı map of given degree.It is often useful, and certainly more intuitive, to consider the subcategories in Proposition 1.2that correspond to Belyı maps f : X → P1 whose source is connected (and accordingly, we saythe map is connected). A Belyı map is connected if and only if the corresponding permutationtriple σ is transitive, i.e., the subgroup σ0,σ1,σ∞ is a transitive group. Restricting totransitive permutations gives a further equivalent category of finite index subgroups of F2:the objects are subgroups H ≤ F2 of finite index and morphisms H → H are restrictions ofinner automorphisms of F2 that map H to H . The category of finite index subgroups of F2

is equivalent to that of finite transitive F2-sets (to a subgroup H of F2, one associates theF2-set F2/H). Proposition 1.2 now becomes the following.


(i) connected Belyı maps;

(ii) transitive permutation triples;

(iii) transitive finite F2-sets;

(iii) subgroups of F2 of finite index; and

(iv) dessins whose underlying graph is connected.



Unless stated otherwise (e.g., Section 7), in the rest of this article we will assume without fur-ther mention that a Belyı map is connected ; this is no loss of generality, since any disconnectedBelyı map is the disjoint union of its connected components.

Geometric properties and invariants. — Let f : X → P1 be a (connected) Belyı mapover Q. If the cover f is Galois, which is to say that the corresponding extension of functionfields Q(X)/Q(P1

) is Galois, then we call f a Galois Belyı map. More geometrically, thisproperty boils down to the demand that a subgroup of Aut(X) act transitively on the sheetsof the cover; and combinatorially, this is nothing but saying that Mon(f) ⊆ Sd has cardinality#Mon(f) = d. Indeed, the monodromy group of a Belyı map can also be characterized as theGalois group of its Galois closure, which is the smallest Galois cover of which it is a quotient.The genus of X can be calculated by using the Riemann–Hurwitz formula. If we define theexcess e(τ) of a cycle τ ∈ Sd to be its length minus one, and the excess e(σ) of a permutationto be the sum of the excesses of its constituent disjoint cycles (also known as the index of thepermutation, equal to n minus the number of orbits), then the genus of a Belyı map of degreen with monodromy σ is

(1.5) g = 1− n+e(σ0) + e(σ1) + e(σ∞)

2.

In particular, we see that the genus of Belyı map is zero if and only if e(σ0)+e(σ1)+e(σ∞) =

2n− 2.We employ exponential notation to specify both ramification types and conjugacy classes inSd. So for example, if d = 10, then 3

22112 denotes both the conjugacy class of the permutation

(1 2 3)(4 5)(6 7 8) and the corresponding ramification type; two points of ramification index3, one of index 2, and two (unramified) of index 1.The passport of a Belyı map f : X → P1 is the triple (g,G,C) where g is the genus of X andG ⊆ Sd is the monodromy group of f , and C = (C0, C1, C∞) is the triple of conjugacy classesof (σ0,σ1,σ∞) in Sd, respectively [99, Definition 1.1.7]. Although the genus of the Belyı mapis determined by the conjugacy classes by equation (1.5), we still include it in the passportfor clarity and ease. The size of a passport (g,G,C) is the number of equivalence classes oftriples σ = (σ0,σ1,σ∞) such that σ = G and σi ∈ Ci for i = 0, 1,∞.We will occasionally need slightly altered notions of passport. The ramification passport off is the pair (g, C) with conjugacy classes in Sd. Another version of the passport will beconsidered in Section 7. The passport has the following invariance property [84].

Theorem 1.6. — The passport and the ramification passport of a Belyı map are invariant

under the action of Gal(Q/Q).

One can calculate the set of isomorphism classes of permutation triples with given passportusing the following lemma with G = Sd.

Lemma 1.7. — Let G be a group and let C0, C1 be conjugacy classes in G represented by

τ0, τ1 ∈ G. Then the map

CG(τ0)\G/CG(τ1) → (σ0,σ1) : σ0 ∈ C0,σ1 ∈ C1/∼G

CG(τ0)gCG(τ1) → (τ0, gτ1g−1

)



is a bijection, where CG(τ) denotes the centralizer of τ in G and ∼G denotes simultaneous

conjugation in G.

The virtue of this lemma is that double-coset methods in group theory are quite efficient; byusing this bijection and filtering appropriately [91, Lemma 1.11], this allows us to enumerateBelyı maps with a given passport relatively quickly up to moderate degree d. One can alsoestimate the size of a passport using character theory; for more on this, see Section 7.

Applications. — Having introduced the basic theory, we now mention some applicationsof the explicit computation of Belyı maps.We began in the introduction with the motivation to uncover the mysterious nature of theaction of Gal(Q/Q) on dessins following Grothendieck’s Esquisse. Dessins of small degree tendto be determined by their passport in the sense that the set of dessins with given passportforms a full Galois orbit. However, even refined notions of passport do not suffice to distinguishGalois orbits of dessins of high degree in general: a first example was Schneps’ flower [138,§IV, Example I]. Some further examples of distinguishing features of non-full Galois orbitshave been found by Wood [174] and Zapponi [175], but it remains a challenge to determinethe Galois structure for the set of dessins with given passport. Even statistics in small degreeare not known yet; an important project remains to construct full libraries of dessins. Theoriginal “flipbook” of dessins, due to Bétréma–Péré–Zvonkin [15], contained only dessins thatwere plane trees but was already quite influential, and consequently systematic tabulationpromises to be just as inspiring.Further applications of the explicit study of Belyı maps have been found in inverse Galoistheory, specifically the regular realization of Galois groups over small number fields: see thetomes of Matzat [120], Malle–Matzat [107], and Jensen–Ledet–Yui [80]. Upon specialization,one obtains Galois number fields with small ramification set: Roberts [131, 132, 133], Malle–Roberts [112], and Jones–Roberts [86] have used the specialization of three-point covers toexhibit number fields with small ramification set or root discriminant. The covering curvesobtained are often interesting in their own right, spurring further investigation in the studyof low genus curves (e.g., the decomposition of their Jacobian [127]). Finally, a Belyı mapf : P1 → P1, after precomposing so that 0, 1,∞ ⊆ f−1

(0, 1,∞), is an example of a rigidpost-critically finite map, a map of the sphere all of whose critical points have finite orbits.(Zvonkin calls these maps dynamical Belyı functions [176, §6].) These maps are objects ofcentral study in complex dynamics [7, 129]: one may study the associated Fatou and Juliasets.Belyı maps also figure in the study of Hall polynomials, (also called Davenport-Stothers triples)which are those coprime solutions X(t), Y (t), Z(t) ∈ C[t] of the equations in polynomials

X(t)3 − Y (t)2 = Z(t)

with deg(X(t)) = 2m, deg(Y (t)) = 3m and deg(Z(t)) = m + 1. These solutions areextremal in the degree of Z and are analogues of Hall triples, i.e. integers x, y ∈ Z for which|x3 − y2| = O(

|x|). Hall polynomials have been studied by Watkins [166] and by Beukers–

Stewart [17]; Montanus [124] uses the link with dessins (X3(t)/Y 2

(t) is a Belyı map) to finda formula for the number of Hall polynomials of given degree. Hall polynomials also lead tosome good families of classical Hall triples [50], as the following example illustrates.



Example 1.8. — Taking m = 5 above, one obtains the following Hall polynomials due toBirch:

X(t) =1

9(t10 + 6t7 + 15t4 + 12t),

Y (t) =1

54(2t15 + 18t12 + 72t9 + 144t6 + 135t3 + 27),

Z(t) = −1

108(3t6 + 14t3 + 27).

Choosing t ≡ 3 mod 6, we get some decent Hall triples, notably|384242766

3− 7531969451458

2| = 14668

|3906200823− 7720258643465

2| = 14857

for t = ±9; remarkably, in both cases the constant factor |x3 − y2|/|x| is approximately

equal to the tiny number 3/4.

Belyı maps also give rise to interesting algebraic surfaces. The Belyı maps of genus 0 anddegree 12 (resp. 24) with ramification indices above 0, 1 all equal to 3, 2 correspond to ellipticfibrations of rational (resp. K3) surfaces with only 4 (resp. 6) singular fibers; given such afibration, the associated Belyı map is given by taking its j-invariant. By work of Beauville[9] (resp. Miranda and Persson [122]), there are 6 (resp. 112) possible fiber types for thesefamilies. This result comes down to calculating the number of Belyı maps of given degreeswith specified conjugacy classes with cycle type (3, . . . , 3) and (2, . . . , 2) for σ0 and σ1.Especially in the degree 24 case, the explicit calculation of these Belyı maps is quite a chal-lenge. By developing clever methods specific to this case, this calculation was accomplishedby Beukers–Montanus [16]. They find 191 Belyı maps, exceeding the 112 ramification typesdetermined by Miranda and Persson: this is an instance of the phenomenon mentioned above,that the passport may contain more than one Belyı map, so that to a given ramification triplethere may correspond multiple isomorphism classes of Belyı maps.One can also specialize Belyı maps to obtain abc triples: this connection is discussed by Elkies[45] and Frankenhuysen [55] to show that the abc conjecture implies the theorem of Faltings,and it is also considered by van Hoeij–Vidunas [157, Appendix D].Modular curves and certain Shimura curves possess a natural Belyı map. Indeed, Elkies hascomputed equations for Shimura curves in many cases using only the extant structure of aBelyı map [46, 47]. Another such computation was made by Hallouin in [65], where a moreelaborate argument using Hurwitz spaces of four-point covers is used. Explicit equations areuseful in many contexts, ranging from the resolution of Diophantine equations to cryptography[140]. Reducing these equations modulo a prime also yields towers of modular curves thatare useful in coding theory. Over finite fields of square cardinality q, work of Ihara [78] andTsfasman–Vlăduț–Zink [155] shows that modular curves have enough supersingular points thattheir total number of rational points is asymptotic with (

√q − 1)g as their genus grows; this

is asymptotically optimal by work of Drinfeld–Vlăduț [163]. By a construction due to Goppa[59], one obtains the asympotically best linear error-correcting codes known over square fields.But to construct and use these codes we need explicit equations for the curves involved. Afew of these modular towers were constructed by Elkies [49]. There are extensions to otherarithmetic triangle towers, using the theory of Shimura curves, which give other results over



prime power fields of larger exponent [41]. For the cocompact triangle quotients, the modularcovers involved are Belyı maps, and in fact many congruence towers are unramified (andcyclic) after a certain point, which makes them particularly pleasant to work with.There are also applications of explicit Belyı maps to algebraic solutions of differential equations[100]: as we will see in Section 4, subgroups of finite index of triangle groups correspond tocertain Belyı maps, and the uniformizing differential equations for these groups (resp. theirsolutions) can be obtained by pulling back suitable hypergeometric differential equations (resp.hypergeometric functions). Kitaev [89] and Vidunas–Kitaev [162] consider branched coversat 4 points with all ramification but one occuring above three points (“almost Belyı coverings”)and apply this to algebraic Painlevé VI functions. Vidunas–Filipuk [161] classify coveringsyielding transformations relating the classical hypergeometric equation to the Heun differentialequation; these were computed by van Hoeij–Vidunas [157, 158].There are applications to areas farther from number theory. Eyral–Oka [52] explicitly usedessins (and their generalizations to covers of P1 branched over more than 3 points) in theirclassification of the fundamental groups of the complement in the projective plane of certainjoin-type sextic curves of the form a

i(X − αiZ) = b

j(X − βjZ). Boston [20] showed

how three-point branched covers arise in control theory, specifically with regards to a certaincontroller design equation. Finally, dessins appear in physics in the context of brane tilings [68]and there is a moonshine correspondence between genus 0 congruence subgroups of SL2(Z),associated with some special dessins, and certain representations of sporadic groups, withconnections to gauge theory [69, 70, 71].

2. Gröbner techniques

We now begin our description of techniques for computing Belyı maps. We start with theone that is most straightforward and easy to implement, involving the solutions to an explicitset of equations over Q. For Belyı maps of small degree, this method works quite well,and considerable technical effort has made it work in moderate degree. However, for morecomplicated Belyı maps, it will be necessary to seek out other methods, which will be describedin the sections that follow.

Direct calculation. — The direct method has been used since the first Belyı maps werewritten down, and in small examples (typically with genus 0), this technique works wellenough. A large number of authors describe this approach, with some variations relevantto the particular case of interest. Shabat–Voevodsky [144] and Atkin–Swinnerton-Dyer [5]were among the first. Birch [18, Section 4.1] computes a table for covers of small degree andgenus. Schneps [138, III] discusses the case of clean dessins of genus 0 and trees. Malle [111]computed a field of definition for many Belyı maps of small degree and genus 0 using Gröbnermethods, with an eye toward understanding the field of definition of regular realizations ofGalois groups and a remark that such fields of definition also give rise to number fields ramifiedover only a few very small primes. Malle–Matzat [107, §I.9] use a direct method to computeseveral Belyı maps in the context of the inverse Galois problem, as an application of rigidity.Granboulan studied the use of Gröbner bases for genus 0 Belyı maps in detail in his Ph.D.thesis [60]. Elkies [46] used this technique to compute equations for Shimura curves. Otherauthors who have used this method are Hoshino [75] (and Hoshino–Nakamura [76]), who



computed the non-normal inclusions of triangle groups (related to the Belyı-extending mapsof Wood [174]). Couveignes [32, §2] also gives a few introductory examples.We explain how the method works by example in the simplest nontrivial case.

Example 2.1. — Take the transitive permutation triple σ = ((1 2), (2 3), (1 3 2)) from S3,with passport (0, S3, (2111, 2111, 31)). Since these permutations generate the full symmetricgroup S3, the monodromy group of this Belyı map is S3. The Riemann–Hurwitz formula (1.5)gives the genus as

g = 1− 3 +1

2(1 + 1 + 2) = 0.

So the map f : X ∼= P1 → P1 is given by a rational function f(t) ∈ K(t) where K ⊂ Q is anumber field. There are two points above 0, of multiplicities 2, 1, the same holds for 1, andthere is a single point above ∞ with multiplicity 3. The point above ∞ is a triple pole off(t); since it is unique, it is fixed by Gal(K/K); therefore we take this point also to be ∞,which we are free to do up to automorphisms of P1

K , and hence f(t) ∈ K[t]. Similarly, theramified points above 0 and 1 are also unique, so we may take them to be 0 and 1, respectively.Therefore, we have

f(t) = ct2(t+ a)

for some a, c ∈ K \ 0 and

f(t)− 1 = c(t− 1)2(t+ b)

for some b ∈ K \ 0,−1. Combining these equations, we get

ct2(t+ a)− 1 = c(t3 + at2)− 1 = c(t− 1)2(t+ b) = c(t3 + (b− 2)t2 + (1− 2b)t+ b)

and so by comparing coefficients we obtain b = 1/2, c = −2, and a = −3/2. In particular, wesee that the map is defined over K = Q and is unique up to Aut(P1

Q)∼= PGL2(Q). Thus

f(t) = −t2(2t− 3) = −2t3 + 3t2, f(t)− 1 = −(t− 1)2(2t+ 1).

If we relax the requirement that the ramification set be 0, 1,∞ and instead allow 0, r,∞

for some r = 0,∞, then the form of f can be made more pleasing. For example, by takingf(t) = t2(t+ 3) and r = 4 we obtain f(t)− 4 = (t− 1)

2(t+ 2).

