ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE …math.uga.edu/~pete/psl2four.pdf · 2008-12-10 ·...

ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCESUBGROUPS OF TRIANGLE GROUPS

PETE L. CLARK

Abstract. Here we give the first serious consideration of a family of algebraic

curves which can be characterized (essentially) either as the compact Riemannsurfaces uniformized by congruence subgroups of the Fuchsian triangle groups

∆(a, b, c) or as the algebraic curves X/H, where X admits a G ∼= PSL2(Fq)-

Galois Belyi map and H is a subgroup of G. This family contains in particularthe classical modular curves and XX commensurability classes of quaternionic

Shimura curves.

Our perspective is that all of these curves ought to be viewed, in someways, as “generalized Shimura curves,” notwithstanding the fact that the so-

called non-arithmeticity of most of their uniformizing Fuchsian groups means

that the Hecke algebra of modular correspondences will be too small to allowus to employ some of the usual automorphic techniques. Nevertheless the

theory of Cohen and Wolfart provides a modular embedding of each of ourcurves as a cycle in a higher-dimensional quaternionic Shimura variety, and this

embedding allows the theory of these curves to retain at least an automorphic

flavor.Our work here is in two parts. The first part is foundational: our curves can

be defined and studied using their canonical Belyi maps (using the arithmetic

theory of branched coverings), in terms of Takeuchi’s theory of Fuchsian groupswith trace ring an order in a totally real number field, or as subvarieties of

quaternionic Shimura varieties via the Cohen-Wolfart embedding. These three

perspectives give quite complementary information on the properties of thecurves, but it requires some care to check that the various constructions do

in fact yield the same algebraic curves (or compact Riemann surfaces), and

our first order of business is to establish this compatibility. Secondly, we areinterested in determining both the minimal field of definition of our curves as

well as the field of rationality of the automorphism group G ∼= PSL2(Fq). This

latter field F is of obvious interest, both intrinsically and because by Hilbertirreducibility one gets PSL2(Fq) as a Galois group over F . Alas one almost

never has F = Q and in fact our analysis shows what researchers on the InverseGalois Problem must have already suspected or known: apart from one or two

exceptions, it is impossible to attain PSL2(Fq) as the Galois group of a Galois

cover of the projective line which is ramified at only three points! Neverthelesswe are able to show in many cases that F is an abelian number field and

conjecture that this is always the case, a strange sort of converse to Belyi’soriginal work which attains (in particular) PSL2(Fq) as a Galois group over

Qab. We have no really satisfactory explanation for this abelian phenomenon

– as we explain, it is easy to construct, for any number field K, a curve Xacted upon by a finite group G, such that X → X/G is a Belyi map, and suchthat every field of definition of X and G contains K – but it confirms that ourrestriction to congruence subgroups of possibly non-arithmetic triangle groups

gives us a family with distinguished arithmetic-geometric properties. We hopeto persuade the reader that these curves are unusually worthy of further study.

1

2 PETE L. CLARK

1. Introduction

The aim of this paper is to introduce a certain class of complex algebraic curves(of genus g ≥ 2) which include the classical elliptic modular curves and certainShimura curves. There are many indications that the rich geometric, arithmeticand automorphic theories of modular curves should have analogues for our class ofcurves, despite the fact that their uniformizing Fuchsian groups are in general (so-called!) non-arithmetic, i.e., not commensurable with a Fuchsian group derivedfrom a quaternion algebra.

Let us enunciate the property of modular curves that we want to generalize. Letp ≥ 7 be a prime, and let X(p) be the (compactified) modular curve with full levelp-structure. The group G = PSL2(Fp) acts effectively on X(p), and the naturalmap X(p) → X(p)/G has the following property: there exists an isomorphismX(p)/G ∼= P1 such that the composite map X(p) → P1 is ramified only above thepoints 0, 1,∞. Otherwise put, there exists a subgroup G ∼= PSL2(Fp) ⊂ Aut(X(p))such that quotienting by G gives a Galois Belyi map.

Here we will be interested in the class of algebraic curves X/C of genus g ≥ 2with the property that there exists a subgroup G ∼= PSL2(Fq) ⊂ Aut(X) such thatthe map X → X/G is a Belyi map (again, this means that X/G has genus zero andthere are at most three ramification points).

Why is this an appealing class of algebraic curves? First of all, such curves can bedefined over Q. Indeed, if k ⊂ K is an inclusion of algebraically closed fields, Y/k isan algebraic curve, and ϕ : X → Y is a branched covering defined over K but withk-rational branch points, then both X and ϕ can be defined over k. Conversely,Belyi proved that every algebraic curve defined over Q admits a map to P1 with atmost 3 ramification points. In other words, the class of algebraic curves admittingBelyi maps is enormously vast.

On the other hand, the class of algebraic curves X admitting Galois Belyi mapsX → P1 is much more specialized: there are only finitely many such curves of anygiven genus g ≥ 2 or with any given automorphism group.1 We shall call such curvesWolfart curves, after J. Wolfart, who has studied them extensively over the com-plex numbers. For N ≥ 4, the Fermat curve XN +Y N = ZN is also a Wolfart curve.

Wolfart curves admit many characterizations: for instance, the locus on the modulispace Mg of curves of genus g at which the function C 7→ # Aut(C) attains a strictlocal maximum consists precisely of the genus g Wolfart curves. Thus in particularthey include the class of curves attaining the Hurwitz bound # Aut(C) ≤ 84(g−1)(Hurwitz curves), which have been much studied in the literature.2

Or, all-importantly for us, a Riemann surface C is a Wolfart curve if and only ifits uniformizing Fuchsian group Γ is a (necessarily torsionfree) finite-index normal

1If G ⊂ Aut(X) is such that X → X/G is a Belyi map, then so too is X → X/ Aut(X); thusthe distinction between subgroups of automorphisms and the entire automorphism group is nota critical one. Moreover, that X → X/G is a Belyi map means that G is already very large; inpractice, this usually forces G = Aut(X).

2In fact we find the Wolfart condition to be so much more natural than the Hurwitz conditionas to find it slightly odd that Hurwitz curves in particular have been so extensively studied.

ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS3

subgroup of a Fuchsian triangle group ∆(a, b, c), i.e., the group generated by theelements τb ◦ τa, τc ◦ τb, τa ◦ τc, where τa, τb, τc are reflections in the sides of ahyperbolic triangle with angles 2π

a , 2πb , 2π

c (with a, b, c ∈ Z+). Then

Γ\H → ∆(a, b, c)\Hgives the Galois Belyi map.

Viewed as groups of automorphisms of H (i.e., as subgroups of PSL2(R)), forany positive integers a, b , c such that 1

a + 1b + 1

c < 1, there exists a triangle group∆(a, b, c), unique up to conjugacy. One often allows any of a, b, c to take the value∞, which corresponds to allowing the vertices of the triangle to lie on the boundaryRP1 of H. E.g. ∆(2, 3,∞) = PSL2(Z) the standard modular group.

As Wolfart notes in his work, it would be very interesting to undertake an arith-metic study of this distinguished class of curves

The aim of this paper is to begin the explicit arithmetic study of a certain classof complex algebraic curves.

Namely, we consider (always smooth, projective geometrically connected) alge-braic curves X/C of genus g ≥ 2 such that the natural map q : X → X/Aut(X)is a Belyi map: that is, X/Aut(X) has genus zero and q is ramified over threepoints, so that with an appropriate choice of coordinate function, we may identifyX/Aut(X) with P1 and the ramification points with {0, 1,∞}. We dub such curvesWolfart curves, after J. Wolfart, who has studied them extensively.

Wolfart curves can also be characterized as the points X on the moduli spaceMg at which the function X 7→ # Aut(X) attains a strict local maximum; forthis reason they are often called Riemann surfaces with many automorphisms.(In particular, they include the curves X of genus g for which Hurwitz’s bound# Aut(X) ≤ 84(g − 1) is attained (Hurwitz curves), which are themselves muchstudied in the literature.) On the other hand, these special curves (and their quo-tients by subgroups H of Aut(X)) include many if not most of the examples ofcurves which have been most intensively studied in the literature, e.g. the Fermatcurves, Klein’s quartic curve, the classical modular curves, and certain Shimuracurves.

As for any curve admitting a Belyi map, a Wolfart curve can be defined over Q,or equivalently over some number field. This suggests that Wolfart curves shouldbe studied from an arithmetic perspective (e.g., rational points, places of good andbad reduction), and of course the above examples also furnish the list of curvesmost familiar to arithmetic algebraic geometers.

Of course, given a Wolfart curve X/C, before embarking upon such an arithmeticstudy we need a model of X over some number field K. In this sort of business,there are several different ways of construing the statement “X can be defined overK” and it is critically important to distinguish carefully between them. The weak-est possible notion is that of the field of moduli M(X) of X, which is the fixedfield of the largest subgroup A of Aut(C) such that σ ∈ A =⇒ X ⊗σ C ∼= X.Clearly, if there exists some field K and a curve defined over K whose basechangeto C is isomorphic to X/C, then K contains the field of moduli M(X) . In general,the converse is false. However, it can be repaired as follows: any Galois coveringt : X → P1 of compact Riemann surfaces can be defined over its field of moduliM(X, t) [Kock, Thm. 2.2]. Now for any X/C be any curve (of genus g ≥ 2) such

4 PETE L. CLARK

that X/Aut(X) ∼= P1, it is easy to see that the field of moduli of X as a curve isequal to the field of moduli of the covering t : X → X/Aut(X), so it follows thatany Wolfart curve can be defined over its field of moduli.

On the other hand, even if X can be defined over M(X), there will in general notbe a unique model defined over M(X): indeed, after fixing a single model X/M(X),the set of M(X)-models is given by the Galois cohomology set H1(M(X),Aut(X)).Note that here Aut(X/Q) is being viewed as a module over the absolute Galois groupof M(X). This action cuts out a finite Galois extension F (X)/M(X) which is theminimal field such that the covering X → P1 is Galois over M(X).

Definition: An aut-canonical model for an algebraic curve X/Q is given by anumber field K and a K-model X for X/Q such that:(i) All the automorphisms of X are defined over K.(ii) For any number field L for which there exists an L-model X ′ for XQ such thatall automorphisms of X ′ are defined over L, there exists a field homomorphismι : K ↪→ L such that X ⊗ι L ∼= X ′.

In particular, there exists at most one autcanonical model up to isomorphism.

Example: For a prime number p ≥ 7, consider the modular curve X(p) = Γ(p)\H.Then G = Aut(X(p)) ∼= PSL2(Fp) and the quotient map X → X/G is the naturalmap corresponding to the inclusion of Fuchsian groups Γ(p) ⊂ Γ(1) = PSL2(Z).Thus X/G ∼= P1 = C(j), and the map is ramified over j = 1728, 0, ∞ with indices2, 3 and p. (So in particular, X(p) is a Wolfart curve.) Then the field of moduli ofX(p) is Q, and indeed X(p) admits models over Q.3 It turns out that X(p) does not

admit a Q-model with all of its automorphisms defined. Letting p∗ = (−1)p2−1

2 p,K.-y. Shih showed that X(p) has an autcanonical model over Q(

√p∗).

Remark 1: The standard moduli interpretation of X(p) gives a model over Q(ζp)with all automorphisms defined. This model is derived from the autcanonical modelby basechange (and indeed Q(

√p∗) embeds in Q(ζp)).

Remark 2:We assume chosen on the rational curve X/Aut(X) a Q-model so that each

of the three ramification points are Q-rational, or equivalently so that the Galoisaction on the ramification divisor D is trivial. (In work on the Inverse Galois Prob-lem, it is often convenient to equip D with a nontrivial Galois module structure insuch a way as to “cancel out” certain other nontrivial Galois actions. We shall notdo this here.)

As we shall recall later, this class contains many of the families of curves whichhave received the most intense arithmetic study: e.g. certain Shimura curves includ-ing the classical modular curves and Klein’s quartic curve, and the Fermat curves.On the other hand, there are only finitely many Wolfart curves of any given genus(equivalently, with any fixed automorphism group). They can be characterized

3Every elliptic curve E/Q gives rise to such a model, two such models being isomorphic iff

E[p] ∼= E′[p] as Galois modules. This exhibits infinitely many distinct Q-models, and there areothers besides.


as the points on the moduli space Mg of genus g curves at which the functionX 7→ # Aut(X) has a strict local maximum, so are in a sense the “maximally spe-cial points” on Mg. In particular they include the family of curves attaining theHurwitz bound Aut(X) = 84(g − 1) which have been intensely studied.4

Having said all this, in this paper our focus of attention is the Wolfart curveswith automorphism group isomorphic to PSL2(Fq), a class which has decidedlynicer arithmetic properties. Especially, we are interested in the following

Question 1. Let X/C be a Wolfart curve with Galois group G ∼= PSL2(Fq). Is itthe case that X, together with all its automorphisms, can be defined over an abeliannumber field?

We shall answer this question affirmatively in many cases. Note well that this isnot a general property of Wolfart curves:

Proposition 2. For any number field K, there exists a Wolfart curve X/C suchthat any field of definition of X and Aut(X) contains K.

Proof: Let E be an elliptic curve with j-invariant j(E) ∈ K \ OK . Since E can bedefined over K, by Belyi’s theorem there exists a Belyi map β : E → P1 (possiblydefined over a larger field). We claim that β is not a Galois cover, and indeed thatthe only elliptic curves admitting Galois Belyi maps are those with j-invariant 0 or1728. Indeed, suppose G ⊂ Aut(E) is a (finite!) subgroup such that β : E → E/Gis a Belyi map. Then G has a maximal normal subgroup Gτ of translations (bytorsion points); thus G′ = G/Gτ is a group of automorphisms fixing the origin ofthe isogenous elliptic curve E′ := E/Gτ and such that E′ → E′/G′ is a GaloisBelyi map. But unless j(E) ∈ {0, 1728}, we have G′ ⊂ {±1}, and the quotient by−1 is certainly not a Belyi map, being ramified over the four 2-torsion points.

Let X → P1 be the Galois Belyi map obtained by taking the Galois closure ofβ : E → P1. If X were an elliptic curve it would again have j-invariant 0 or 1728,and then the natural map X → E would be an isogeny, so that E would have com-plex multiplication, contradicting the non-integrality of j(E). Thus X has genusat least 2 so is a Wolfart curve in our sense. If X is defined together with all itsautomorphisms over some field L, then the subgroup H ⊂ G such that X/H = Eacts L-rationally, so that E is defined over L. Thus K ⊂ L, completing the proof.