It is hopefully clear from this example (see Schneps [138, Definition 8]) how to set up thecorresponding system of equations for a Belyı map on a curve of genus g = 0: with variablecoefficients, we equate the two factorizations of a rational map with factorization specifiedby the cycle types in the permutations triple σ. We illustrate this further in the followingexample; for a large list of examples of this kind, see Lando–Zvonkin [99, Example 2.3.1].

Example 2.2. — To get a small taste of how complicated the equations defining a passportcan get, consider the case G = PGL2(F7) with permutation triple σ = (σ0,σ1,σ∞) given by

σ0 =

−1 0

0 1

, σ1 =

−1 1

−1 0

, σ∞ =

0 1

1 1

.



The permutation representation of G acting on the set of 8 elements P1(F7) is given by the

elements

(1 6)(2 5)(3 4), (0 ∞ 1)(2 4 6), (0 1 4 3 2 5 6 ∞).

The corresponding degree 8 Belyı map f : X → P1 has passport

(0,PGL2(F7), (2312, 3212, 81)).

After putting the totally ramified point at ∞, the map f is given by a polynomial f(t) ∈ Q[t]such that

(2.3) f(t) = ca(t)2b(t) and f(t)− 1 = cd(t)3e(t)

where c ∈ Q× and a(t), b(t), d(t), e(t) ∈ Q[t] are monic squarefree polynomials with deg a(t) =3 and deg b(t) = deg d(t) = deg e(t) = 2. We write a(t) = t3 + a2t2 + a1t+ a0, etc.Equating coefficients in (2.3) we obtain the following system of 8 vanishing polynomials in 10

variables:

a20b0c− cd30e0,

2a1a0b0c+ a20b1c− 3cd1d20e0 − cd30e1,

2a2a0b0c+ a21b0c+ 2a1a0b1c+ a20c− 3cd21d0e0 − 3cd1d20e1 − cd30 − 3cd20e0,

2a2a1b0c+ 2a2a0b1c+ a21b1c+ 2a1a0c+ 2a0b0c− cd31e0 − 3cd21d0e1 − 3cd1d20

− 6cd1d0e0 − 3cd20e1,

a22b0c+ 2a2a1b1c+ 2a2a0c+ a21c+ 2a1b0c+ 2a0b1c− cd31e1 − 3cd21d0 − 3cd21e0

− 6cd1d0e1 − 3cd20 − 3cd0e0,

a22b1c+ 2a2a1c+ 2a2b0c+ 2a1b1c+ 2a0c− cd31 − 3cd21e1 − 6cd1d0 − 3cd1e0 − 3cd0e1,

a22c+ 2a2b1c+ 2a1c+ b0c− 3cd21 − 3cd1e1 − 3cd0 − ce0,

2a2c+ b1c− 3cd1 − ce1.

Using a change of variables t ← t− r with r ∈ Q we may assume that b1 = 0, so b0 = 0. Notethat if f(t) ∈ K[t] is defined over K then we may take r ∈ K, so we do not unnecessarilyincrease the field of definition of the map. Similarly, if d1 = 0, then with t ← ut and u ∈ K×

we may assume d1 = b0; similarly if e1 = 0, then we may take e1 = b0. If d1 = e1 = 0, thenf(t) = g(t2) is a polynomial in t2, whence a0 = 0 so a1 = 0, and thus we may take a1 = b0.This gives a total of three cases: (i) d1 = b0 = 0, (ii) d1 = 0 and e1 = b0 = 0, and (iii)d1 = e1 = 0 and a1 = b0 = 0. We make these substitutions into the equations above, addingc = 0 and b1 = 0 in all cases. Note that the equation c = 0 can be added algebraically byintroducing a new variable c and adding the equation cc = 1.These equations are complicated enough that they cannot be solved by hand, but not socomplicated that they cannot be solved by a Gröbner basis. There are many good referencesfor the theory of Gröbner bases [1, 34, 35, 64, 97].In the degenerate cases (ii) and (iii) we obtain the unit ideal, which does not yield anysolutions. In the first case, we find two conjugate solutions defined over Q(

√2). After some



simplification, the first of the solutions becomes

f(t) =2

√2t3 − 2(2

√2 + 1)t2 + (−4 + 7

√2)t+ 1

214t2 + 6(

√2 + 4)t− 8

√2 + 31

with

f(t)− 432(4

√2− 5) =

2t2 − 2

√2 + 1

314t2 − 8(

√2 + 4)t− 14

√2 + 63

.

The direct method does not give an obvious way to discriminate among Belyı maps by theirmonodromy groups, let alone to match up which Galois conjugate corresponds to whichmonodromy triple: all covers with a given ramification type are solutions to the above systemof equations.To set up a similar system of equations in larger genus g ≥ 1, one can for example writedown a general (singular) plane curve of degree equal to degϕ and ask that have sufficientlymany nodal singularities so that it has geometric genus g; the Belyı map can then be takenas one of the coordinates, and similar techniques apply, though many non-solutions will stillbe obtained in this way by cancellation of numerator and denominator.

Remark 2.4. — Any explicitly given quasiprojective variety X with a surjective map to themoduli space Mg of curves of genus g will suffice for this purpose; so for those genera gwhere the moduli space Mg has a simpler representation (such as g ≤ 3), one can use thisrepresentation instead. The authors are not aware of any Belyı map computed in this waywith genus g ≥ 3.

The direct method can be used to compute the curves X with Belyı maps of small degree.The curve P1 is the only curve with a Belyı map of degree 2 (the squaring map), and theonly other curve that occurs in degree 3 is the genus 1 curve with j-invariant 0 and equationy2 = x3 + 1, for which the Belyı map is given by projecting onto the y-coordinate. In degree4, there is the elliptic curve of j-invariant 1728 with equation y2 = x3 − x with Belyı mapgiven by x2 and one other given by the the elliptic curve y2 = 4(2x + 9)(x2 + 2x + 9) andregular function y + x2 + 4x+ 18. Both were described by Birch [18].In the direction of tabulating the simplest dessins in this way, all clean dessins (i.e. those forwhich all ramification indices above 1 are equal to 2) with at most 8 edges were computedby Adrianov et al. [2]. Magot–Zvonkin [106] and Couveignes–Granboulan [33] computed thegenus 0 Belyı maps corresponding to the Archimedean solids, including the Platonic solids,using symmetry and Gröbner bases. For a very complete discussion of trees and Shabatpolynomials and troves of examples, see Lando–Zvonkin [99, §2.2].In general, we can see that these Gröbner basis techniques will present significant algorithmicchallenges. Even moderately-sized examples, including all but the first few of genus 1, do notterminate in a reasonable time. (In the worst case, Gröbner basis methods have runningtime that is doubly exponential in the input size, though this can be reduced to singlyexponential for zero-dimensional ideals; see the surveys of Ayad [6] and Mayr [116].) Onefurther differentiation trick, which we introduce in the next section, allows us to computein a larger range. However, even after this modification, another obstacle remains: theset of solutions can have positive-dimensional degenerate components. These componentscorrespond to situations where roots coincide or there is a common factor and are often calledparasitic solutions [95, 96]. The set of parasitic solutions have been analyzed in some cases by



van Hoeij–Vidunas [158, §2.1], but they remain a nuisance in general (as can be seen alreadyin Example 2.2 above).

Remark 2.5. — Formulated more intrinsically, the naive equations considered in this sectiondetermine a scheme in the coefficient variables that is a naive version of the Hurwitz schemesthat will be mentioned in Section 9. Besides containing degenerate components, this naivescheme is usually very non-reduced. We will revisit this issue in Remark 2.10.When calculating a Belyı map f : P1 → P1, one usually fixes points on the source and thetarget. As we saw most elaborately when working out equation (2.3), this reduces the problemof calculating a Belyı map in genus 0 to finding the points on an affine scheme. The familiesof solutions in which numerator and denominator cancel give rise to some of the degeneratecomponents mentioned in the previous paragraph.

The ASD differentiation trick. — There is a trick, due to Atkin–Swinnerton-Dyer [5,§2.4] that uses the derivative of f to eliminate a large number of the indeterminates (“thenumber of unknowns c can be cut in half at once by observing that dj/dζ has factorsF 23F2”). Couveignes [32] implies that this trick was known to Fricke; it has apparently been

rediscovered many times. Hempel [73, §3] used differentiation by hand to classify subgroupsof SL2(Z) of genus 0 with small torsion and many cusps. Couveignes [29, §2,§10] used this tocompute examples in genus 0 of clean dessins. Schneps [138, §III] used this trick to describe ageneral approach in genus 0. Finally, Vidunas [160] applied the trick to differential equations,and Vidunas–Kitaev [162] extended this to covers with 4 branch points.

Example 2.6. — Again we illustrate the method by an example. Take

σ = ((1 2), (2 4 3), (1 2 3 4))

with passport (0, S4, (2112, 3111, 41)). Choosing the points 0 and 1 again to be ramified, thistime of degrees 2, 3 above 0, 1 respectively, and choosing ∞ to be the ramified point above∞, we can write

f(t) = ct2(t2 + at+ b)

and

f(t)− 1 = c(t− 1)3(t+ d).

The trick is now to differentiate these relations, which yields

f (t) = ct

2(t2 + at+ b) + t(2t+ a)

= c(t− 1)

2((t− 1) + 3(t+ d))

t(4t2 + 3at+ 2b) = (t− 1)2(4t+ (3d− 1)) .

By unique factorization, we must have 4t2+3at+2b = 4(t− 1)2 and 4t = 4t+(3d− 1), so we

instantly get a = −8/3, b = 2, and d = 1/3. Substituting back we see that c = 3, and obtain

f(t) = t2(3t2 − 8t+ 6) = (t− 1)3(3t+ 1) + 1.

More generally, the differentiation trick is an observation on divisors that extends to highergenus, as used by Elkies [47] in genus g = 1.



Lemma 2.7. — Let f : X → P1be a Belyı map with ramification type σ. Let

div f =

P

ePP −

R

eRR and div(f − 1) =

Q

eQQ−

R

eRR

be the divisors of f and f − 1. Then the divisor of the differential df is

div df =

P

(eP − 1)P +

Q

(eQ − 1)Q−

R

(eR + 1)R.

Proof. — Let

D =

P

(eP − 1)P +

Q

(eQ − 1)Q−

R

(eR + 1)R.

Then div df ≥ D by the Leibniz rule. By Riemann–Hurwitz, we have

2g − 2 = −2n+

P

(eP − 1) +

Q

(eQ − 1) +

R

(eR − 1)

so

deg(D) = 2g − 2 + 2n− 2

R

eR = 2g − 2

since

R eR = n. Therefore div df can have no further zeros.

Combined with unique factorization, this gives the following general algorithm in genus 0.Write

f(t) =p(t)

q(t)= 1 +

r(t)

q(t)

for polynomials p(t), q(t), r(t) ∈ Q[t]. Consider the derivatives p(t), q(t), r(t) with respectto t and let p0(t) = gcd(p(t), p(t)) and similarly q0(t), r0(t). Write

P (t) =p(t)

p0(t)and P (t) =

p(t)

p0(t)

and similarly Q, etc. Then by unique factorization, and the fact that P,Q,R have no commondivisor, evaluation of the expressions p(t) − q(t) = r(t) and p(t) − q(t) = r(t) yieldsthat Q(t) R(t) − Q(t)R(t) is a multiple of p0(t), and similarly P (t) R(t) − P (t)R(t) (resp.P (t) Q(t)− P (t)Q(t)) is a multiple of q0(t) (resp. r0(t)).These statements generalize to higher genus, where they translate to inclusions of divisors;but the usefulness of this for concrete calculations is limited and do not pass to relations offunctions, since the coordinate rings of higher genus curves are usually not UFDs. Essentially,one has to be in an especially agreeable situation for a statement on functions to fall out,and usually one only has a relation on the Jacobian (after taking divisors, as in the lemmaabove). A concrete and important situation where a relation involving functions does occuris considered by Elkies [47]. The methods in his example generalize to arbitrary situationswhere the ramification is uniform (all ramification indices equal) except at one point of theBelyı curve: Elkies himself treats the Belyı maps with passport (1,PSL2(F27), (3911, 214, 74)).



The differentiation trick does not seem to generalize extraordinarily well to higher derivatives;we can repeat the procedure above and further differentiate p(t), q(t), r(t), but experimen-tally this not seem to make the ideal grow further than in the first step.

Question 2.8. — Is the ideal obtained by adding all higher order derivatives equal to the one

obtained from just adding equations coming from first order derivatives (in genus 0)?

However, Shabat [143, Theorem 4.4] does derive some further information by consideringsecond-order differentials; and Dremov [40] calculates Belyı maps using the quadratic differ-ential

MP (f) =df2

f(1− f)

for a regular function f and considering the equalities following from the relation

MP (f−1) = −MP (f)/f.

It is not immediately clear from these paper how to use this strategy in general, though.

Question 2.9. — How generally does the method of considering second-order differentials

apply?

The additional equations coming from the differentiation trick not only speed up the processof calculating Belyı maps, but they also tend to give rise to a Jacobian matrix at a solutionthat is often of larger rank than the direct system. This is important when trying to Hensellift a solution obtained over C or over a finite field, where the non-singularity of the Jacobianinvolved is essential. (We discuss these methods in sections that follow.)

Remark 2.10. — Phrased in the language of the naive moduli space in Remark 2.5, theadditional ASD relations partially saturate the corresponding equation ideal, so that the largerset of equations defines the same set of geometric points, but with smaller multiplicities. (Wethank Bernd Sturmfels for this remark.) Reducing this multiplicity all the way to 1 is exactlythe same as giving the Jacobian mentioned above full rank.

Example 2.11. — The use of this trick for reducing multiplicities is best illustrated by somesmall examples.The first degree d in which the ASD differentiation trick helps to give the Jacobian ma-trix full rank is d = 6; it occurs for the ramification triples (2

3, 23, 32), (2212, 32, 4121),

(32, 31211, 312111), (3113, 4121, 4121), (4121, 4112, 312111), and (4

121, 312111, 312111), where it

reduces the multiplicity of the corresponding solutions from 9, 3, 3, 3, 4, 3 respectively to 1.Note the tendency of Belyı maps with many automorphisms to give rise to highly singularpoints, as for curves with many automorphisms in the corresponding moduli spaces.On the other hand, there are examples where even adding the ASD relations does not lead toa matrix of full rank. Such a case is first found in degree 7; it corresponds to the ramificationtriples (412111, 312112, 4131), and throwing in the ASD relations reduces the multiplicity from8 to 2. Unfortunately, iterating the trick does not make the ideal grow further in this case.More dramatically, for the ramification triples (2

4, 3221, 3221) and (2312, 42, 3221), differen-

tiation reduces some multiplicities from 64 to 1 (resp. 64 to 4). In the latter case, thesemultiplicities are in fact not determined uniquely by the corresponding ramification type, so



that considering these multiplicities gives a way to split the solutions into disjoint Galoisorbits.

Question 2.12. — How close is the ideal obtained from the differentiation trick (combined

with the direct method) to being radical? Can one give an upper bound for the multiplicity of

isolated points?