So an affirmative answer to Question 1 would exhibit a curious property of thegroups PSL2(Fq) calling out not just for verification but for some sort of expla-nation. The answer seems to be that every PSL2(Fq)-Wolfart curve is a Shimuracurve. Taken literally this claim asserts the arithmeticity of infinitely many Fuch-sian triangle groups, so is known to be false by work of Takeuchi. However, work ofBeasley-Cohen and Wolfart nevertheless allows us to obtain our Galois Belyi mapX → X/PSL2(Fq) by embedding P1 as a cycle in a higher-dimensional quater-nionic Shimura variety and pulling back to the PSL2(Fq)-covering associated toan appropriate congruence subgroup. In this way the abelian phenomenon can berelated to the Shimura-Deligne theory of canonical models.

But in fact things are not so simple, as the work of Shimura and Deligne furnishes

4In fact Wolfart curves are so similar in their arithmetic-geometric behavior to Hurwitz curvesthat it seems rather artificial to restrict attention to the latter class.

6 PETE L. CLARK

us with a canonical model over an abelian extension not of Q but over a certaintotally real number field. This field of definition is well-known to be the minimalone consistent with a certain moduli interpretation, but it is also known that it canbe a proper extension of the minimal field of definition of the automorphisms ofX: in the special case of the PSL2(Fp)-covering X(p) → X(1) of classical modularcurves, this discrepancy goes back (at least) to work of Shih.

Nowadays Shih’s work can also be understood in the context of the rigidity-rationality theory developed by workers on the Inverse Galois Problem. In thegeneral case “rigidity” does not hold, and using related criteria of Volklein leavesus within a quadratic extension of fully determining the minimal field of definitionof Aut(X). In the special case where X can as a curve be defined over Q, we areable to put together all of these considerations to find the precise minimal fieldof definition, an abelian number field. The general case seems to require a morecomplete theory of canonical models than we currently possess.

2. Statements of the Main Results

Let G be a finite group. By a G-Wolfart curve we mean an algebraic curve withAut(X) ∼= G and such that q : X → X/G is a Belyi map.

Theorem 3. Let X be a complex algebraic curve admitting a PSL2(Fq)-GaloisBelyi map map β : X → P1, unramified outside {0, 1,∞}. Then:a) The field of moduli of X is an abelian number field.b) There exists an abelian number field K and an extension L/K with [L : K] ≤ 2such that X and all of its automorphisms can be defined over L.

More precise results are possible: we can compute the minimal field of definitionand the Galois orbit of the corresponding point on the moduli space. The followingresult is representative.

For more detailed investigations, we shall find it useful to restrict our attentionto a particular family of curves. Namely, suppose p is a prime number and N ≥ 7is a positive integer which is either equal to p or coprime to p. We define a positiveinteger a = a(N, p) as 1 if N = p and as the order of p in (Z/2NZ)×/(±1) when(N, p) = 1. We also make the following auxiliary hypothesis:

• a(N, p) is an odd number.

Theorem 4. There exists a curve X(N, p)/Q with the following properties:i) G = Aut(X(N, p)) ∼= PSL2(Fpa(N,p)).ii) X → X/G is a Belyi map with ramification indices (2, 3, N).iii) The field of moduli Fw of X(N, p) is the fixed field of Q(ζ2N ) under the actionof H = 〈−1, p〉 ⊂ (Z/2NZ)×, and X(N, p) can be defined over Fw.iv) Every Wolfart curve with Galois group PSL2(Fpa(N,p)) and ramification indices(2, 3, N) lies in the same Galois orbit as X(N, p), and the Galois orbit has size[Fw : Q].

Remarks: It would be more accurate to say, “There is a Galois orbit of curves. . .”since our construction does not single out a unique curve. This could be remediedfor instance by defining X(N, p) to be the disjoint union of all [Fw : Q] distinctcurves in the Galois orbit. But the abuse of notation simplifies matters so we will


stick with it.This theorem generalizes the main result of [Streit], which considered (by differ-

ent methods) the special cases N = 7 (Hurwitz curves) and a = 1.The curves X(p, p) are the classical modular curves X(p), and the covering

X(p, p) → P1 is the one associated to the congruence subgroup Γ(p) ⊂ GL2(Z),i.e., to the kernel of the reduction map GL2(Z) → GL2(Fp). When

(1) N ∈ {7, 9, 11, 14, 18},let FN = Q(ζ2N + ζ−1

2N ), and let XN/FN be the Shimura curve associated to a max-imal order O in the quaternion algebra over FN which is split at every finite placeand ramified at all but one infinite place. Since O is split at every prime above p,O ⊗ Fp

∼= M2(OFN) ⊗ Fp = M2(Fpa(N,p)), so the analogous congruence subgroup

Γ(p) ⊂ O× gives rise to the PSL2(Fa(N,p)p )-covering X(N, p). Note that (1) is not

the complete list of all Shimura curves uniformized by Fuchsian triangle groups (seeTakeuchi); we have only taken those with ϕ [FIGURE OUT WHETHER THIS ISCONSISTENT WITH TAKEUCHI]. The proposal of the introduction can now bestated in a more concrete form: we suggest that all the curves X(N, p) be viewedas Shimura curves.

Of course, Theorem 3 applies to the curves X(N, p) to give that their full auto-morphism group can be defined over an at-most-quadratic extension of an abeliannumber field. In certain cases we can show that the minimal field of definition isitself abelian:

Theorem 5. Under any of the following assumptions, the minimal field of defini-tion of X(N, p) and all its automorphisms is F2N (

√p∗):

a) N = p (classical modular case).b) (6, N) > 1 (rigid cases).c) [F2N : Q] = ϕ(2N)

2 is odd and Fw = Q.

[AFTER THIS POINT, WE REVISIT AN EARLIER DRAFT OF THE PAPER]

To be more precise, we can show the following.

Theorem 6. Let X/Q be a Belyi-Wolfart curve with Galois group G = PSL2(Fq)with q > 3. Then the field of moduli of X is an abelian number field, and thereexists an abelian number field L, and extension K/L with [K : L] ≤ 2 such that Kis a field of definition for all the automorphisms of X.

We have some reason to believe that the field extension K/L, if nontrivial, is ob-tained by adjoining

√D for some rational number D. If this is true, then the

conjecture follows immediately. At least the conjecture holds for infinitely manycases among the Belyi-Wolfart curves X(N, p) to be introduced presently, namelywhenever p generates (Z/2NZ)×/(±1). In this case the field of moduli is Q, so thatK/Q is a Galois number field, whose Galois group is an extension of a cyclic groupby a group of order at most two. By elementary group theory, all such groups areabelian.

Thus one perspective on the present work is to understand fields of definition ofautomorphism groups of Belyi-Wolfart curves. But there are other ways to look atwhat is being done here:

8 PETE L. CLARK

Group theory: we use essentially the rigidity - rationality results developed byresearchers on the Inverse Galois Problem. Indeed, around the same time as theproof of Belyi’s theorem a fundamental advance on the inverse Galois problem wasmade by Belyi, Fried and Thomson, who found a purely algebraic condition on afinite group sufficient for its regular Galois realizability over an explicit cyclotomicfield, namely rigidity. This is a certain simple transitivity condition on triples ofelements in G (recalled explicitly, for what it’s worth, in Section 2.3). Many finitegroups have been realized as Galois groups over Q (and many more over Qab) usingrigid triples, but at a certain point one has to deal with the fact that the existenceof a rigid triple is rather restrictive. Much less restrictive (but still too much tohope for in general) is a condition of weak rigidity (also Section 2.3), and (despitemy very limited knowledge in this area) I think it is fair to characterize the greatestpart of the work of the last ten years on the Inverse Galois Problem as studyingwhen the condition of weak rigidity is enough to ensure Galois realizations overQ or over Qab (see [Malle-Matzat] and especially [Volklein]). In this context wecan mention the key fact, which is that the groups PSL2(Fq) possess weakly rigidtriples which, while not rigid in general, are very close. Indeed, results of [Macbeath1969] imply that (as we shall make more precise later!) we are at most a singleouter automorphism away from rigidity, so in what I call a semirigid situation.This is “the same factor of 2” giving rise to the mysterious quadratic extension inTheorem 1.

I have not seen this semirigidity property of PSL2(Fq) in the literature, 5 butat the least it seems to be a useful way to understand some known Galois realiza-tions of groups PSL2(Fq). Indeed, while in the standard references [Malle-Matzat]and [Volklein] one finds Galois realizations of PSL2(Fq) over Qab using only re-duction to the rigid case, the semi-rigidity implies that we are within a factor of2 of understanding all possible regular realizations of groups PSL2(Fq) as Galoisgroups via triples.6 There is an infinite family of groups PSL2(Fp3) in which thefield L of Theorem 2 is Q. In case p = 2 we have not only semirigidity but rigidity –i.e., PSL2(F8) occurs regularly as a Galois group over Q, a result due to G. Malle.This underscores the importance of determining the mysterious at-most quadraticextension K: can it be trivial?!?

Generalized Shimura curves: A final perspective – to which we hew the most closelyin the present draft – is obtained by specializing to the case of ramification degrees(2, 3, N). In this case we get an explicit two parameter family of “generalizedShimura curves” X(N, p)/Q where p is a prime and N ≥ 7 is a positive integereither equal to p or prime to p. We impose an additional constraint on N and p,as follows: define a positive integer a = a(N, p) as

• 1, if N = p,

5One should note the carefully phrased weakness of this statement.6The last two words are important: conjecturally, every finite group is the Galois group of

a finite branched covering of P1/Q, but certainly more than three branch points are required ingeneral: e.g. (Z/2Z)n requires n + 1 branch points even over C.


• the order of p in (Z/2NZ)×/(±1), if gcd(p, N) = 1.

Note that a(N, p) will be odd for all primes p if N is itself prime and congru-ent to 3 modulo 4; to fix ideas, the reader might like to keep this case in mind.(Indeed, keep in mind the case N = 7 no matter what.)

We illustrate Theorem 1 in this case:

Theorem 7. Let N ≥ 7 be a positive integer, p a prime number such that eitherN = p or (N, p) = 1, and such that the integer a = a(N, p) defined above is odd.Then there is an algebraic curve X(N, p)/Q which is not necessarily connected. Ithas the following properties (we assume here that gcd(6N, p) = 1; the other cases,which are easier, are treated in Section 3.5):a) Each connected component of X(N, p) has full automorphism group G = PSL2(Fpa),and the canonical map

X(N, p) → X(N, p)/G ∼= P1

is ramified only over (0, 1,∞) with indices (2, 3, N).b) The field of moduli Fw of each connected component is the unique subfield ofQ(ζ2N ) which is of degree ϕ(2N)

2a over Q. The curve X(N, p) itself is the disjointunion of the distinct Galois conjugates of any of its connected components, so admitsQ-rational models.c) Let K be the splitting field of the G-action on any component of X(N, p). ThenF2N ⊂ K and [K : F2N ] ≤ 2. Thus PSL2(Fpa) occurs regularly as a Galois groupover the number field K, which has degree equal to either ϕ(2N)

2 or ϕ(2N).

This underscores the last point that we want to make: the general theory ofShimura varieties as exposed in [Deligne 1971] gives disconnected canonical modelsfor Shimura varieties X (p) with “level p structure,” such that the degree of the fieldof definition of each component over the field of definition F of the base Shimuravariety X goes to infinity with p. But the semirigidity of PSL2(Fq) points to the ex-istence of models of X (p) over at-most quadratic extensions K of F which still havea canonicity property, not in the sense of moduli interpretations or fields generatedby special points, but in the more elementary sense of being characterized by theproperty that K(X (p)) → K(X ) is a Galois extension. We might tentatively callthese models subcanonical models to emphasize their smaller field of definition.

3. Background on Wolfart curves

Such Riemann surfaces have been studied by J. Wolfart [Wolfart 1997], [Wolfart 2000].In particular, he gives the following intriguing characterization.

Theorem 8. (Wolfart) For a compact Riemann surface X of genus at least two,the following are equivalent:a) X admits a Galois Belyi map.b) The canonical map q : X → X/Aut(X) is a Belyi map.c) X is uniformized by a Fuchsian group Γ which is a finite index normal subgroupof a hyperbolic triangle group ∆(a, b, c).d) There exists an open neighborhood U of X in the complex manifold Mg(C) suchthat # Aut(X) > # Aut(Y ) for all Y ∈ U \ {X}.

10 PETE L. CLARK

In order of Wolfart’s work, we shall call a curve satisfying the equivalent conditionsof Theorem 8 a Belyi-Wolfart curve.

Theorem 9. (Properties of Belyi-Wolfart Curves) a) If X is a G-Belyi-Wolfartcurve with ramification degrees (a, b, c), then

g(X) = 1 +#G

2(1− 1

a− 1

b− 1

c).

b) There exist at most finitely many Belyi-Wolfart curves of a given genus g or withfull automorphism group any given finite group G.c) A Belyi-Wolfart curve can be defined over its field of moduli.d) Suppose Γ is a finite index normal subgroup of ∆(a, b, c), where (a, b, c) is notof one of the following forms, said to be non-maximal:

(2, b, 2b), (3, b, 3b), (a, a, c).

ThenAut(Γ\H) = ∆(a, b, c)/Γ.

Remark: By part a), #G as a function of g is maximized when (a, b, c) = (2, 3, 7).Combining with Theorem 8 we recover the Hurwitz bound,

# Aut(X) ≤ 84(g(X)− 1).

A curve attaining the Hurwitz bound is called a Hurwitz curve, and the automor-phism group of a Hurwitz curve is called a Hurwitz group. The problem of deter-mining all Hurwitz groups is purely algebraic: we are asking which finite groupsare generated by an element x of order 2 and an element y of order 3 such that xyhas order 7.

Here we shall be concerned with the following questions on a Belyi-Wolfart curveX with Aut(X) = G. First, we wish to compute its field of moduli Fw, which is theunique minimal field of definition for X. Second, we wish to compute the field ofdefinition of G: that is, Aut(X/Q) is naturally a g = GalQ/Fw

-module, so cuts out aGalois extension K/Fw, the unique minimal field such that the automorphisms aredefined over K. This is also the minimal field K such that the corresponding ex-tension of function fields K(X) → K(X/G) = K(P1) = K(t) is a Galois extension,with Galois group G. Thus there is an application to the regular Inverse GaloisProblem: applying Hilbert’s irreducibility theorem, we get that the finite group Goccurs as a Galois group over K. Thus one is interested in conditions to ensurethat K is as small as possible: ideally, such that K = Q, or more realistically, suchthat K is an abelian number field.