Further extensions. — There can be several reasons why a Gröbner basis calculationfails to terminate. One problem is coefficient blowup while calculating the elimination ide-als. This can be dealt by first reducing modulo a suitable prime p, calculating a Gröb-ner basis for the system modulo p, then lifting the good solutions (or the Gröbner ba-sis itself) p-adically, recognizing the coefficients as rational numbers, and then verifyingthat the basis over Q is correct. This was used by Malle [110, 113] to compute cov-ers with passports (0,Hol(E8), (412112, 412112, 6121)) and (0,PGL(F11), (2512, 43, 11111)) andsimilarly Malle–Matzat [108] to compute covers for (0,PSL2(F11), (2413, 613121, 613121)) and(0,PSL2(F13), (27, 4312, 6212)). This idea was also used by Vidunas–Kitaev [162, §5]. For fur-ther developments on p-adic methods to compute Gröbner bases, see Arnold [4] or Winkler[169]. One can also lift a solution modulo p directly, and sometimes such solutions can beobtained relatively quickly without also p-adically lifting the Gröbner bases: this is the basicidea presented in Section 5.In the work of van Hoeij–Vidunas [157, 158] mentioned in Section 1, genus 0 Belyı functionsare computed by using pullbacks of the hypergeometric differential equation and their solu-tions. This method works well when the order of each ramification point is as large as possible,e.g., when the permutations σ0,σ1,σ∞ contain (almost) solely cycles of order n0, n1, n∞ say,and only a few cycles of smaller order. For example, this occurs when the cover is Galois,or slightly weaker, when it is regular, that is to say, when the permutations σ0,σ1,σ∞ are aproduct of disjoint cycles of equal cardinality.The method of van Hoeij–Vidunas to calculate a Belyı map f : X → P1 is to consider then exceptional ramification points in X of f whose ramification orders do not equal the usualorders a, b, c. One then equips the base space P1 with the hypergeometric equation whoselocal exponents at 0, 1,∞ equal a, b, c. Pulling back the hypergeometric equation by f , oneobtains a Fuchsian differential equation with singularities exactly in the n exceptional points.The mere fact that this pullback exists implies equations on the undetermined coefficients off .For example, when the number of exceptional points is just n = 3, the differential equationcan be renormalized to a Gaussian hypergeometric differential equation, which completelydetermines it. When n = 4, one obtains a form of Heun’s equation [125, 157]. Heun’sequation depends on the relative position of the fourth ramification point, as well as on anaccessory parameter; still, there are only two parameters remaining in the computation.One shows that for fixed n and genus g (taken as g = 0 later), there are only finitely manyhyperbolic Belyı functions with n exceptional points. For small n, van Hoeij and Vidunasshow that this differential method is successful in practice, and they compute all (hyperbolic)examples with n ≤ 4 (the largest degree of such a Belyı map was 60).



Question 2.13. — Are there other sources of equations (such as those arising from differ-

ential equations, algebraic manipulation, etc.) that further simplify the scheme obtained from

the direct method?

3. Complex analytic methods

In this section we consider complex analytic methods for finding equations for Belyı maps.These methods are essentially approximative; a high precision solution over C is determined,from which one reconstructs an exact solution over Q.

Newton approximation. — We have seen in the previous section how to write downa system of equations which give rise to the Belyı map. These equations can be solvednumerically in C using multidimensional Newton iteration, given an approximate solutionthat is correct to a sufficient degree of precision and a subset of equations of full rank whoseJacobian has a good condition number (determinant bounded away from zero). Then, givena complex approximation that is correct to high precision, one can then use the LLL lattice-reduction algorithm [102] (as well as other methods, such as PSLQ [53]) to guess algebraicnumbers that represent the exact values. Finally, one can use the results from Section 8to verify that the guessed cover is correct; if not, one can go back and iterate to refine thesolution.

Remark 3.1. — We may repeat this computation for each representative of the Galois orbitto find the full set of conjugates for each putative algebraic number and then recognize thesymmetric functions of these conjugates as rational numbers using continued fractions instead.For example, one can compute each representative in the passport, possibly including severalGalois orbits. The use of continued fractions has the potential to significantly reduce theprecision required to recognize the Belyı map exactly.

Example 3.2. — Consider the permutation triple

σ0 = (1 3 2)(4 6 5), σ1 = (1 5 2)(3 4)(6 7), σ∞ = (1 3 5 2 6 7 4).

From the Riemann–Hurwitz formula, we find that the associated Belyı curve X has genusg = 1. The ramification point of index 7 on X (over ∞) is unique, so we take it to be theorigin of the group law on X. Moreover, since there is a unique unramified point above 0, wecan use a normal form (due to Tate) of an elliptic curve with a marked point. This is givenby an equation

y2 + p3y = q(x) = x3 + p2x2+ p4x(3.3)

with marked point (0, 0). The equation (3.3) is unique up to scaling the coefficients by u = 0

according to (p2, p3, p4) → (u2a2, u3a3, u4a4), showing that the moduli spaces M1,2 of genus1 curves with two marked points is isomorphic to the weighted projective space P(2, 3, 4).Since the origin of the group law of X maps to ∞ and (0, 0) maps to 0, the Belyı mapf : X → P1 of degree 7 is of the form

f(x, y) = (a3x3+ a2x

2+ a1x) + (b2x

2+ b1x+ b0)y = a(x) + b(x)y.



The ramification above f = 0 leads to the equation

NC(x,y)/C(x)(f(x, y)) = a(x)(a(x)− p3b(x))− b(x)2q(x) = −b22xc(x)3

where c(x) is the monic polynomial x2 + c1x+ c0. Consideration of the ramification above 1

yields

NC(x,y)/C(x)(f(x, y)− 1) = (a(x)− 1)(a(x)− 1− p3b(x))− b(x)2q(x) = −b22d(x)3e(x)2

where d(x) = x+ d0 and e(x) = x2 + e1x+ e0.This yields 13 equations in 14 unknowns. The reason for this is that we are still free to scalethe pi. Here we have to distinguish cases. We first suppose that the point (0, 0) in (3.3) isnot 2-torsion, or equivalently, that p3 = 0: this is the “generic” case. We can then distinguishtwo further cases, namely p2 = 0 and p4 = 0. Accordingly, we may then ensure p2 = p3 orp3 = p4 by scaling over the ground field, so that we do not needlessly enlarge the coefficientof the Belyı map. In either case, plugging in random choices for the vector of unknowns(a, b, c, d, e, p) ∈ C14 and applying multivariate Newton iteration fails to yield a solution.To improve the convergence, we now proceed to remove some degenerate cases from this set ofequations. Applying the trick from Example 2.2, we impose that c0d0e0 = 0, as we may sincethe ramification points are distinct and (0, 0) is a ramification point. (This in fact assumesthat none of the other ramification points is (0,−1), which leads to a subcase that turnsout not to yield a solution.) We further insist that c and e do not have a double root, so(c21 − c0)(e21 − e0) = 0. This adds 2 more variables and equations.Finally, we saturate our equations using the Atkin–Swinnerton-Dyer trick in Lemma 2.7. Thedifferential dx/(2y+p3) is holomorphic and has no zeros or poles, so denoting derivation withrespect to x by , we see that

df

dx/(2y + p3)= (2y + p3)

df

dx= a(x)(2y + p3) + b(x)(2y + p3)y + b(x)(2y + a3)y

= (2b(x)q(x) + b(x)q(x) + p3a(x)) + (2a(x)− p3b

(x))y

satisfies

N(((2y + p3)(df/dx)) = 49b22c(x)2d(x)2e(x).

This differentation trick thus yields another 8 equations. But even after adding these and thenondegeneracy conditions, random choices for an initial approximation fail to converge to asolution for the new system of 23 equations in 15 unknowns.So we are led to consider the case where p3 = 0, so that the unramified point above 0 is2-torsion. (Here, there is some extra ambiguity, since the moduli space X0(2) = X1(2) isnot a fine moduli space.) If we write div(f) = (0, 0) + 3P1 + 3P2 − 7∞ and div(f − 1) =

2Q1 + 2Q2 + 3Q3, then we have

div(df) = 2P1 + 2P2 +Q1 +Q2 + 2Q3 − 8∞

and so we obtain the relations

Q1 +Q2 = 3Q3 = 0, 3Q3 = 0, (0, 0) + (P1 + P2) = −Q3, 2(0, 0) = 0

in the group law of X. In particular, P1 + P2 is a 6-torsion point on X. Relations such asthese can be used to find extra equations for X and f by using division polynomials. But



again, the new system fails to yield any solutions; perhaps one can prove non-existence ofsolutions directly.Here, we look ahead to the methods of this section and Section 4 that allow us to find anapproximation to the solution. It turns out that we only need 3 decimal places to get theNewton method converging to a real solution with p3 = 0 and p2 = p3, approximated by thesolution

(a, b, c, d, e, p) ≈

(182.7513294, 146.8290694, 29.38993410,−308.3482399,−244.0552479

− 48.11742858, 0.7992141684, 0.1613326212, 0.1482181605, 0.9764940118,

0.2561882114, 1.165925608, 0.4430649844, 163.2364906, 3.003693522)

in C13. The condition number of the system without the additional Atkin–Swinnerton-Dyerrelations is approximately 3.3·107; but by adding some of these relations, this can be decreasedto approximately 1.2 · 105.Using LLL, we recognize this as a putative solution over Q(α) with α3 − 3α + 12 = 0;then we verify that the recognized solution is correct using the methods of Section 8. Thissolution thereby gives rise to two more complex (conjugate) solutions. Since there are onlythree permutation triples with the given ramification passport, we see that we have found alldessins of the given ramification type, so we need not consider the other cases further.As mentioned in Remark 3.1, the standard algorithms to recognize algebraic dependency workbetter after symmetrizing over these conjugate solutions. For the most difficult algebraicnumber to recognize (which is b2) using a single solution requires the knowledge of 161 digits,whereas recognition as an algebraic number needs only 76 digits.If we drop the demand that the unramified point is at (0, 0), then we can simplify the solutionsomewhat, as in Section 8. In Weierstrass form, we can take X to be given by the curve

y2 = x3 + (−541809α2+ 898452α+ 2255040)x

+ (−2929526838α2+ 5759667648α− 11423888784).

and the function f = a(x) + b(x)y by

21331455a(x) = (1491α2

+ 6902α+ 10360)x3

+ (1410885α2+ 2033262α− 4313736)x2

+ (731506545α2+ 15899218650α+ 32119846920)x

− (7127713852353α2+ 3819943520226α+ 62260261739784)

and

21331655b(x) = (−197α2

− 240α+ 528)x2

+ (906570α2− 546840α− 8285760)x

− (715988241α2− 2506621464α− 1458270864).

We thank Marco Streng for his help with reducing these solutions. Applying the methodsin Section 4 already gives equations that are better than those in the normalized forms



(3.3) considered above; at least experimentally, using the modular method also tends togive equations of relatively small height.

As we have seen in the preceding example, in order for this procedure to work, one needsa good starting approximation to the solution. In the non-trivial examples that we havecomputed so far, it seems that often this approximation must be given to reasonably highprecision (at least 30 digits for moderately-sized examples) in order for the convergence tokick in. The required precision seems difficult to estimate from above or below. And indeedthe dynamical system arising from Newton’s method has quite delicate fractal-like propertiesand its study is a subject in itself [128].

Question 3.4. — Is there an explicit sequence of Belyı maps with the property that the

precision required for Newton iteration to converge tends to infinity?

One way to find a starting approximation to the solution is explained by Couveignes–Granboulan [29, 60, 33]. They inductively use the solution obtained from a simpler map:roughly speaking, they replace a point of multiplicity ν with two points of multiplicities ν1, ν2with ν1 + ν2 = ν. One can use any appropriate base case for the induction, such as a maphaving simple ramification. Couveignes [29] gives a detailed treatment of the case of trees,corresponding to clean Belyı polynomials f(t), i.e. those with f(t)− 1 = g(t)2: geometrically,this means that the corresponding dessin can be interpreted as a tree with oriented edges.In this case, after an application of the differentiation trick, one is led to solve a system ofequations where many equations are linear. See Granboulan [60, Chapter IV] for an examplewith monodromy group Aut(M22).

Remark 3.5. — There is a misprint in the example of Couveignes [29, §3, pg. 8] concerningthe discriminant of the field involved, corrected by Granboulan [60, p. 64].

So far, it seems that the inductive numerical method has been limited to genus 0 Belyı mapswith special features. A similar method was employed by Matiyasevich [119] for trees: herecursively transforms the initial polynomial 2tn − 1 (corresponding to a star tree) into apolynomial representing the desired planar tree.

Question 3.6. — Can an inductive complex analytic method be employed to compute more

complicated Belyı maps in practice?

In particular, the iterative method by Couveignes and Granboulan to find a good starting valueseems to rely on intuition involving visual considerations; can these be made algorithmicallyprecise?

Circle packing. — Another complex analytic approach is to use circle packing methods. Thistechnique was extensively developed in work of Bowers–Stephenson [22], with a correspondingJava script CirclePack available for calculations.Given a dessin (i.e., the topological data underlying a Belyı map), one obtains a triangulationof the underlying surface by taking the inverse image of P1

(R) ⊂ P1(C) together with the

corresponding cell decomposition. Choosing isomorphisms between these triangle and thestandard equilateral triangle in C and gluing appropriately, one recovers the Riemann surfacestructure and as a result a meromorphic description of the Belyı map.



However, the Riemann surface structure is difficult to determine explicitly, starting from thedessin. As an alternative, one can pass to discrete Belyı maps instead. To motivate thisconstruction, note that a Riemann surface structure on a compact surface induces a uniquemetric of constant curvature 1, 0,−1 (according as g = 0, 1,≥ 2) so that one can then speakmeaningfully about circles on such a surface. In particular, it makes sense to ask whetheror not there exists a circle packing associated with the triangulation, a pattern of circlescentered at the vertices of this triangulation satisfying the tangency condition suggested by thetriangulation. Satisfyingly enough, the circle packing theorem, due Koebe–Andreev–Thurston[93, 114, 154], states that given a triangulation of a topological surface, there exists a uniquestructure of Riemann surface that leads to a compatible circle packing. This then realizes thetopological map to the Riemann sphere as a smooth function.In summary, starting with a dessin, one obtains a triangulation and hence a circle packing.The corresponding discrete Belyı map will in general not be meromorphic for the Riemannsurface structure induced by the circle packing; but Bowers and Stephenson prove that it doesconverge to the correct solution as the triangulation is iteratively hexagonally refined.The crucial point is now to compute the discrete approximations obtained by circle packing inan explicit and efficient way. Fortunately, this is indeed possible; work by Collins–Stephenson[26] and Mohar [123] give algorithms for this. The crucial step is to lift the configurationof circles to the universal cover H (which is either the sphere P1

(C), the plane C, or theupper half-plane H) and perform the calculation in H. In fact, this means that the circlepacking method also explicitly solves the uniformization problem for the surface involved; fortheoretical aspects, we refer to Beardon–Stephenson [8]. Upon passing to H and using theappropriate geometry, one then first calculates the radii of the circles involved from thecombinatorics, before fitting the result into H, where it gives rise to a fundamental domainfor the corresponding curve as a quotient of H.An assortment of examples of the circle packing method is given by Bowers–Stephenson [22,§5], and numerical approximations are computed to a few digits of accuracy. This includesgenus 0 examples of degree up to 18, genus 1 examples of degree up to 24, and genus 2 examplesof degree up to 14. For determining the conformal structure, this approach is therefore muchmore effective indeed than the naive method from Section 2. Even better, one can proceedinductively from simpler dessins by using so-called dessin moves [22, §6.1], which makes thisapproach quite suitable for calculating large tables of conformal realizations of dessins.On the other hand, there are no theoretical results on the number of refinements needed toobtain given accuracy for the circle packing method [22, §7]. In examples, it is possible forthe insertion of a new vertex to drastically increase the accuracy needed [22, Figure 25] andthereby the number of discrete refinements needed, quite radically increasing the complexityof the calculation [22, §8.2]. However, the method is quite effective in practice, particularlyin genus 0.More problematically, it seems difficult to recover equations over Q for the Belyı map fromthe computed fundamental domain if the genus is strictly positive. One can compute theperiods of the associated Riemann surface to some accuracy, but one still needs to recover thecurve X and transfer the Belyı map f on X accordingly. Moreover, is also not clear that theaccuracy obtained using this method is enough to jump start Newton iteration and therebyobtain the high accuracy needed to recognize the map over Q. In Section 4, we circumventthis problem by starting straightaway with an explicit group Γ of isometries of H so that



Γ\H ∼= X and then finding equations for X by numerically computing modular forms (i.e.,differential forms) on X.