Even this latter condition certainly is not satisfied for every Belyi-Wolfart curve.In the Appendix we give a simple argument showing that for any number field L,there exists a Belyi-Wolfart curve X such that the minimal field of definition of(X, Aut(X)) contains L. The construction, however, tells us absolutely nothingabout G = Aut(X). Here we study the special case of G = PSL2(Fq) (actually,we restrict attention to the case of q = pa for a odd in order to better analyze aparticular subclass of curves, but this restriction is not necessary for the presentparagraph) and get results which seem to indicate that the case of PSL2(Fq) isespecially well behaved.


4. The proof of Theorem 3

4.1. PSL2(Fq): elementary results. In this section we recall the basic facts onconjugacy classes and automorphisms of the groups PSL2(Fq), which will be neededin our subsequent analysis.

As is traditional, we represent elements of PSL2(Fq) by matrices in SL2(Fq)and must keep in mind the twofold ambiguity arising from the map SL2(Fq) →PSL2(Fq) = SL2(Fq)/(±1) (note that there is no such ambiguity when q is even:±1 = 1).

First we recall the classification of conjugacy classes, first in SL2(Fq) and thenin PSL2(Fq). Let g ∈ SL2(Fq), with characteristic polynomial X2 − tr(g)X + 1.Suppose first that g 6= 1 is semisimple, i.e., its minimal polynomial has distinctroots, so accordingly here coincides with the characteristic polynomial. (Equiva-lently, g can be diagonalized over Fq, although not necessarily over Fq. This occursif and only if tr(g) 6= ±2. Then the rational canonical form of g is[

0 −11 tr(g)

].

Notice that −R(g) has minimal polynomial X2 + tr(g) + 1, so to any α ∈ k \ 2we associate a unique conjugacy class C(α) in SL2(Fq) of matrices of trace α,such that C(α), C(−α) become identified in PSL2(Fq). Any element g ∈ C(α)is diagonalizable over Fq if and only if its eigenvalues λ, λ−1 are in Fq. Otherwiseg is diagonalizable over Fq2 . It follows that the order of any semisimple elementof SL2(Fq) divides q2 − 1. This bound can be improved: suppose g ∈ SL2(Fq)has order N prime to p. Then tr(g) = ζN + ζ−1

N ∈ Fq, and conversely if Fq con-tains ζN + ζ−1

N we will get an element of order N . That is, the inertial degree ofFN = Q(ζN + ζ−1

N ) at p must divide a = logp(q), i.e., a must be a multiple ofthe order of p in (Z/NZ)×/(±1). It follows (by looking at some congruences) thatthe maximal order of a semisimple element in SL2(Fq) is q + 1. When q is even,SL2(Fq) = PSL2(Fq), and when q is odd, quotienting out by −1 halves the order ofany even order conjugacy class, and we get that the maximal order of a semisimpleelement in PSL2(Fq) is q+1

2 .

Consider now the conjugacy classes of nontrivial elements g of SL2(Fq) of trace±2; by multiplying by ±1, we may look only at the trace 2 case, in which we aregetting unipotent matrices, every one of which is conjugate to a matrix

U(u) =[

1 u0 1

],

for some u ∈ F×q . (By convention – and with apologies to Lincoln7, we do not callthe identity matrix unipotent.) Moreover, one can check that U(u) is conjugate toU(v) if and only if u ∼= v (F×2

q ). That is, if q is odd there are exactly two conju-gacy classes of nontrivial unipotent elements, and if q is even there is exactly one.These elements always have order p, and hence still have order p after descent toPSL2(Fq). Note then that in PSL2(Fp) there is an element of order N if and only

7“Q: How many legs does a dog have if you count its tail as a leg? A: Four. Calling its tail aleg doesn’t make it one.” – Abraham Lincoln.

12 PETE L. CLARK

if N = p or N is prime to p.

Finally, we will need to know the outer automorphism group of PSL2(Fq) – re-call that the outer automorphism group of any group G is the quotient of thefull automorphism group by the subgroup of inner automorphisms G/Z(G). Oneway to get an automorphism of PSL2(Fq) is to restrict an inner automorphismof PGL2(Fq) to PSL2(Fq) via the normal embedding PSL2(Fq) ↪→ PGL2(Fq).Since the quotient is isomorphic to F×q /F×2

q (induced by the determinant map), thesubgroup has index 2 if q is odd and index 1 if q is even, so we get at most onenontrivial outer automorphism this way and only then if q is odd. Conversely, if qis odd, conjugation by the nontrivial coset induces a nontrivial outer automorphismof order two on PSL2(Fq), denoted τ . On the other hand we can produce outerautomorphisms using the Galois theory of Fpa/Fp: i.e., the Frobenius fp : x 7→ xp

acts entrywise on SL2(Fpa) has order a, and descends to the projectivization withthe same order. And we’re done: indeed

Out(PSL2(Fpa)) = 〈τ〉 × 〈fp〉,e.g. [Suzuki]. In particular it is abelian, and of order 2a if q is odd and a if p iseven.

4.2. PSL2(Fq): Macbeath’s theory. We summarize here some of the results of[Macbeath 1969], which lead to a complete classification of subgroups of PSL2(Fq)generated by two elements.

Consider triples of elements (g1, g2, g3) ∈ SL2(Fq) satisfying g1g2g3 = 1. Let (a, b, c)be, respectively, the orders of g1, g2 and g3. Note that, contrary to elsewhere inthis paper, we do not make any assumption on the subgroup generated by g1, g2

and g3: indeed the point is to find conditions to ensure that the projective imageof 〈g1, g2, g3〉 is all of PSL2(Fq).

We define tr(g1, g2, g3) := (tr(g1), tr(g2), tr(g3)) ∈ F3q. (Notice that this is very close

to just recording the conjugacy classes of g1, g2 and g3.) Also, if (α, β, γ) ∈ F3q, put

T (α, β, γ) to the set of all triples (g1, g2, g3) such that tr(g1, g2, g3) = (α, β, γ).We say that an Fq-triple (α, β, γ) is commutative if there exists some triple(g1, g2, g3) ∈ T (α, β, γ) which is commutative: such that g1g2 = g2g1. Note thatthis is only possible if the order of g3 divides the lcm of the orders of g1 and g2, soin particular no Fq-triple consisting of elements of orders 2, 3, N with N ≥ 7 (orN = 5) can be commutative.

We say a triple is exceptional if its sequence (a, b, c) of orders is in the follow-ing list:(2)(2, 2, n), (2, 3, 3), (3, 3, 3), (3, 4, 4), (2, 3, 4), (2, 5, 5), (5, 5, 5), (3, 3, 5), (3, 5, 5), (2, 3, 5)

The exceptional triples are precisely the orders of triples of elements in the finitespherical triangle groups. No (2, 3, N) for N ≥ 7 is exceptional.

Finally, let k ⊂ Fq be a subfield. Then clearly PSL2(k) is a subgroup of PSL2(Fq).This is called a regular projective subgroup. If moreover the unique quadraticextension k2 of k is contained in Fq, then PGL2(k) is also a subgroup of PSL2(Fq)


– since we can rescale by the squareroot of the determinant – an irregular pro-jective subgroup. Suppose (α, β, γ) is an Fq-triple with Fq = Fp(α, β, γ). If(g1, g2, g3) ∈ T (α, β, γ) generate an irregular projective subgroup, then that sub-group is PGL2(F√q). Especially, if q is an odd power of p, then irregular subgroupsdo not exist. (This is indeed our reason for restricting to odd a.)

Here now is Macbeath’s classification of subgroups of PSL2(Fq) generated by twoelements.

Theorem 10. (Macbeath) a) Every triple (g1, g2, g3) ∈ SL2(Fq) is either excep-tional, commutative or generates a projective subgroup of PSL2(Fq). In particular,if q = pa with a odd, then any triple of elements with orders (2, 3, N) with N ≥ 7generates a subgroup PSLq(k) of PSL2(Fq).b) For any k-triple (α, β, γ), the set T (α, β, γ) of triples of elements of SL2(Fq)with traces α, β, γ is nonempty.c) PSL2(Fq) acts on noncommutative triples by simultaneous conjugacy and pre-serves each T (α, β, γ) (clearly). The number of orbits of this action on T (α, β, γ)is 1 if p = 2 and 2 if p > 2. In all cases, the inner automorphism group ofSL2(Fp) acts transitively on T (α, β, γ), and it follows that in the generating case,Aut(PSL2(Fq)) acts transitively on PT (α, β, γ).

For the proof see [Macbeath 1969, Theorem 5]. We shall explain the “it followsthat” in part c), since this is a key point for us that is not stated explicitly inhis paper. Indeed, if g and h are two triples with common trace triple (α, β, γ)and g (hence also h) generates PSL2(Fq), then the matrix m ∈ PSL2(Fp) whichconjugates g to h carries a generating set for PSL2(Fq) into a generating set forPSL2(Fq), hence m must stabilize PSL2(Fq) and thus induces an automorphismof PSL2(Fq).

Here are some immedidate consequences of this remarkable theorem: applying partsb) and c) together, we can just choose three elements (α, β, γ) ∈ F3

q, and there existsa triple (g1, g2, g3) ∈ SL2(Fq) with the given elements as traces. Moreover, sinceAut(PSL2(Fq)) acts transitively on the projectivized triples PT (α, β, γ), any twosuch triples generate isomorphic subgroups which are, except in very special degen-erate cases, isomorphic either to PGL2(F√q′) or to PSL2(Fq′) for some Fq′ ⊂ Fq.In particular, if logp(q) is odd, the first possibility does not occur.

Suppose we have (g1, g2, g3) ∈ T (α, β, γ) such that 〈g1, g2, g3〉 = PSL2(Fq′) ⊂PSL2(Fq). Let k := Fp(α, β, γ) be the subfield of Fq generated by the traces of theelements of the triple. The theorem implies that the triples, unique up to automor-phisms, could also be constructed in k, and it follows that F′q ⊂ k. (This is really notobvious: each element gi can certainly be conjugated into a matrix with k-rationalentries, but it was not clear that all three elements could be simultaneously conju-gated into k, which is what has just been shown.) The converse statement k ⊂ Fq′

is obvious, so we conclude that the projective group generated by any (nonex-ceptional, noncommutative, regular) (g1, g2, g3) ∈ T (α, β, γ) is PSL2(Fp(α, β, γ)).Since PSL2(Fp) always has elements of orders 2 and 3 (use either the above de-scription of conjugacy classes or even the fact that 6 | (p

2−1)(p2−p)2

gcd(2,p) (p−1)= #PSL2(Fp))

the field generated is the trace field of the element of order N , which recall, exists

14 PETE L. CLARK

in PSL2(Fp) when N = p or gcd(N, p) = 1.

The following is an easy consequence of Macbeath’s theorem.

Proposition 11. Fix N and p as above, and assume that a = a(N, p) is odd [e.g.this holds for all p when N ≡ −1(4) is prime]. Then q = pa is the unique power ofp for which there exists a nondegenerate epimorphism

ϕ : ∆(2, 3, N) → PSL2(Fq).

Proof: It follows immediately that there is at most one power pa of p such thata triple of elements of orders (2, 3, N) can generate PSL2(Fpa). If N = p thenthe element of order N is unipotent of order p, so its trace, 2, generates Fp. Ifp = 2 then PSL2(Fq) = SL2(Fq) and we want the field generated by the trace ofan element of order N in SL2(Fq), i.e., Fp(ζN + ζ−1

N ). If p > 2 is prime to N , thenour element of order N comes from an element of order 2N in SL2(Fp), so has traceζ2N + ζ−1

2N .8 This completes the proof.

Remark: I do not claim that it is necessary to consider only odd-degree field ex-tensions of Fp to get groups PSL2(Fq), only that it is sufficient. Indeed, Macbeathremarks that by taking a triple of elements with orders (2, 3, q + 1) in Fq (such anelement is shown to exist in Section 3.1), then because GL2(F√q) does not have anelement of order q + 1, it must be that the associated triples generate projectivesubgroups isomorphic to PSL2(Fq). In this way Macbeath showed that PSL2(Fq)is generated by elements of order 2 and 3 for every q ≥ 13. The same result canbe shown for the smaller values of q, except q = 9, in which case the result is false.Rather than try to write down more complicated necessary and sufficient conditionsfor the triple to generate PSL2(k) rather than PGL2(k), we will be content to lookat odd-degree extensions of the prime subfield.

4.3. The Weak Rigidity - Weak Rationality Lemma. Let now G be a finitegroup with trivial center, and let g = (g1, . . . , gn) be a tuple of elements in G:from now on this means that g1 · · · gn = 1 and G = 〈g1, . . . , gn〉. To g we associatethe tuple of conjugacy classes C = (C1, . . . , Cn).

C is rigid if G = Inn(G) acts simply transitively on the set of tuples with conju-gacy class tuple C. In fact the “simple” is redundant here, under the requirementthat G has trivial center, since any inner automorphism which fixes each gi in atuple corresponds to an element h of g which commutes with every gi, hence iscentral in G since the gi form a generating set, hence is trivial.

C is weakly rigid if Aut(G) acts transitively on the set of tuples of C.

Also, for any n, gQ, the absolute Galois group of Q, acts on the set of n-tuplesof conjugacy classes in a finite group G: if G has order N , then the action by def-inition factors through GQ(ζN )/Q ∼= (Z/NZ)× and then Cσ := (Cσ

1 , . . . , Cσn) is just

8When N is odd, the passage to 2N is not necessary: we could also find an element of order

N in SL2 whose associated cyclic group maps injectively into the projectivization. On the other

hand it is harmless, since when N is odd, (Z/2NZ)× ∼= (Z/NZ)×. When N is even one mustindeed use ζ2N instead of ζN (unless p = 2).


interpreted as raising each conjugacy class to the appropriate power modulo N .9

This allows us to define the field of rationality Fr(C) of C as the field extensioncut out by this gQ-action: note that Fr ⊂ Q(ζN ). Similarly we have the field ofweak rationality Fw(C), which is the subfield of Q(ζN ) fixed by the subgroupof (Z/NZ)× consisting of all exponents i modulo N with the property that thereexists an automorphism ϕ of G with Ci = ϕ(C) (note that the automorphism ϕcan depend on the exponent i but must be uniform for all n conjugacy classes inC). Clearly we have Fw ⊂ Fr.

Now all of our results about fields of definition and fields of moduli for curvesX(N, p) and for their automorphism groups will use the following lemma.