Example 3.7. — In Figure 3.8, we give an example from an alternate implementation byWestbury, which is freely available [168] for the case of genus 0. In the figure, an outer polygonis inserted instead of a circle to simplify the calculation of the radii. We show the conformaltriangulation induced by the second barycentric subdivision of the original triangulation forone of the exactly 2 covers in Example 7.9 that descend to R.

Figure 3.8: A second subdivision for M23

Several more subdivisions would be needed to get the solution close enough to apply Newton’smethod.

Puiseux series. — Couveignes–Granboulan [33, §6] proposed an alternative method usingPuiseux series expansions to get a good complex approximation to the solution so that againmultidimensional Newton iteration can kick off.At every regular point P in the curve X, the Belyı map has an analytic expansion as apower series in a uniformizer z at P that converges in a neighborhood of P . Similarly, at aramification point P , there is an expansion for f that is a Puiseux series in the uniformizer z;more specifically, it is a power series in z1/e = exp(2πi log(z)/e) where e is the ramificationindex of P and log is taken to be the principal logarithm. Now, these series expansionsmust agree whenever they overlap, and these relations between the various expansions giveconditions on their coefficients. More precisely, one chooses tangential base points, calledstandards, and the implied symbolic relations are then integrated with respect to a measurewith compact support. Collecting the relations, one obtains a block matrix, the positioningof whose blocks reflects the topology of the overlaps of the cover used.



Unfortunately, Couveignes and Granboulan do not give an example of this method in practice,and the most detail they give concerns iterative ad hoc methods [33, §7].

Question 3.9. — How effective is the method of Puiseux series in finding a good starting

approximation? Can one prove rigorously that this method gives a correct answer to a desired

precision?

Homotopy methods. — One idea that has yet to be explored (to the authors’ knowledge)is the use of techniques from numerical algebraic geometry, such as polyhedral homotopymethods [10, 159], to compute Belyı maps. The success of homotopy methods in solvingextremely large systems of equations, including those with positive-dimensional components,has been dramatic. In broad stroke, one deforms the solution of an easier system to thedesired ones and carefully analyzes the behavior of the transition matrix (Jacobian) to ensureconvergence of the final solution. Because these methods are similar in spirit to the onesabove, but applied for a more general purpose, it is natural to wonder if these ideas can bespecialized and then combined into a refined technique tailored for Belyı maps.

Question 3.10. — Can the techniques of numerical algebraic geometry be used to compute

Belyı maps efficiently?

A potential place to start in deforming is suggested by the work above and by Couveignes [32,§6]: begin with a stable curve (separating the branch points) and degenerate by bringing to-gether the genus 0 components. The difficulty then becomes understanding the combinatorialgeometry of this stable curve, which is an active area of research.

Zipper method. — Complex analytic techniques can also be brought to bear on Belyı mapsof extremely large degree, at least for the case of trees, using an extension of the zipper method

due to Marshall–Rohde [117, 118]. The zipper method finds a numerical approximation of theconformal map of the unit disk onto any Jordan region [115]. In its extension, this amountsto constructing the conformal map onto the domain of the exterior of the desired dessin,which can be done quite simply for trees even with thousands of branches. For example,Marshall and Rohde have computed the dessins associated to the Belyı maps fn

(z) wheref(z) = (3z3 − 9z− 2)/4, giving a sequence of Belyı trees (under the preimage of [−2, 1]), andby extension one can obtain complex approximation to Belyı maps of extremely large degree:trees with tens of thousands of edges, far beyond the reach of other methods.

Question 3.11. — Does the zipper method extend to higher genus?

In the latter extension, one would need to consider not only the convergence of the Belyı mapbut also the associated Belyı curve X, so it appears one will have to do more than simply solvethe Dirichlet problem. See also work by Larusson and Sadykov [100], where the connectionwith the classical Riemann-Hilbert problem is discussed in the context of trees.

4. Modular forms

In this section we continue with the general strategy of using complex analytic methods butshift our focus in the direction of geometry and consideration of the uniformization theorem;



we work explicitly with quotients of the upper half-plane by Fuchsian groups and recast Belyımaps in this language. This point of view is already suggested by Grothendieck [62]:

In more erudite terms, could it be true that every projective non-singular algebraiccurve defined over a number field occurs as a possible “modular curve” parametris-ing elliptic curves equipped with a suitable rigidification? . . . [T]he Soviet mathe-matician Belyı announced exactly that result.

As in the last section, the method here uses numerical approximations; however, the use ofmodular functions adds considerable more number-theoretic flavor to the analytic techniquesin the previous section.

Classical modular forms. — Let F2 be the free group on two generators as in (1.1). Recallthat the map that considers the permutation action of x, y, z on the cosets of a subgroup yieldsa bijection

(4.1)

transitive permutation triples σ = (σ0,σ1,σ∞) ∈ S3

d

/∼

1:1

subgroups of F2 of index d /∼ ;

here the equivalence relation on triples is again uniform conjugation, and the equivalencerelation on subgroups is conjugation in F2. In particular, by Proposition 1.4, isomorphismclasses of (connected) Belyı maps are in bijection with the conjugacy classes of subgroups F2

of finite index.The key observation is now that F2 can be realized as an arithmetic group, as follows.The group Γ(1) = PSL2(Z) = SL2(Z)/±1 acts on the completed upper half-plane H∗

=

H ∪ P1(Q) by linear fractional transformations

z →az + b

cz + d, for ±

a bc d

∈ PSL2(Z).

The quotient X(1) = Γ(1)\H∗ can be given the structure of a Riemann surface of genus 0 bythe uniformizing map j : X(1)

∼−→ P1

(C) (often called the modular elliptic j-function),

j(q) =1

q+ 744 + 196884q + 21493760q2 + 864299970q3 + . . .

where q = exp(2πiz).For an integer N , we define the normal subgroup Γ(N) as the kernel of the reduction mapPSL2(Z) → PSL2(Z/NZ). We will be particularly interested in the subgroup

Γ(2) =

±

a bc d

∈ PSL2(Z) : b ≡ c ≡ 0 (mod 2)

of index 6, with quotient isomorphic to Γ(1)/Γ(2) ∼= PSL2(F2) = GL2(F2)∼= S3. The

group Γ(2) is in fact isomorphic to the free group F2∼= Γ(2): it is freely generated by

±

1 2

0 1

,±

1 0

2 1

, which act on H by z → z + 2 and z → z/(2z + 1), respectively; the

corresponding action on the upper half plane is free as well.



The quotient X(2) = Γ(2)\H∗ is again a Riemann surface of genus 0; the action of Γ(2) onP1

(Q) has three orbits, with representatives 0, 1,∞ ∈ P1(Q). We obtain another uniformizing

map λ : X(2)∼−→ P1

(C) with expansion

λ(z) = 16q1/2 − 128q + 704q3/2 − 3072q2 + 11488q5/2 − 38400q3 + . . . .

As a uniformizer for a congruence subgroup of PSL2(Z), the function λ(z) has a modularinterpretation: there is a family of elliptic curves over X(2) equipped with extra structure.Specifically, given λ ∈ P1

(C) \ 0, 1,∞, the corresponding elliptic curve with extra structureis given by the Legendre curve

E : y2 = x(x− 1)(x− λ),

equipped with the isomorphism (Z/2Z)2 ∼−→ E[2] determined by sending the standard gener-

ators to the 2-torsion points (0, 0) and (1, 0).There is a forgetful map that forgets this additional torsion structure on a Legendre curve andremembers only isomorphism class; on the algebraic level, this corresponds to an expressionof j in terms of λ, which is given by

(4.2) j(λ) = 256(λ2 − λ+ 1)

3

λ2(λ− 1)2;

indeed, the map X(2) → X(1) given by j/1728 is a Galois Belyı map of degree 6 withmonodromy group S3, given explicitly by (4.2). This map is the Galois closure of the mapcomputed in Example 2.1.The cusp ∞ plays a special role in the theory of modular forms, and marking it in ourcorrespondence will allow a suitable rigidification. With this modification, the correspondence(4.1) becomes a bijection

(4.3)

transitive permutation triples σ ∈ Sd

with a marked cycle of σ∞

/∼

1:1

subgroups of F2∼= Γ(2) of index d /∼

with equivalence relations as follows: given Γ,Γ ≤ Γ(2), we have Γ ∼ Γ if and only if

gΓg−1= Γ

for g an element of the subgroups of translations generated by z → z + 2; andtwo triples σ,σ ∈ S3

d with marked cycles c, c in σ∞,σ∞ are equivalent if and only if they are

simultaneously conjugate by an element τ with τcτ−1= c.

It is a marvelous consequence of either of the bijections (4.1) and (4.3), combined withBelyı’s theorem, that any curve X defined over a number field is uniformized by a subgroupΓ ≤ Γ(2) < PSL2(Z), so that there is a uniformizing map Γ\H∗ ∼

−→ X(C). This is themeaning of Grothendieck’s comment: the rigidification here corresponds to the subgroup Γ.In general, the group Γ is noncongruence, meaning that it does not contain a subgroup Γ(N),so membership in the group cannot be determined by congruences on the coordinate entriesof the matrices. This perspective of modular forms is taken by Atkin–Swinnerton-Dyer [5]and Birch [18, Theorem 1] in their exposition of this subject: they discuss the relationshipbetween modular forms, the Atkin–Swinnerton-Dyer congruences for noncongruence modularforms, and Galois representations in the context of Belyı maps. For more on the arithmetic



aspects of this subject, we refer to the survey by Li–Long–Yang [104] and the referencestherein.The description (4.3) means that one can work quite explicitly with the Riemann surfaceassociated to a permutation triple. Given a triple σ, the uniformizing group Γ is given as thestabilizer of 1 in the permutation representation Γ(2) → Sd given by x, y, z → σ0,σ1,σ∞ asin (4.3). A fundamental domain for Γ is given by Farey symbols [98], including a reductionalgorithm to this domain and a presentation for the group Γ together with a solution to theword problem in Γ. These algorithms have been implemented in the computer algebra systemsSage [137] (in a package for arithmetic subgroups defined by permutations, by Kurth, Loeffler,and Monien) and Magma [19] (by Verrill).Once the group Γ has been computed, and the curve X = Γ\H∗ is thereby described, theBelyı map is then simply given by the function

λ : X → X(2) ∼= P1,

so one immediately obtains an analytic description of Belyı map. In order to obtain explicitequations, one needs meromorphic functions on X, which is to say, meromorphic functions onH that are invariant under Γ.We are led to the following definition. Let Γ ≤ PSL2(Z) be a subgroup of finite index. Amodular form for Γ ≤ PSL2(Z) of weight k ∈ 2Z is a holomorphic function f : H → C suchthat

(4.4) f(γz) = (cz + d)kf(z) for all γ = ±

a bc d

∈ Γ

and such that the limit limz→c f(z) = f(c) exists for all cusps c ∈ Q ∪ ∞ = P1(Q) (with

the further technical condition that as z → ∞, we take only those paths that remain in abounded vertical strip). A cusp form is a modular form where f(c) = 0 for each cusp c. Thespace Sk(Γ) of cusp forms for Γ of weight k is a finite-dimensional C-vector space. If Γ istorsion-free or k = 2, then there is an isomorphism

(4.5)Sk(Γ)

∼−→ Ω

k/2(X)

f(z) → f(z) (dz)⊗k/2

where Ωk/2

(X) is the space of holomorphic differential (k/2)-forms on X. In any case,evaluation on a basis for Sk(Γ) defines a holomorphic map ϕ : X → Pr−1 where r =

dimC Sk(Γ), whenever r ≥ 1. Classical theory of curves yields a complete description of themap ϕ; for example, for generic X of genus g ≥ 3, taking k = 2 (i.e., a basis of holomorphic1-forms) gives a canonical embedding of X as an algebraic curve of degree 2g − 2 in Pg−1, bythe theorem of Max Noether.Selander–Strömbergsson [141] use this analytic method of modular forms to compute Belyımaps; this idea was already present in the original work of Atkin–Swinnerton-Dyer [5] andwas developed further by Hejhal [72] in the context of Maass forms. Starting with the analyticdescription of a subgroup Γ ≤ Γ(2), they compute a hyperelliptic model of a curve of genus2 from the knowledge of the space S2(Γ) of holomorphic cusp forms of weight 2 for Γ. These



cusp forms are approximated to a given precision by truncated q-expansions

(4.6) f(z) =N

n=0

anqn,

one for each equivalence class of cusp c and corresponding local parameter q under the actionof Γ. These expansions (4.6) have undetermined coefficients an ∈ C, and the equation (4.4)implies an approximate linear condition on these coefficients for any pair of Γ-equivalentpoints z, z. These linear equations can then be solved using the methods of numerical linearalgebra. This seems to work well in practice, and once complex approximations for the cuspforms are known, the approximate algebraic equations that they satisfy can be computed, sothat after a further Newton iteration and then lattice reduction one obtains an exact solution.Atkin–Swinnerton-Dyer say of this method [5, p. 8]:

From the viewpoint of numerical analysis, these equations are of course very ill-conditioned. The power series converge so rapidly that one must be careful not totake too many terms, and the equality conditions at adjacent points in a subdivisionof the sides are nearly equivalent. However, by judicious choice of the number ofterms in the power series and the number of subdivision points, for which we cangive no universal prescription, we have been able to determine the first 8 or socoefficients [...] with 7 significant figures in many cases.

Question 4.7. — Does this method give rise to an algorithm to compute Belyı maps? In

particular, is there an explicit estimate on the numerical stability of this method?

For Belyı maps such that the corresponding subgroup Γ is congruence, methods of modular

symbols [36, 149] can be used to determine the q-expansions of modular forms using exactmethods. The Galois groups of congruence covers are all subgroups of PGL2(Z/NZ) for someinteger N , though conversely not all such covers arise in this way; as we will see in the nextsubsection, since PSL2(Z) has elliptic points of order 2 and 3, a compatibility on the ordersof the ramification types is required. Indeed, “most” subgroups of finite index in PSL2(Z) (ina precise sense) are noncongruence [81].

Example 4.8. — To give a simple example, we consider one of the two (conjugacy classesof) noncongruence subgroups of index 7 of PSL2(Z), the smallest possible index for a non-congruence subgroup by Wohlfarht [170]. The cusp widths of this subgroup are 1 and 6. Theinformation on the cusps tells us that the ramification type of the Belyı map above ∞ is givenby (6, 1), whereas the indices above 0 (resp. 1) have to divide 3 (resp. 2). This forces thegenus of the dessin to equal 0, with ramification triple (6

111, 3211, 2311).

There are exactly two transitive covers with this ramification type, both with passport(0, G, (2311, 3211, 6111)). Here the monodromy group G is the Frobenius group of order 42;the two covers correspond to two choices of conjugacy classes of order 6 in G. For one suchchoice, we obtain the following unique solution up to conjugacy:

σ0 = (1 2)(3 4)(6 7), σ1 = (1 2 3)(4 5 6), σ∞ = (1 4 7 6 5 3).



A fundamental domain for the action of Γ = Γ7 is as follows.

s1 s1

s2 s2

s3 s3s4 s4 s5 s5

1

2

34 567

Label Coset Representative

1

1 0

0 1

2

0 −1

1 0

3

1 5

0 1

4

1 1

0 1

5

1 4

0 1

6

1 3

0 1

7

1 2

0 1

Label Side Pairing Element

s1

1 6

0 1

s2

1 0

1 1

s3

5 −6

1 −1

s4

3 −7

1 −2

s5

4 −17

1 −4

Figure 4.9: A fundamental domain and side pairing for Γ7 ≤ Γ(1) of index 7

We put the cusp of Γ(1) at t = ∞ and the elliptic point of order 3 (resp. 2) at t = 0 (resp.zt = 1). After this normalization, the q-expansion for the Hauptmodul t for Γ is given by

t(q) =1

ζ+ 0 +

9 +√−3

2134ζ +

−3− 5√−3

2235ζ2 +

1− 3√−3

2137ζ3 + . . .

where ζ = ηq1/6 and

η6 =310

77(−1494 + 3526

√−3).



From this, we compute using linear algebra the algebraic relationship between t(q) and j(q),expressing j(q) as a rational function in t(q) of degree 7:

j = −26(1 +

√−3)

(5−√−3)7

(54√−3t2 + 18

√−3t+ (5− 3

√−3))

3(6√−3t− (1 + 3

√−3))

(6√−3t− (1 + 3

√−3))

.

We will compute this example again using p-adic methods in the next section (Example 5.5).

Modular forms on subgroups of triangle groups. — There is related method thatworks with a cocompact discrete group Γ ≤ PSL2(R), reflecting different features of Belyımaps. Instead of taking the free group on two generators, corresponding to the fundamentalgroup of P1 \ 0, 1,∞, we instead consider orbifold covers arising from triangle groups, asubject of classical interest (see e.g. Magnus [105]). For an introduction to triangle groups,including their relationship to Belyı maps and dessins, see the surveys of Wolfart [172, 173].Let a, b, c ∈ Z≥2 ∪ ∞. We define the triangle group

∆(a, b, c) = δ0, δ1, δ∞ | δa0 = δb1 = δc∞ = δ0δ1δ∞ = 1

where infinite exponents a, b, c are ignored in the relations. Let χ(a, b, c) = 1/a+1/b+1/c−1 ∈

Q. For example, we have ∆(2, 3,∞) ∼= PSL2(Z) and ∆(∞,∞,∞) ∼= F2∼= Γ(2), so this

construction generalizes the previous section. The triangle group ∆(a, b, c) is the index 2

orientation-preserving subgroup of the group generated by the reflections in the sides of atriangle T (a, b, c) with angles π/a,π/b,π/c drawn in the geometry H, where H = P1,C,Haccording as χ(a, b, c) is positive, zero, or negative.Associated to a transitive permutation triple σ from Sd is a homomorphism

∆(a, b, c) → Sd

δ0, δ1, δ∞ → σ0,σ1,σ∞

where a, b, c ∈ Z≥2 are the orders of σ0,σ1,σ∞, respectively. (Here we have no index ∞, so∆(a, b, c) is cocompact, which is where this method diverges from that using classical modularforms.) The stabilizer of a point Γ ≤ ∆(a, b, c) has index d, and the above homomorphism isrecovered by the action of ∆ on the cosets of Γ. The quotient map

ϕ : X = Γ\H → ∆\H

then realizes the Belyı map with monodromy σ, so from this description we have a way ofconstructing the Belyı map associated to σ. More precisely, as in (4.1), the bijection (1.3)generalizes to

(4.10)

permutation triples σ = (σ0,σ1,σ∞) ∈ S3

dsuch that a, b, c are multiples of the orders of σ0,σ1,σ∞

/∼

1:1←→

subgroups of ∆(a, b, c) of index n /∼,

where the equivalences are as usual: conjugacy in the group ∆(a, b, c) and simultaneousconjugacy of triples (σ0,σ1,σ∞). (In particular, these triples are not marked, as by contrastthey are in (4.3), though certainly our construction could be modified in this way if so desired.)Explicitly, one obtains the Riemann surfaces corresponding to a subgroup Γ < ∆(a, b, c)under the bijection (4.10) by gluing together triangles T (a, b, c) and making identifications.



This gives a conformally correct way to draw dessins and a method for computing the coversthemselves numerically.This method has been developed in recent work of Klug–Musty–Schiavone–Voight [91]. Al-gorithms are provided for working with the corresponding triangle group ∆, determiningexplicitly the associated finite index subgroup Γ, and then drawing the dessin on H togetherwith the gluing relations that define the quotient X = Γ\H. From this explicit descriptionof the Riemann surface (or more precisely, Riemann 2-orbifold) X one obtains equations forthe Belyı map f numerically. The main algorithmic tool for this purpose is a generalizationof Hejhal’s method replacing q-expansions with power series expansions, due to Voight–Willis[164]. This method works quite well in practice; as an application, a Belyı map of degree 50

of genus 0 regularly realizing the group PSU3(F5) over Q(√−7) is computed.

Example 4.11. — Consider the permutation triple σ = (σ0,σ1,σ∞), whereσ0 = (1 7 4 2 8 5 9 6 3)

σ1 = (1 4 6 2 5 7 9 3 8)

σ∞ = (1 9 2)(3 4 5)(6 7 8).

Then σ0σ1σ∞ = 1 and these permutations generate a transitive subgroupG ∼= Z/3Z Z/3Z ≤ S9

of order 81 and give rise to a Belyı map with passport (0, G, (91, 91, 33)). The correspondinggroup Γ ≤ ∆(9, 9, 3) = ∆ of index 9 arising from (4.10) has signature (3;−), i.e., the quotientΓ\H is a (compact) Riemann surface of genus 3. The map X(Γ) = Γ\H → X(∆) = ∆\H ∼= P1

gives a Belyı map of degree 9, which we now compute.First, we compute a coset graph, the quotient of the Cayley graph for ∆ on the generatorsδ±0 , δ

±1 by Γ with vertices labelled with coset representatives Γαi for Γ\∆. Given a choice

of fundamental domain D∆ for ∆ (a fundamental triangle and its mirror, as above), sucha coset graph yields a fundamental domain DΓ =

ni=1 αiD∆ equipped with a side pairing,

indicating how the resulting Riemann orbifold is to be glued. We consider this setup in theunit disc D, identifying H conformally with D taking a vertex to the center w = 0; the resultis Figure 4.12. We obtain in this way a reduction algorithm that takes a point in z ∈ H (orD) and produces a representative z ∈ DΓ and γ ∈ Γ such that z = γz.We consider the space S2(Γ) of cusp forms of weight 2 for Γ, defined as in (4.4) (but notethat since no cusps are present we can omit the corresponding extra conditions). As in (4.5),we have an isomorphism S2(Γ)

∼= Ω1(X) of C-vector spaces with the space of holomorphic

1-forms on X. Since X has genus 3, we have dimC S2(Γ) = 3. We compute a basis of formsby considering power series expansions

f(w) = (1− w)2∞

n=0

bnwn

for f ∈ S2(Γ) around w = 0 in the unit disc D. (The presence of the factor (1 − w)2

makes for nicer expansions, as below.) We compute with precision = 10−30, and so

f(w) ≈ (1−w)2N

n=0 bnwn with N = 815. We use the Cauchy integral formula to isolate each

coefficient bn, integrating around a circle of radius ρ = 0.918711 encircling the fundamentaldomain. This integral is approximated by summing the evaluations at O(N) points on this



circle, which can be explictly represented by elements in the fundamental domain DΓ afterusing the reduction algorithm.

s1

s1

s2

s2

s3

s3

s4

s4

s5

s5

s6

s6

s7

s7s8

s8

s9

s9

×

×

×

×

×

×

×

×

×

1

2

3

4

5

6

7

8

9


1 1

2 δ3a3 δ−1

a

4 δ2a5 δ−4

a

6 δ−2a

7 δa8 δ4a9 δ−3

a


s1 δbδ−2a

s2 δ−1b δ−4

a

s3 δaδbδ3a

s4 δaδ−1b δ4a

s5 δ−1a δbδ

−4a

s6 δ−1a δ−1

b δ3as7 δ2aδbδ

2a

s8 δ−2a δbδ

−3a

s9 δ3aδbδ4a


1 12 δ303 δ−1

04 δ205 δ−4

06 δ−2

07 δ08 δ409 δ−3

0


s1 δ1δ−20

s2 δ−11 δ−4

0s3 δ0δ1δ

30

s4 δ0δ−11 δ40

s5 δ−10 δ1δ

−40

s6 δ−10 δ−1

1 δ30s7 δ20δ1δ

20

s8 δ−20 δ1δ

−30

s9 δ30δ1δ40

Figure 4.12: A fundamental domain and side pairing for Γ ≤ ∆(9, 9, 3) of index 9



We find the echelonized basis

x(w) = (1− w)21−

40

6!(Θw)6 +

3080

9!(Θw)9 −

1848000

12!(Θw)12 +O(w15

)

y(w) = (1− w)2(Θw) +

4

4!(Θw)4 +

280

7!(Θw)7 −

19880

10!(Θw)10 +O(w13

)

z(w) = (1− w)2(Θw)3 −

120

6!(Θw)6 −

10080

9!(Θw)9 −

2698080

12!(Θw)12 +O(w15

)

where Θ = 1.73179 . . . + 0.6303208 . . .√−1. The algebraicity and near integrality of these

coefficients are conjectural [91], so this expansion is only numerically correct, to the computedprecision.We now compute the image of the canonical map

X(Γ) = Γ\H → P2

w → (x(w) : y(w) : z(w));

we find a unique quartic relation

216x3z − 216xy3 + 36xz3 + 144y3z − 7z4 = 0

so at least numerically the curve X is nonhyperelliptic. Evaluating these power series atthe ramification points, we find that the unique point above f = 0 is (1 : 0 : 0), the pointabove f = 1 is (1/6 : 0 : 1), and the three points above f = ∞ are (0 : 1 : 0) and((−1± 3

√−3)/12 : 0 : 1).

The uniformizing map f : X(Γ) → X(∆) ∼= P1 is given by the reversion of an explicit ratioof hypergeometric functions:

f(w) = −1

8(Θw)9 −

11

1280(Θw)18 −

29543

66150400(Θw)27 +O(w36

).

Using linear algebra, we find the expression for f in terms of x, y, z:

f(w) =−27z3

216x3 − 108x2z + 18xz2 − 28z3.

Having performed this numerical calculation, we then verify on the curve X(Γ) that thisrational function defines a three-point cover with the above ramification points, as in Section 8.

An important feature of methods using modular forms is that it allows a much more directalgebraic approach to determining the algebraic structure on the target Riemann surface.There are no “parasitic” solutions to discard, just as when using the more advanced analyticmethod of Section 3. Moreover, the equation for the source surface are much easier to findthan with the analytic method, where one typically needs to compute period matrices to highprecision.

Question 4.13. — What are the advantages of the noncocompact (q-expansions) and co-compact (power series expansions) approaches relative to one another? How far (degree,

genus) can these methods be pushed? Can either of these methods be made rigorous?



5. p-adic methods

As an alternative to complex analytic methods, we can use p-adic methods to find a solution;in this section we survey this method, and give a rather elaborate example of how thisworks in practice. It is simply the p-adic version of the complex analytic method, withthe big distinction that finding a suitable approximation and then Hensel lifting can be mucheasier; usually finding a solution over a finite field suffices to guarantee convergence of Newtonapproximation.

Basic idea. — The p-adic method begins by finding a solution in a finite field of smallcardinality, typically by exhaustive methods, and then lifts this solution using p-adic Newtoniteration. Again, lattice methods can be then employed to recognize the solution over Q.Turning the ‘p-adic crank’, as it is called, has been a popular method, rediscovered many timesand employed in a number of contexts. Malle [109] used this method to compute polynomialswith Galois groups M22, Aut(M22), and PSL3(F4) : 2 over Q. Elkies [47] computed a degree28 cover f : X → P1 with group G = PSL2(F27) via its action on P1

(F27) modulo 29, andother work of Elkies [48], Watkins [166] and Elkies–Watkins [50] have also successfully usedp-adic methods to compute Belyı maps. Elkin–Siksek [51] used this method and tabulatedBelyı maps of small degree. Van Hoeij–Vidunas [157] used this approach to compute a listof examples whose branching is nearly regular, before extending the direct method [158] asexplained in Section 2. More recently, Bartholdi–Buff–von Bothmer–Kröker [7] computed aBelyı map in genus 0 that is of degree 13 and which arises in a problem of Cui in dynamicalsystems; they give a relatively complete description of each of the steps involved.A foundational result by Beckmann indicates which primes are primes of good reduction forthe Belyı map; which primes, therefore, can be used in the procedure above.

Theorem 5.1 (Beckmann [11]). — Let f : X → P1be a Belyı map and let G be the

monodromy group of f . Suppose that p #G. Then there exists a number field L such that pis unramified in L and f is defined over L with good reduction at all primes p of L lying over

p.

Remark 5.2. — In fact, Beckmann proves as a consequence that under the hypotheses ofthe theorem, the prime p is unramified in the field of moduli K of f . (For the definition ofthe field of moduli, see Section 7.)

If one works with a pointed cover instead, then the statement of Beckmann’s theorem issimpler [18, Theorem 3]. In the notation of this theorem, if p divides the order of one ofthe permutations σ then f has bad reduction at p [18, Theorem 4]. But for those p thatdivide #G but not any of the ramification indices, it is much harder to find methods (beyondexplicit calculation) to decide whether or not a model of f with good reduction over p exists.Important work in this direction is due to Raynaud [130] and Obus [126].

Question 5.3. — Can one perform a similar lifting procedure by determining solutions mod-

ulo primes where f has bad reduction?

As the matrix of derivatives of the equations used is almost always of full rank (see Section 2),the most time-consuming part is usually the search for a solution over a finite field. In order forthis method to be efficient, one must do better than simply running over the potential solutions



over Fq. Bartholdi–Buff–van Bothmer–Kröker describe [7, Algorithm 4.7] a more carefulmethod for genus 0, working directly with univariate polynomials (and rational functions)with coefficients in Fq. In the example below, we show an approach that is similar in spiritto theirs and that works for hyperelliptic curves as well.When the field of definition is “generic” in some sense, then there is often a split prime ofsmall norm, so this method is often efficient in practice. The following question still meritscloser investigation.