Lemma 12. (“WRWR Lemma”) Let G be a finite group of order N with trivialcenter, let g = (g1, . . . , gn) be a (generating!) tuple with associated conjugacy classtuple C = (C1, . . . , Cn). Assume that C is weakly rigid. Then:a) There exists a unique pair (X, β) with X = X(G, C)/Q and subgroup G ⊂Aut(X) such that

β : X(G, C) → X/G ∼= P1

is a branched covering with ramification type C.b) X(G, C) can be defined over its field of moduli, which is the field of weak ratio-nality Fw.c) There is a canonical bijection between the gQ-orbits {Xσ} and {Cσ}, the latterbeing merely the set of Ci as i ranges over elements of (Z/NZ)×.d) There is a unique minimal field of definition K for the group of automorphismsG, and we have:• Fr ⊂ K• a natural embedding ρ : GK/Fr

↪→ StabOut(G)(C),where the latter term is the subgroup of outer automorphisms of G which stabilizeeach of the conjugacy classes in C. In particular, K = Fr if G is rigid, and ingeneral we have [K : Fr] ≤ #Out(G).e) Finally, X(G, C) admits a model over Fr which is canonical in the sense of beinguniquely determined by the condition that the automorphisms of G are Fr-rational.

9It is reasonable to ask what right gQ really has to act on the conjugacy classes of our poor finite

group G. This has a good answer: let X = P1Q\D be the projectiv line over Q minus any Q-rational

divisor D of degree n (the nth roots of unity are a popular choice, although {∞, 0, 1, . . . , n − 2}would work just as well). Then the etale fundamental group of X fits in a short exact sequence

1 → π1(X) → π1(X) → gQ → 1,

with the first term being the profinite completion of the geometric fundamental group, i.e., the

profinite completion of the discrete group with the canonical presentation〈g1, . . . , gn | g1 · · · gn = 1〉. So a tuple g in G corresponds to an epimorphism π1(X) → G, whence,

by the Riemann existence theorem, an algebraic curve X covering the projective line and branched

only over the points of D. Now the short exact sequence gives rise to a continuous homomorphismρ : gQ → Out(π1(X)) – i.e., there is a compatible inverse system of actions of gQ on each finitequotient, so in particular we get ρ : gQ → Out(G). To try to say exactly what ρ is (for each

G compatibly) is an entire field of mathematics – see e.g. [Deligne 1989] – but the first step isto determine the induced action on the conjugacy classes of G, and this action is indeed by the

cyclotomic character. Note by the way, that the geometric meaning of weak rigidity is that the

resulting branched covering depends, up to isomorphism, only on C and not on the choice of tupleg.

16 PETE L. CLARK

This lemma follows almost immediately from results found in [Volklein], althoughit is not stated therein explicitly. One needs to combine Remark 3.9, parts a) - d)with Proposition 9.2b). It would be nice to have a complete self-contained proof ofthe WRWR Lemma in which all terms are expressed both group-theoretically andgeometrically. Probably I will write this up some day, but not today.

Important Remark: In view of our intended application to the (2, 3, N)-case,we have in our account of the definitions of weakly rigid and weakly rational n-tuple made a simplifying assumption: we have implicitly assumed that the branchdivisor in P1 consists of n points P1, . . . , Pn with each Pi ∈ P1(Q). But in generalfor a branched covering X → P1/Q we can say only that the ramification divisor Dis defined over Q as a divisor. To such a choice of D = P1 + . . .+Pn ∈ Divn(P1)(Q)and an n-tuple C = (C1, . . . , Cn) we assign the ramification type (G, C,D), andinclude the gQ-action on the points of D in the definition of (weak) rationality:e.g. the field of rationality of the type is the subfield of Q(ζN ) fixed by elementsσ ∈ GQ(ζN )/Q such that for all 1 ≤ j ≤ n,

CσPj

= Cσ(Pj),

with an analogous definition for weak rationality. But observe that the gQ-actioncan only permute ramification points with equal ramification indices, so when theramification indices are pairwise distinct – e.g. (2, 3, N) for N ≥ 7 – the ramifica-tion divisor is forced to be pointwise Q-rational.

In practice, this means that one will usually get more economical realizations ofGalois groups by using tuples with at least some coincident orders. In particular,Galois realizations of nonexceptional groups PSL2(Fq) over Q using triples withdistinct orders (a, b, c) do not exist (as can be shown easily using the results ofSection 3.2 and Lemma 5). On the other hand, since a favorable choice of thedivisor D can only cut down the degree of the field of rationality of the type by afactor of n!, one can deduce an absolute bound on the exponent a such that anygroup PSL2(Fpa) can be the Galois group of a covering X → P1/Q ramified at threebranch points. Indeed, Macbeath’s theorem shows that the traces of the triple mustgenerate Fpa , and therefore the Galois orbit of C = (C1, C2, C3) must have size atleast a (with inequality occuring only if there are unipotent conjugacy classes). Ifthe orders of two of the conjugacy classes coincide, then perhaps we can choosethe divisor D such that the field of rationality of the type has degree a/2. If allthree orders coincide, then the degree of the field of rationality is still at least a/6.In fact, since the gQ action on conjugacy classes factors through Qab, it is clearlyimpossible for gQ to permute the three conjugacy classes via S3, and the degree ofthe field of rationality is at least a/3. It follows that only PSL2(Fp), PSL2(Fp2) orPSL2(Fp3) can be Galois groups of Belyi-Wolfart curves with all automorphismsdefined over Q. The first two cases occur for sets of primes p of positive density:e.g. PSL2(Fp) can be realized if any of 2, 3 or 5 is a quadratic nonresidue mod p,and PSL2(Fp2) can be realized if 5 is a quadratic nonresidue mod p. For exponentsa ≥ 3, exactly one group PSL2(Fpa) is known to occur as a Galois group over Q(by any means): pa = 23 = 8, via a rigid (9, 9, 9) triple [Malle-Matzat].

4.4. Semirigidity in PSL2(Fpa). Let g be any generating triple in PSL2(Fq)which is not exceptional, i.e., the triple of orders of the elements (a, b, c) is not one


of the small orders found in the list (1). It is automatically non-commutative, asPSL2(Fp) is never an abelian group. Then Macbeath’s implies that the associatedconjugacy class triple C is weakly rigid, so that the WRWR Lemma may be applied.Theorem 2c) also tells us that C is in certain cases rigid but generically not: werecord the following result.

Proposition 13. Let g be a generating nonexceptional triple in G = PSL2(Fpa) ofelements having orders (a, b, c) with associated conjugacy class triple C. Then C isweakly rigid, and is rigid if and only if p = 2 or p |abc. In these cases the minimalfield of definition of G is Fr (which can easily be computed in terms of (a, b, c)).Otherwise we are in a “semirigid” situation: StabOut(G)(C) = 〈τ〉 ∼= Z/2Z, and itfollows that the minimal field of definition K of the automorphisms of G is an atmost quadratic extension of Fr.

Proof: We saw above that if the triple generates PSL2(Fq) then the traces ofthe elements generate Fq. It follows that the smallest power of Frobenius fP thatstabilizes C is fa

P . So the only nontrivial outer automorphism that could possiblystabilize C is τ . Note first that τ exists only if p is odd, disposing of the case p = 2.Moreover, if p divides abc, then at least one of a, b or c is equal to p – i.e., oneof the conjugacy classes is unipotent – and τ is easily seen to interchange the twounipotent conjugacy classes (if p > 2). Therefore the stabilizer is trivial in thiscase, which is equivalent to rigidity. In the general case, the automorphism τ doesindeed stabilizer C and is needed in order to get a transitive action of Aut(G) onthe triples of C. This completes the proof.