Question 5.4. — How efficiently can a Belyı map be computed modulo a prime p? How far

can one reduce the dimension of the affine space employed in the enumeration?

In particular, can a “partial projection” (partial Gröbner basis) be computed efficiently toreduce the number of looping variables?

Example 5.5. — We return to the Belyı maps with ramification type (6111, 3211, 2311)

considered in Example 4.8.Theorem 5.1 suggests to reduce modulo 5 first. We put the ramification type (6, 1) over ∞

and the corresponding points at ∞ and 0; we can do this without risking an extension of thefield of definition since these points are unique. In the same way, we put the type (3

2, 1) over0 and the single point in this fiber at 1. This defines a reasonably small system over F5 ofdimension 7, which could even be checked by enumeration. We get the solutions

f(t) =α8

(t− 2)3(t+ α)3(t− 1)

tand its conjugate, where α is a root of the Conway polynomial defining F52 over F5, i.e.,α2 − α+ 2 = 0. At the prime 13, we get two solutions defined over F13:

f(t) =−3(t2 + 3t+ 8)

3(t− 1)

t, f(t) =

2(t2 + 6)3(t− 1)

t.

In both cases, the derivative matrices of the equations (with or without ASD) are non-singular,so we can lift to the corresponding unramified p-adic fields. After a few iterations of the secondpair of solutions, we get the 13-adic approximations

f(t) = (−3− 5 · 13− 132+ . . . )(t− 1)t−1

· (t2 + (3 + 8 · 13− 2 · 132+ . . . )t+ (8− 3 · 13− 6 · 13

2+ . . . ))3

f(t) = (2− 3 · 13 + 3 · 132+ . . . )(t− 1)t−1

· (t2 + (−4 · 13 + 6 · 132+ . . . )t+ (6− 3 · 13

2+ . . . ))3.

We continue, with quadratically growing accuracy, in order to use LLL in the end. Thissuggests a pair of solutions over Q(

√−3) given by

f(t) =−1 +

√−3

4√−3

3(√−3 + 2)7

(162t2 + 18(−√−3− 6)t+ (

√−3 + 3))

3(t− 1)

t

and its conjugate. One verifies as in Section 8 that this yields a solution over Q(√−3) to

the given equations and that they are the requested Belyı maps. Though we stop here, onecould further simplify the equation even further by suitable scalar multiplications in t, or evenbetter, the general methods described in Section 8.



Example 5.6. — We now illustrate the complexities involved in employing the above methodin an example. It arose during a study of Galois Belyı maps with monodromy group PSL2(Fq)

or PGL2(Fq), undertaken by Clark–Voight [25].Consider the passport with uniform ramification orders 3, 5, 6 and monodromy group G =

PSL2(F11) ≤ S11. Here the embedding of G in S11 results from its conjugation action on thecosets of its exceptional subgroup A5 (and indeed #G/#A5 = 660/60 = 11).Let f : E → P1 be the degree 11 Belyı map defined by the above data, and let ϕ : X → P1

be its Galois closure, with Galois group G. We anticipate [25] that ϕ with its Galois actionis defined over an at most quadratic extension of Q(

√3,√5), in which case by the Galois

correspondence the quotient map f will be defined over the same field. We confirm this bydirect computation.Using the representation of G above, we find that f has passport

(1,PSL2(F11), (3312, 5211, 613121));

in accordance with the construction above, the ramification orders are divisors of 3, 5, 6, andE has genus 1.We distinguish the point of ramification degree 6 above ∞ and obtain a corresponding grouplaw on E. We fix two more points by taking the other points above ∞ (with ramification 3

and 2, respectively) to be (0, 1) and (1, y1). We write the equation

y2 = π3x3+ π2x

2+ (y21 − π3 − π2 − 1)x+ 1 = π(x)

for the curve E. The Belyı function f has the form

f(x, y) =q(x) + r(x)y

(x− 1)2x3

where q(x) = q8x8 + · · ·+ q0 and r(x) = r6x6 + · · ·+ r0 have degree 8, 6 respectively and thenumerator fnum(x, y) = q(x) + r(x)y vanishes to degree 3 at (0,−1) and 2 at (0,−y1).By the ramification description above 0, we must have

(5.7)NQ(x,y)/Q(x)(fnum(x, y)) = q(x)2 − r(x)2π(x)

= q28x3(x− 1)

2s(x)3t(x)

where s(x) = x3 + s2x2 + s1x+ s0 and t(x) = x2 + t1x+ t0, and similarly above 1 we shouldhave

(5.8)NQ(x,y)/Q(x)((f(x, y)− 1)num) = (q(x)− (x− 1)

2x3)2 − r(x)2π(x)

= q28x3(x− 1)

2u(x)v(x)

where u(x) = x2 + u1x+ u0 and v(x) = x+ v0.An approach using Gröbner basis techniques utterly fails here, given the number of variablesinvolved. This calculation is also made more difficult by the possibility that other Belyıcovers will intervene: the Mathieu group M11 → S11 also has a (3, 5, 6) triple of genus 1,and it is a priori conceivable that S11 occurs as well. Discarding these parasitic solutions is anontrivial task until one has already computed all of them along with the correct ones, justas in Section 2.As explained above, we search for a solution in a finite field Fq, lift such a solution usingHensel’s lemma (if it applies), and then attempt to recognize the solution p-adically as an



algebraic number using the LLL lattice reduction algorithm. The primes of smallest normin the field Q(

√3,√5) that are relatively prime to #PSL2(F11) have norm q = 49, 59, so

there is no hope of simply running over all the Fq-rational values in the affine space iny1,π, q, r, s, t, u, v, which is 28-dimensional.We speed up the search with a few tricks. Subtracting the two equations (5.7)–(5.8), we have

q28s(x)3t(x)− 2q(x) + (x− 1)

2x3 = r28u(x)5v(x).

Comparing coefficients on both sides, by degree we see that the coefficients of x9 and x10 ofs(x)3t(x) and u(x)5v(x) must agree. So we precompute a table of the possible polynomials ofthe form u(x)5v(x); there are O(q3) such, and we sort them for easy table lookup. Then, foreach of the possible polynomials of the form s(x)3t(x), of which there are O(q5), we matchthe above coefficients. Typically there are few matches. Then for each q28 ∈ F×2

q , we computeq(x) as

q(x) =1

2

q28s(x)

3t(x)− q28u(x)5v(x)− (x− 1)

2x3.

From equation (5.7) we have

q(x)2 − q28(x− 1)2x3s(x)3t(x) = π(x)r(x)2,

so we compute the polynomial on the right and factor it into squarefree parts. If thecorresponding π(x) has degree 3, then we find r(x) as well, whence also our solution.Putting this on a cluster at the Vermont Advanced Computing Center (VACC) using Magma

[19], after a few days we have our answer. We find several solutions in F49 but only one solutionlifts p-adically without additional effort; it turns out the Jacobian of the corresponding systemof equations is not of full rank. After some effort (see also Section 8), we recognize this coveras an M11-cover with ramification (3, 5, 6), defined over the number field Q(α) where

α7− α6

− 8α5+ 21α4

+ 6α3− 90α2

+ 60α+ 60 = 0.

We find 62 solutions in F59. Note that the M11-covers above do not reappear since there is noprime of norm 59 in Q(α). Only 8 of these solutions yield covers with the correct ramificationdata; our above conditions are necessary, but not sufficient, as we have only considered thex-coordinates and not the y-coordinates. These 8 covers lift to a single Galois orbit of curvesdefined over the field Q(

√3,√5,√b) where

b = 4

√3 +

11 +√5

2;

with N(b) = 112; more elegantly, the extension of Q(

√3,√5) is given by a root β of the

equation

T 2−

1 +√5

2T − (

√3 + 1) = 0.



The elliptic curve E has minimal model

y2 + ((12(13

√5 + 33)

√3 +

12(25

√5 + 65))β + (

12(15

√5 + 37)

√3 + (12

√5 + 30)))xy

+ (((8

√5 + 15)

√3 +

12(31

√5 + 59))β + (

12(13

√5 + 47)

√3 +

12(21

√5 + 77)))y

= x3 + ((12(5

√5 + 7)

√3 +

12(11

√5 + 19))β + (

12(3

√5 + 17)

√3 + (2

√5 + 15)))x2

+ ((12(20828483

√5 + 46584927)

√3 +

12(36075985

√5 + 80687449))β

+ (12(21480319

√5 + 48017585)

√3 +

12(37205009

√5 + 83168909)))x

+ (((43904530993

√5 + 98173054995)

√3 +

12(152089756713

√5 + 340081438345))β

+ ((45275857298

√5 + 101240533364)

√3 + (78420085205

√5 + 175353747591))).

The j-invariant of E generates the field Q(√3,√5,β), so this is its minimal field of definition.

This confirms that ϕ : X → P1 as a G-cover is defined over an at most quadratic extensionof Q(

√3,√5) contained in the ray class field of conductor 11∞, as predicted by the results of

Clark–Voight [25].

6. Galois Belyı maps

In this short section we sketch some approaches for calculating Galois Belyı maps, i.e.,those Belyı maps f : X → P1 corresponding to Galois extensions of function fields. Theflavor of these computations is completely different from those in the other sections, as therepresentation-theoretic properties of the Galois group involved are used heavily. In light ofthe Galois correspondence, all Belyı maps are essentially known once the Galois Belyı mapsare known; however, the growth in degree between the degree of the Belyı map and that ofits Galois closure makes it very difficult in general to make this remark a feasible approachto computing general Belyı maps. We therefore consider the subject only in itself, and evenhere we limit ourselves to the general idea: exploiting representations and finding invariantfunctions.The Galois Belyı maps in genus 0 correspond to the regular solids, and can be computed usingthe direct method (see the end of Section 2). The most difficult case, that of the icosahedron,was calculated first by Klein [90]. The Galois Belyı maps in genus 1 only occur on curveswith CM by either Q(

√−3) or Q(

√−1), and can therefore be calculated by using explicit

formulas for isogenies; see work of Singerman–Syddall [147].So it remains to consider the case of genus ≥ 2, where Belyı maps are related with hyperbolictriangle groups (see Section 4). In genus ≥ 2, Wolfart [173] has shown that Galois Belyımaps can be identified with quotient maps of curves with many automorphisms, that is, thosecurves that do not allow nontrivial deformations that leave the automorphism group intactand whose automorphism group therefore defines a zero-dimensional subscheme of the modulispace of curves Mg of genus g ≥ 2. Wolfart [171] compares these Belyı maps with therelated phenomenon of Jacobians of CM type, which define zero-dimensional subschemes ofthe moduli space of principally polarized abelian varieties Ag. In particular, the CM factorsof the Jacobians of the Galois Belyı curves are essentially known; they come from Fermatcurves [171, §4].



A fundamental technique for proving these theorems is to determine the representation ofthe automorphism group on the space of differentials, first considered by Chevalley and Weil[24]; this is elaborated by Berry–Tretkoff [14] and Streit [150]. Once this is done, onetypically recovers the curve by determining the shape of its canonical embedding, often anintersection of quadrics. (When the canonical embedding is not injective, the situation is evensimpler; since the hyperelliptic involution is central in the automorphism group, this reduceto the calculations in genus 0 mentioned above.) The particular form of the equations is thendetermined by being fixed under the action of the automorphism group, which acts by lineartransformations.

Question 6.1. — Can the representation of the automorphism group G on the space of

differentials be used to give a rigorous algorithm for the computation of G-Galois Belyı maps

(with a bound on the running time)?

Put another way, computing a Galois Belyı map amounts to determining G-invariant poly-nomials of a given degree; in some cases, there is a unique such polynomial with given degreeand number of variables, and so it can be found without any computation.

Example 6.2. — We illustrate the invariant theory involved by giving an example of acalculation of a quotient map X → X/Aut(X) ∼= P1 that turns out to be a Belyı map;the example was suggested to us by Elkies.Consider the genus 9 curve X defined by the following variant of the Bring equations:

v + w + x+ y + z = 0,

v2 + w2+ x2 + y2 + z2 = 0,

v4 + w4+ x4 + y4 + z4 = 0.

This curve is known as Fricke’s octavic curve, and it was studied extensively by Edge [44].There is an obvious linear action by S5 on this curve by permutation of coordinates. To findcoordinates on the quotient X/S5 it therefore suffices to look at the symmetric functions inthe variables v, w, x, y, z. We see that the power sums with exponents 1, 2, 4 vanish on X.Since the ring of invariants function for S5 is generated by the power sums of degree at most5, this suggests that we cook up a function from the power sums p3 and p5 of degree 3 and 5.These functions do not have the same (homogeneous) degree; to get a well-defined function,we consider their quotient f = (p53 : p

35) as a morphism from f : X → P1.

The intersection of the hyperplanes defined by p3 = 0 and p5 = 0 with X are finite; indeed, thisis obvious since the corresponding functions do not vanish indentically on X. By Bézout’stheorem, these zero loci are of degrees 24, 40. But whereas in the former case one indeedobtains 24 distinct geometric points in the intersection, one obtains only 20 geometric pointsin the latter case. This shows that the ramification indices over 0 and ∞ of the degree 120

morphism f are 6 and 5.This is in fact already enough to conclude that there is only one other branch point for q.Indeed, the orbifold X/Aut(X) is uniformized by the upper half plane H since the genus 9

curve X is, so X/Aut(X) is a projective line with at least 3 branch points for the quotientby the action of S5. On the other hand, the Riemann–Hurwitz formula shows that adding asingle minimal contribution of 2 outside the contributions 5 and 6 already known from ∞ and



0 already makes the genus grow to 9, so additional ramification is impossible. The additionalbranch point of f can be found by considering the divisor of df on X; this point turns out tobe −(15/2)2. So the morphism f : X → P1 defined by

f(v, w, x, y, z) =−2

2p53152p35

= −

2

15

2(v3 + w3

+ x3 + y3 + z3)5

(v5 + w5 + x5 + y5 + z5)3

realizes the quotient X → X/S5 as a Belyı map. Moreover, we see that the Galois action isdefined over Q, since it is given by permuting the given coordinate functions on X.In fact we have an isomorphism Aut(X) ∼= S5 since Aut(X) cannot be bigger than S5; sucha proper inclusion would give rise to a Fuchsian group properly containing the triangle group∆(2, 5, 6), whereas on the other hand this group is maximal (by work of Takeuchi [153], ormore generally see Singerman [146] or Greenberg [63, Theorem 3B]).We therefore have found a Galois cover realizing S5 with ramification indices 2, 5, 6. Itturns out that this is the only such cover up to isomorphism. Considering the exceptionalisomorphism PGL2(F5)

∼= S5, we see that our calculation also yields a Galois cover realizinga projective linear group.

7. Field of moduli and field of definition

Considering Grothendieck’s original motivation for studying dessins, it is important to con-sider the rather delicate issue of fields of definition of Belyı maps. In fact this is not only anengaging question on a theoretic level, but it is also interesting from a practical point of view.Indeed, as we have seen in our calculations above, it is often necessary to determine equationsfor Belyı maps by recognizing complex numbers as algebraic numbers. A bound on the degreeK is an important part of the input to the LLL algorithm that is typically used for this.Moreover, having an estimate for the degree of K is a good indication of how computable agiven cover will be—if the estimate for the size is enormous, we are very unlikely to succeedin practice!

Field of moduli. — For a curve X defined over Q, the field of moduli M(X) of X is thefixed field of the group τ ∈ Gal(Q/Q) : Xτ ∼= X on Q, where as before Xτ is the basechange of X by the automorphism τ ∈ Gal(Q/Q) (obtained by applying the automorphismτ to the defining equations of an algebraic model of X over Q). One similarly defines thefield of moduli of a Belyı map: M(f) is the fixed field of τ ∈ Gal(Q/Q) : f τ ∼= f withisomorphisms as defined in Section 1.Now let f : X → P1 be a Belyı map with monodromy representation σ : F2 → Sd andmonodromy group G. By Theorem 1.6, the monodromy group G of f , considered as aconjugacy class of subgroups of Sd, is invariant under the Galois action. Therefore, givenτ ∈ Gal(Q/Q), the conjugated morphism f τ

: Xτ → P1 is a Belyı map, and its monodromyrepresentation στ

: F2 → Sd, which is well-defined up to conjugation, can be taken to haveimage G. Because the Galois action preserves the monodromy group up to conjugation andthe ramification indices [84], the Belyı map f τ has the same passport P as f . We thereforeget an action of Gal(Q/Q) on the set S of Belyı maps with passport P . Since the stabilizerof an element of S under this action has index at most #S in Gal(Q/Q), we get the followingresult.



Proposition 7.1. — Let f be a Belyı map with passport P and field of moduli K. Then the

degree [K : Q] is bounded above by the size of P .

As mentioned at the end of Section 1, finding better bounds than in Proposition 7.1 is farfrom trivial and a subject of ongoing research. Experimentally, the bound is often an equalityfor generic (non-Galois) Belyı maps.

Rigidified categories. — Working with Galois Belyı maps and the additional structurecoming from their automorphism group naturally leads one to consider a new, more rigidifiedcategory [38]. A G-Belyı map is a pair (f, i), where f : X → P1 is a Galois Belyı map andi : G

∼−→ Mon(f) is an isomorphism of the monodromy group of f with G. A morphism of

G-Belyı maps from (f, i) to (f , i) is an isomorphism of Belyı maps h : X∼−→ X that identifies

i with i, i.e., such that

(7.2) h(i(g)x) = i(g)h(x) for all g ∈ G and x ∈ X.

A G-permutation triple is a triple of permutations (σ0,σ1,σ∞) in G such that σ0σ1σ∞ = 1

and such that σ0,σ1,σ∞ generate G. A morphism of G-permutation triples is an isomorphismof permutation triples induced by simultaneous conjugation by an element in G. The mainequivalence is now as follows.


(i) G-Belyı maps;

(ii) G-permutation triples;

(iii) surjective homomorphisms F2 → G.

We leave it to the reader to similarly rigidify the notion of dessins; it will not be needed inwhat follows.We will need a slight weakening of this notion in the following section. A weak G-Belyı map isa pair (f, i), where f : X → P1 is a Galois Belyı map and i : H → Mon(f) is an isomorphismof the monodromy group of f with a subgroup H of G. A morphism of weak G-Belyı mapsfrom (f, i) to (f , i) is an isomorphism of Belyı maps h : X

∼−→ X such that (7.2) holds up

to conjugation, i.e., such that there exists a t ∈ G such that h(i(g)x) = i(tgt−1)h(x) for all

g ∈ G and x ∈ X.A weak G-permutation triple is a triple of permutations (σ0,σ1,σ∞) in G such that σ0σ1σ∞ = 1.A morphism of weak G-permutation triples is an isomorphism of permutation triples inducedby simultaneous conjugation by an element in G. The equivalence of Proposition 7.3 nowgeneralizes to the following result.


(i) weak G-Belyı maps;

(ii) weak G-permutation triples;

(iii) homomorphisms F2 → G.

The set of Belyı maps of degree d can be identified with the set of weak Sd-Belyı maps. Inparticular, whereas G-Belyı maps are always connected, weak G-Belyı maps need not be.



The absolute Galois group Gal(Q/Q) acts on the set of (weak) G-Belyı maps, so we can againdefine the field of moduli of these rigidified Belyı maps.Having introduced weak G-Belyı maps, it makes sense to consider passports up to the actionof the monodromy group G ⊂ Sd instead of the full group Sd. We accordingly define therefined passport of a (not necessarily Galois) Belyı map f : X → P1 to be the triple (g,G,C),where g is the genus of X, the group G is the monodromy group of f , and C = (C0, C1, C∞)

are the conjugacy classes of σ0,σ1,σ∞ in G, not the conjugacy classes in Sd included in the(usual) passport.Fried [57] shows how the conjugacy classes Ci change under the Galois action. Let σ ∈

Gal(Q/Q), let n = #G, and let ζn ∈ Q be a primitive n-th root of unity. Then σ sends ζn toζan for some a ∈ (Z/nZ)×. We obtain new conjugacy classes Ca

i by raising a representative ofCi to the ath power. Then for any character χ of G we have

σ(χ(Ci)) = χ(Cai ).(7.5)

Let Q(χ(Ci)) be the field generated by the character values of the conjugacy classes Ci. Wehave Q(χ(Ci)) = Q if and only if all conjugacy classes of G are rational, as for instance in thecase G = Sd.

Proposition 7.6. — Let (f, i) be a weak G-Belyı map with refined passport R and field of

moduli K as a weak G-Belyı map. Then the degree [K : Q(χ(Ci))] is bounded above by the

size of R.

Calculating in the category of weak G-Belyı maps can be useful even when considering Belyımaps without this additional structure. More precisely, this is useful when using formulasthat approximate the size of a passport. To this end, let G be a finite group and let C0, C1, C∞be conjugacy classes in G. Let S be the set of isomorphism classes of weak G-Belyı mapsσ = (σ0,σ1,σ∞) with the property that σi ∈ Ci for i ∈ 0, 1,∞. Then a formula that goesback to Frobenius (see Serre [142, Theorem 7.2.1]) shows that

(f,i)∈S

1

AutG(f, i)=

#C0#C1#C∞(#G)2

χ

χ(C0)χ(C1)χ(C∞)

χ(1).(7.7)

Here the automorphism group AutG(f, i) is the group of automorphisms of (f, i) as a weak G-Belyı map. The sum on the left of (7.7) runs over all weak Belyı maps with the aforementionedproperty; in particular, one often obtains non-transitive solutions that one does not care aboutin practice.When working with mere Belyı maps (without rigidification as a weak G-Belyı map), it canstill be useful to consider the estimate (7.7) when the monodromy group of the Belyı map isquestion is included in G. We illustrate this by a few concrete examples.

Example 7.8. — We start by taking G to be a full symmetric group and give the above-mentioned estimate for the number of genus 0 Belyı maps with ramification passport

(0, (3223, 51412111, 614121)),

Before giving it, we calculate the possible permutation triples up to conjugacy directly usingLemma 1.7. This shows that the number of solutions is 583, of which 560 are transitive. The



transitive solutions all have monodromy group S23 and hence trivial automorphism group.On the other hand, the Serre estimate (7.7) equals 5671

4 , which more precisely decomposes as

5671

4=

560

1+

1

1+

3

2+

19

4;

of the 23 nontransitive solutions, there is only one with trivial automorphism group, whereasthere are 3 (resp. 19) with automorphism group of cardinality 2 (resp. 4). For each of thenontransitive solutions, the associated Belyı maps are disjoint unions of curves of genus 1,such as those corresponding to the products of the genus 1 Belyı maps with ramification types(2

3, 5111, 61) (which always have trivial automorphism group) with those with ramificationtypes (3

2, 4121, 4121) (which have either 1 or 2 automorphisms, depending on the solution).

Example 7.9. — Another example is the case (0, H, (442213, 442213, 5413)) with H ≤ M23.We can identify M23-conjugacy classes with S23-conjugacy classes for these groups, as theconjugacy classes of S23 do not split upon passing to M23.The calculations are much more rapid working with M23 than for the full group S23. We obtainthe estimate 909, which fortunately enough equals the exact number of solutions because thecorresponding subgroups of M23 all have trivial centralizer; this is not the case when theyare considered as subgroups of S23. Of these many solutions, it turns out that only 104 aretransitive.As mentioned before, this estimate only gives the number of weak M23-Belyı maps; accord-ingly, permutation triples are only considered isomorphic if they are conjugated by an elementof M23 rather than S23. However, since M23 coincides with its own normalizer in S23, thiscoincides with the number of solutions under the usual equivalence. Directly working withthe group M23 indeed saves a great deal of computational overhead in this case.

An explicit (but complicated) formula, using Möbius inversion to deal with the disconnectedBelyı maps, was given by Mednykh [121] this vein; in fact, his formula can be used to countcovers with specified ramification type of an arbitrary Riemann surface.Finally, we mention that in the Galois case, the situation sometimes simplifies: there arecriteria [27, 85] for Galois Belyı maps to have cyclotomic fields of moduli, in which casethe Galois action is described by a simple powering process known as Wilson’s operations.Additionally, Streit–Wolfart [152] have calculated the field of moduli of an infinite family ofBelyı maps whose Galois group is a semidirect product ZpZq of cyclic groups of prime order.

Field of moduli versus field of definition. — We have seen that in all the categoriesof objects over Q considered so far (curves, Belyı maps, etc.) there is a field of moduli forthe action of Gal(Q/Q). Given an object Y of such a category with field of moduli M , itis reasonable to ask whether Y is defined over M , i.e., if there exists an object YM in theappropriate category over M that is isomorphic with Y over Q, in which case M is said to bea field of definition of Y . For example, if Y = (X, f) is a Belyı map over Q, this means thatthere should exist a curve XM over M and a Belyı map fM : XM → P1

M such that (X, f) canbe obtained from (XM , fM ) by extending scalars to Q.We first consider the case of curves. Curves of genus at most 1 are defined over their field ofmoduli. But this ceases to be the case for curves of larger genus in general, as was alreadyobserved by Earle [42] and Shimura [145]. The same is true for Belyı maps and G-Belyı maps.



This issue is a delicate one, and for more information, we refer to work of Coombes–Harbater[28], Dèbes–Ensalem [39], Dèbes–Douai, [38], and Köck [92].The obstruction can be characterized as a lack of rigidification. For example, a curve furnishedwith an embedding into projective space is trivially defined over its field of moduli (as aprojectively embedded curve). Additionally, marking a point on the source X of a Belyı mapand passing to the appropriate category, [18, Theorem 2] states that the field of moduli isa field of definition for the pointed Belyı curve [18, Theorem 2]; however, this issue seemsquite subtle, and in [92] only auxiliary points with trivial stabilizer in Aut(X) are used.Note that the more inclusive version of this rigidification (with possibly non-trivial stabilizer)was considered in Section 4 (see e.g. (4.3)). As mentioned at the beginning of the previoussubsection, this implication can then be applied to give an upper bound on the degree of thefield of definition of a Belyı map, an important bit of information needed when for exampleapplying LLL to recognize coefficients algebraically.Note that for a Belyı map f : X → P1, the curve X may descend to its field of moduli inthe category of curves while f does not descend to this same field of moduli in the categoryof Belyı maps. Indeed, this can be seen already for the example X = P1, as Gal(Q/Q) actsfaithfully on the set of genus 0 dessins. In general, the problem requires careful considerationof obstructions that lie in certain Galois cohomology groups [38].

Remark 7.10. — Although in general we will have to contend with arbitrarily delicateautomorphism groups, Couveignes [31] proved that every curve defined over a number field Kadmits a Belyı map without automorphisms defined over K. This map will then necessarilynot be isomorphic to any of its proper conjugates.

On top of all this, a Belyı map may descend to its field of moduli in the weak sense, i.e., as acover of a possibly non-trivial conic ramified above a Galois-stable set of three points, ratherthan in the strong sense, as a cover of P1\ 0, 1,∞ (i.e., in the category of Belyı maps overthe field of moduli). This distinction also measures the descent obstruction for hyperellipticcurves, as in work of Lercier–Ritzenthaler–Sijsling [103]. For Belyı maps, a deep study ofthis problem in genus 0, beyond the general theory, was undertaken by Couveignes [29, §§4–7]: he shows that for the clean trees, those Belyı maps with a single point over ∞ and onlyramification index 2 over 1, on the set of which Gal(Q/Q) acts faithfully, the field of moduliis always a field of definition in the strong sense. Moreover, he shows that in genus 0, thefield of moduli is always a field of definition in the weak sense as long as the automorphismgroup of the Belyı map is not cyclic of even order, and in the strong sense as long as theautomorphism group is not cyclic.These considerations have practical value in the context of computations. For example,Couveignes [29, §10] first exhibits a genus 0 Belyı map that descends explicitly to Q in thestrong sense. Then, due to the presence of non-trivial automorphisms of this Belyı map, onecan realize it as a morphism f : C → P1 for infinitely many mutually non-isomorphic conics Cover Q. And by choosing C appropriately (not isomorphic to P1 over Q), Couveignes managesto condense his equations from half a page to a few lines. Further simplification techniqueswill be considered in the next section.We mention some results on the field of moduli as a field of definition that are most usefulfor generic (G-)Belyı maps.



1. If a curve or (G-)Belyı map has trivial automorphism group, then it can be defined overits field of moduli, by Weil’s criterion for descent [167].

2. If the center of the monodromy group of a Galois Belyı map is trivial, then it can bedefined over its field of moduli by the main result in the article by Dèbes–Douai [38].

3. A G-Belyı map, when considered in the category of Belyı maps (without extra structure)is defined over its field of moduli as a Belyı map [28].

To give an impression of the subtleties involved, we further elaborate on Example 6.2 fromthe previous section. Along the way, we will illustrate some of the subtleties that arise whenconsidering fields of moduli. As we will see, these subtleties correspond with very naturalquestions on the level of computation.

Example 7.11. — Since ∆(3, 5, 5) is a subgroup of ∆(2, 5, 6) of index 2, we also obtain fromthis example a Belyı map with indices 3, 5, 5 for the group A5 by taking the correspondingquotient. Indeed, ramification can only occur over the points of order 2 and 6, which meansthat in fact the cover is a cyclic degree 2 map of conics ramifying of order 2 over these pointsand under which the point of order 5 has two preimages. An equation for this cover (whichis a Belyı map) can now be found by drawing an appropriate square root of the function(s53/s

35)+ (15/2)2 (which indeed ramifies of order 6 over ∞ and of order 2 over 0) and sending

the resulting preimages ±15/2 of the point of order 5 to 0 and 1, respectively.Alternatively, we can calculate as follows. The full ring of invariant homogeneous polynomi-als for A5 (acting linearly by permutation of coordinates) is generated by the power sumsp1, . . . , p5 and the Vandermonde polynomial

a = (v − w)(v − x)(v − y)(v − z)(w − x)(w − y)(w − z)(x− y)(x− z)(y − z).

One easily determines the expression for a2 in terms of the pi; setting p1 = p2 = p4 = 0, weget the relation

a2 =4

45s53s5 + 5s35.

This suggests that to get a function realizing the quotient X → X/A5, we take the mapg : X → C, where C is the conic

C : 45y2 = 4xz + 225z2

and g is given by

g(v, w, x, y, z) = (s53 : as5 : s35).

Note that Q admits the rational point (1 : 0 : 0).This result is not as strong as one would like. As we have seen when calculating the fullquotient f , the branch points of g of order 5 on C satisfy (x : z) = (0 : 1). But thecorresponding points are only defined over Q(

√5), so this is a descent of a Belyı map in the

weak sense. We explain at the group-theoretical level what other kinds of descent can beexpected.There are actually two Galois covers with ramification indices (3, 5, 5) for A5 up to isomor-phism. The other cover is not found as a subcover of f ; when composing with the samequadratic map, we instead get a Galois Belyı map whose Galois group is the direct product



of A5 and Z/2Z. The corresponding curve is given by taking the hyperelliptic cover ramifiedover the vertices of an icosahedron, leading to the equation

t2 = s20 + 228s15 + 494s10 − 228s5 + 1.

In particular, this means that the Galois orbit of these covers consists of a single isomorphismclass, as their monodromy groups upon composition differ [174]. As mentioned above, anA5-Belyı map, considered as a mere Belyı map, is defined over its field of moduli as a Belyımap, so our equations above can be twisted to a Belyı map over Q, that is, with ramificationat three rational points.However, the Galois cover does not descend as an A5-Belyı map (so in the strong sense, as aGalois cover unramified outside 0, 1,∞). Indeed, the character table of A5 is only definedover Q(

√5). Twisting may therefore give a cover defined over Q, but the Galois action will

then only be defined over Q(√5) and be accordingly more complicated. We therefore forgo

this calculation and content ourselves with the symmetric form above.

For more on the questions considered in this section, see also further work by Couveignes [30],and in a similar vein, the work of van Hoeij–Vidunas on covers of conics [158, §§3.3–3.4], [157,§4]. We again refer to the fundamental paper of Dèbes–Douai [38], in which strong resultsare given for both Belyı maps and G-Belyı maps that suffice in many concrete situations.Admittedly, this subject is a delicate one, and we hope that computations will help to furtherclarify these nuances.

8. Simplification and verification

Once a potential model for a Belyı map has been computed, it often remains to simplify themodel as much as possible and to verify its correctness (independently of the method usedto compute it). The former problem is still open in general; the latter has been solved to asatisfactory extent.

Simplification. — By simplifying a Belyı map f : X → P1, we mean to reduce the total(bit) size of the model. Lacking a general method for doing this, we focus on the following:

1. If X is of genus 0, we mean to find a coordinate on X that decreases the (bit) size ofthe defining coefficients of f .

2. If X is of strictly positive genus, we mean to simplify the defining equations for X. (Inpractice, this will also lead to simpler coefficients of the Belyı map f .)

Problem (1) was considered by van Hoeij–Vidunas [158, §4.2] under the hypotheses thatone of the ramification points has a minimal polynomial of degree at most 4; one tries tofind a smaller polynomial defining the associated number field and changes the coordinateaccordingly, which typically yields one a simpler expression of the Belyı map.Problem (1) is directly related with Problem (2) for hyperelliptic curves, since simplifying theequations for hyperelliptic curves over a field K boils down to finding a small representative ofthe GL2(K)-orbit of a binary form. Typically one also requires the defining equation to haveintegral coefficients. For the case K = Q, this leads one to consider the problem of findingsimpler representations for binary forms under the action of the group of integral matricesSL2(Z). This is considered by Cremona–Stoll [37], using results from Julia [87] to find a binary



quadratic covariant, to which classical reduction algorithms are then applied. The resultingalgorithms substantially reduce the height of the coefficients of the binary form in practice,typically at least halving the bit size of already good approximations in the applications [37].A generalization to, and implementation for, totally real fields is given in Bouyer–Streng [21].In fact, corresponding results for the simplification of Belyı maps can be obtained by taking thebinary form to be the product of the numerator and denominator of the Belyı map. That theresulting binary form may have double roots and hence may not correspond to hyperellipticcurves is no problem; see the discussion by Cremona–Stoll [37, after Proposition 4.5].This problem of reduction is intimately related with the problem of finding a good model of aBelyı map or hyperelliptic curve over Z. Note that even for the case K = Q we have not yetused the full group GL2(Q); the transformations in SL2(Z) considered by Cremona and Stollpreserve the discriminant, but it could be possible that a suitable rational transformationdecreases this quantity while still preserving integrality of the binary form. An approach tothis problem is given by Bouyer–Streng [21, §3.3].In general, Problem (2) is much harder, if only because curves of high genus become moredifficult to write down.

Question 8.1. — Are there general methods to simplifying equations of curves defined over

a number field in practice?

Verification. — Let f : X → P1 be a map defined over a number field K of degree d thatwe suspect to be a Belyı map of monodromy group G, or more precisely to correspond to agiven permutation triple σ or a given dessin D. To show that this is in fact the case, we haveto verify that

(i) f is indeed a Belyı map;(ii) f has monodromy group G; and(iii) f (or its monodromy representation) corresponds to the permutation triple σ; or(iii)’ the pullback under f of the closed interval [0, 1] is isomorphic (as a dessin) to D.This verification step is necessary for all known methods, and especially when using the directmethod from Section 2; the presence of parasitic solutions means that not even all solutions ofthe corresponding system of equations will be Belyı maps, let alone Belyı maps with correctmonodromy group or pullback.Point (i) can be computationally expensive, but it can be accomplished, by using the methodsof computational algebraic geometry. Not even if X = P1 is this point trivial, since althoughverifying that a Belyı map is returned is easy for dessins of small degree, we need bettermethods than direct factorization of the polynomials involved as the degree mounts.As for point (ii), one simple sanity check is to take a field of definition K for f and then tosubstitute different K-rational values of t ∈ 0, 1,∞. One obtains an algebra that is again anextension of K of degree d and whose Galois group H must be a subgroup of the monodromygroup G by an elementary specialization argument. So if we are given a finite number ofcovers, only one of which has the desired monodromy group G, then to eliminate a cover inthe given list it suffices to show that specializing this cover gives a set of cycle type in H thatis not contained in the given monodromy group G when considered as a subgroup of Sd. Suchcycle types can be obtained by factoring the polynomial modulo a small prime of K.



There are many methods to compute Galois groups effectively in this way; a general methodis given by Fieker–Klüners [54]. This method proceeds by computing the maximal subgroupsof Sd and checking if the Galois group lies in one of these subgroups by evaluating explicitinvariants. This method works well if G has small index in Sd. Iterating, this allows on tocompute the monodromy group of a Belyı map explicitly instead of merely giving the maximalgroups in which is it included. To this end, one may work modulo a prime p of good reduction,and in light of Beckmann’s Theorem, we may still reasonably expect a small prime of the ringof integers of K that is coprime with the cardinality #G of the monodromy group to do thejob.Second, one can compute the monodromy by using numerical approximation. This has beenimplemented by van Hoeij [156], though one must be very careful to do this with rigorouserror bounds. This idea was used by Granboulan [61] in the computation of a cover withGalois group M24, first realized (without explicit equation) by Malle–Matzat [107, III.7.5].In particular, Schneps [138, §III.1] describes a numerical method to draw the dessin itself,from which one can read off the mondromy. This method is further developed by Bartholdi–Buff–von Bothmer–Kröker [7], who lift a Delaunay triangulation numerically and read off thepermutations by traversing the sequence of edges counterclockwise around a basepoint. Inparticular this solves (iii): if we express each of the complex solutions obtained by embeddingK → C, we may also want to know which cover corresponds to which permutation triple upto conjugation.A third and final method is due to Elkies [48], who uses an effective version (due to Weil’s proofof the Riemann hypothesis for curves over finite fields) of the Chebotarev density theoremin the function field setting. This was applied to distinguish whether the Galois group of agiven cover was equal to M23 or A23. More precisely, one relies on reduction modulo a primewhose residue field is prime of sufficiently large characteristic (in his case, > 10

9) and usesthe resulting distribution of cycle structures to deduce that the cover was actually M23. Thismethod has the advantage of using exact arithmetic and seems particularly well-suited toverify monodromy of large index in Sd.

9. Further topics and generalizations

This section discuss some subjects that are generalizations of or otherwise closely relatedwith Belyı maps. At the end, we briefly discuss the theoretical complexity of calculatingBelyı maps.

Generalizations. — Over Fp, one can consider the reduction of Belyı maps from charac-teristic 0; this is considered in Section 5 above. Switching instead to global function fieldsmight be interesting, especially if one restricts to tame ramification and compares with thesituation in characteristic 0. As a generalization of Belyı’s theorem, over a perfect field ofcharacteristic p > 0, every curve X has a map to P1 that is ramified only at ∞ by work ofKatz [88]. But this map is necessarily wildly ramified at ∞ if g(X) > 0, so the correspondingtheory will differ essentially from that of Belyı maps over Q.If we view Belyı’s theorem as the assertion that every curve over a number field is an étalecover of P1 \ 0, 1,∞ ∼= M0,4, the moduli space of genus 0 curves with 4 marked points,



then Belyı’s result generalizes to a question by Braungardt [23]: is every connected, quasi-projective variety X over Q birational to a finite étale cover of some moduli space of curvesMg,n? Easton and Vakil also have proven that the absolute Galois group acts faithfully onthe irreducible components of the moduli space of surfaces [43]. Surely some computationsin small dimensions and degree will be just as appealing as in the case of Belyı maps.As mentioned on a naive level in Remarks 2.5 and 2.10, another more general way to lookat Belyı maps is through the theory of Hurwitz schemes, which give a geometric structure tothe set Hn,r(Q) parametrizing degree n morphisms to P1 over Q that are ramified above rpoints. The theorem of Belyı then amounts to saying that by taking the curve associated to amorphism, one obtains a surjective map from the union of the Q-rational points of the spacesHn,3 to the union of the Q-rational points of the moduli spaces of curves Mg of genus g. Werefer to work of Romagny–Wewers [134] for a more complete account.

Origamis. — One generalization of Belyı maps is given by covers called origamis: coversof elliptic curves that are unramified away from the origin. For a more complete accounton origamis, see Herrlich–Schmithüsen [74]; Belyı maps can be obtained from origamis by adegeneration process [74, §8].The reasons for considering origamis are many. First, the fundamental group of an ellipticcurve minus a point is analogous to that of the Riemann sphere, in that it is again freeon two generators. The ramification type above the origin is now given by the image of thecommutator of these two generators. The local information at this single point of ramificationreflects less information about the cover than in the case of Belyı maps. Additionally, thebase curve can be varied, which makes the subject more subtle, as Teichmüller theory makesits appearance.An exciting family of special origamis was considered by Anema–Top [3]: they consider theelliptic curve E : y2 = x3+ ax+ b over the scheme B : 4a3+27b2 = 1 defined by the constantnon-vanishing discriminant 1 of E. Considering the torsion subschemes E[n] over B, oneobtains a family of covers over the base elliptic curve B of j-invariant 0 that is only ramifiedabove the point at infinity and whose Galois groups are subgroups of special linear groups.It would be very interesting to deform this family to treat the case of arbitrary base curves,though it is not clear how to achieve this.

Question 9.1. — How does one explicitly deform special origamis to families with arbitrary

base curves?

Explicit examples of actual families of origamis were found by Rubinstein-Salzedo [135, 136].In particular, by using a deformation argument starting from a nodal cubic, he obtains afamily of hyperelliptic origamis that are totally ramified at the origin. For the case of degree3, this gives a unique cover of genus 2. More precisely, starting with an elliptic curve E withfull 2-torsion in Legendre form

y2 = x(x− 1)(x− λ),

the hyperelliptic curve

y2 =1

2

−4x5 + 7x3 − (2λ− 1)x2 − 3x+ (2λ− 1)



admits a morphism to E given by

(x, y) −→

1

2

−4x3 + 3x+ 1

,y

2

−4x2 + 1

that is only ramified at the points at infinity of these curves.It is important to note here that the field of moduli of these covers is an extension of the fieldof moduli of the base elliptic curve; more precisely, as suggested by the formulas above, thisfield of moduli is exactly the field obtained by adjoint the 2-torsion of the curve. This is avariation on a result in Rubinstein-Salzedo [135], where simpler expressions for similar coversare found in every degree. Amusingly enough, adjoining the full 2-torsion of the base curvesalways suffices to define these covers. This result is appealing and quite different from thecorresponding situation for Belyı maps, and therefore we ask the following question.

Question 9.2. — Which extension of the field of moduli is needed to define similar covers

totally ramified above a single point for general curves?

Specialization. — Covering maps of the projective line with more than 3 ramificationpoints specialize to Belyı maps by having the ramification points coincide. In many cases, thecovers in the original spaces are easier to compute, and this limiting process will then lead tosome non-trivial Belyı maps. This also works in reverse, and provides another application ofcomputing Belyı maps. Hallouin–Riboulet-Deyris [66] explicitly computed polynomials withGalois group An and Sn over Q(t) with four branch points for small values of n; starting froma relatively simple “degenerate” three-point branched cover, the four-point branched coveris obtained by complex approximation (using Puiseux expansions). These methods wereconsiderably augmented by Hallouin in [67] to find another such family with group PSL2(F8).More recently, König [94] similarly computed such an extension of Q(t) with Galois groupPSL5(F2), using a p-adic approximation to calculate the initial three-point degeneration. Inall aforementioned cases, the resulting covers can be specialized to find explicit solution tothe inverse Galois problem for the groups involved, and as mentioned at the end of Section 1,the results from [67] have also found an application in the determination of equations forShimura curves [65].As mentioned in Section 3, Couveignes [32] has used a patching method to describe moregenerally the computation of families of ramified branch covers, using a degeneration to thesituation of three-point covers. More extensive algorithmic methods to deal with this questionshould therefore be in reach of the techniques of numerical algebraic geometry.

Complexity. — In this article, we have been primarily concerned with practical methodsfor computing Belyı maps; but we conclude this section by posing a question concerning thetheoretical complexity of this task.

Question 9.3. — Is there an algorithm that takes as input a permutation triple and produces

as output a model for the corresponding Belyı map over Q that runs in time doubly exponential

in the degree n?

There is an algorithm (without a bound on the running time) to accomplish this task, but itis one that no one would ever implement: there are only countably many Belyı maps, so onecan enumerate them one at a time in some order and use any one of the methods to check



if the cover has the desired monodromy. It seems feasible that the Gröbner method wouldprovide an answer to the above problem, but this remains an open question. Javanpeykar[79] has given explicit bounds on the Faltings height of a curve in terms of the degree of aBelyı map; in principle, this could be used to compute the needed precision to recover theequations over a number field.

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5 décembre 2013

J. Sijsling, Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UKE-mail : [email protected]

J. Voight, Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington,VT 05401, USA; Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH03755, USA • E-mail : [email protected]


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2014/1

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14/1Y. Bugeaud

Around the Littlewood conjecture in Diophantine approximation

Z. Chonoles, J. Cullinann, H. Hausman, A. M. Pacelli, S. Pegado and F. WeiArithmetic Properties of Generalized Rikuna Polynomials

A. GalateauUn théorème de zéros dans les groupes algébriques commutatifs

A. MohamedWeight reduction for cohomological mod p modular forms over imaginaryquadratic fields

J. Sijsling and J. VoightOn computing Belyi maps

Revue du Laboratoire de Mathématiques de Besançon (CNRS UMR 6623)


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