Remark: Along with our Theorem 3, this Proposition also explains the resultsof [Streit] and of [Schmidt-Smith, Section 3].

4.5. The curves X(N, p). Theorem 3 will follow immediately from Propositions5 and 7 as soon as we say what the curves X(N, p) are! Indeed, fix N ≥ 7 andp a prime subject to the conditions of Section 3.2. By Proposition 3, we have atriple (g1, g2, g3) ∈ PSL2(Fpa) of elements of orders (2, 3, N). Changing our nota-tion slightly, we now denote by Ch a conjugacy class of order h. There is alwaysa unique conjugacy class of elements of order 2, so we denote this by C2. For anyconjugacy class Ch of order h prime to p, the Galois orbit of Ch gives all conjugacyclasses of elements of order h (we are just taking the Galois orbit of the trace,ζ2h + ζ−1

2h ), so there are ϕ(2h)/2 such in all, and the field of rationality of such aclass is F2h := Q(ζ2h + ζ2h). In particular, since ϕ(2 ∗ 3)/2 = 1, whenever p 6= 3there is a unique conjugacy class of elements of order 3, which we may denote C3.When p = 3 there are two such unipotent classes, which we denote C ′

3 and C ′′3 .

These two classes are Galois conjugates (see Case 1 below for the generalization tounipotent conjugacy classes in odd characteristic), and the field of rationality foreach is Q(

√−3) = Q(

√3∗).

Now we turn to a consideration of the various rigid cases, followed by the genericsemirigid case.

Case 1 (N = p ≥ 7): There are two unipotent conjugacy classes of order p, de-noted C ′

p and C ′′p , so a (2, 3, N)-triple must have associated conjugacy class triple

either (C2, C3, C′p) or (C2, C3, C

′′N ). Since any unipotent conjugacy class can be

18 PETE L. CLARK

represented by a strictly upper triangular matrix, and since the ith power of sucha matrix corresponds to multiplying the (1, 2)-entry by i, we can get from the qua-dratic residues to the quadratic nonresidues in Fp this way, and C ′

N and C ′′N are in

the same gQ-orbit. So the field of rationality for these classes is Q(√

p∗); since C2

and C3 are Q-rational, the field of rationality of C is Fr = Q(√

p∗). Moreover, aswe saw above, the outer automorphism τ interchanges C ′

N and C ′′N , which means

that these classes are weakly Q-rational, so Fw = Q. Since by Lemma 5 for eachramification type we get a curve X(G, C), and the field of moduli of this curve isFr, in this case we have geometrically one curve X(p, p)/Q, whose field of moduliis Q (so that, as a Belyi-Wolfart curve, it admits models over Q, but not in anycanonical way). Moreover the action of the automorphism group is determined overK = Q(

√p∗).

In fact X(p, p) = X(p), the classical modular curve with full level p structure. Tosee this, because of the weak rigidity, one only needs to find a covering from X(p) tothe projective line, ramified at three points, with Galois group PSL2(Fp) and ram-ification type (C2, C3, C

′N ). Of course the canonical modular cover X(p) → X(1)

has this property, the three points of ramification being – taking j as the coordinatefunction – ∞, 0, 1728.

We want to point out that the Belyi-Wolfart and WRWR theories have some usefulthings to say even in this very classical case. First, we get that, for N = p ≥ 7,PSL2(Fp) is the full automorphism group of X(p): indeed this has come up re-cently in an arithmetic context [Mazur]. More importantly, we have shown thatX(p) admits a model over the imaginary quadratic field K which is canonical inthe sense of Lemma 5e). This is certainly known to some – indeed it was knownto [Shih] who deduced from it that for all p ≥ 7 PSL2(Fp) occurs regularly as aGalois group over K10 However in many standard treatments of modular curves itis adamantly maintained that X(p) is defined minimally over Q(ζn), or at best overQ as a geometrically disconnected curve. This is simply not the case.

Example: We single out the case p = N = 7. Then G = PSL2(F7) is – inaddition to being the modular curve X(7) – Klein’s quartic curve. This is thesmallest nontrivial finite quotient of ∆(2, 3, 7), as is almost immediate to see: since(2, 3, 7) is a perfect triple, the minimal quotients will be simple groups, of whichG is the second-smallest (of order 168). The only smaller one is A5, whose order,60, is not divisible by 7. Since the full automorphism group is simple, the curve isnonhyperelliptic (a hyperelliptic involution gives a nontrivial element of the centerof the automorphism group of a hyperelliptic curve). Since the genus is 3, X(7, 7) iscanonically embedded in P2, i.e., is a smooth plane curve. To see how to derive theequation x3y + y3Z + z3x = 0 from the canonical K = Q(

√−7)-representation of

G on the cotangent space – so that one finds 3× 3 matrices with K-rational entriesleaving the equation invariant – see [Elkies]. This article contains a wealth of otherinformation about X(7, 7) and another family of Hurwitz curves that we will seelater (and was the stimulus for the present work): we mention for now only the fact

10To be sure, this is the easy step of his argument; he went on to show by “X0(p)-methods”

that PSL2(Fp) occurs rationally over Q for certain congruence conditions on p. We have some

unjustified hopes of emulating his method in the cases N = 7, 11.


that the Jacobian of X(7, 7) decomposes as E3, where E is the unique (geometric)elliptic curve with CM by the maximal order of K.

Case 2 (p = 2): As in Proposition 5 this case is also rigid, but for a differentreason: the outer automorphism group of G = PSL2(F2a) lacks the element τ .The possible conjugacy class triples are of the form (C2, C3, C

σN ), where C2 and C3

are fixed but CN has a GalQ orbit of size ϕ(2N)/2, and we get that the field of ra-tionality Fr of any Cσ is F2N . As to the field of weak rationality, Frobenius fp actseffectively on the Galois orbit of CN , so cuts down the degree of the field extensionby a factor of a: that is, Fw is the unique subfield of F2N of degree d(N, a) := ϕ(2N)

2a

over Q – this number is also the order of 2 in (Z/NZ)×/(±1). So we get one Galoisorbit of curves, but not necessarily a unique curve up to moduli. We choose torepair this by defining X(N, 2) :=

⋃σ∈gQ

Xσ, i.e., as a possibly disconnected curve.Note then that X(N, 2) always admits a Q-rational model.

Case 3 (p = 3): Again we are in the rigid case for the same reason as Case 1. Ourpossible conjugacy class triples are (C2, C

′3, C

σN ) and (C2, C

′′3 , Cσ

N ), these classes, forequal σ, being conjugate under the outer automorphism τ . We get then that thefield of rationality is Fr = F2N (

√−3), which is equal to the minimal definition of

the automorphism group G. The field of weak rationality will again be the subex-tension of Fr of degree ϕ(2N)/2a over Q, and we define X(N, 3) :=

⋃σ∈gQ

X(G, C).

Case 4 (gcd(6N, p) = 1): In this case we have no unipotent conjugacy classes,so C2 and C3 are uniquely determined and CN is well-determined up to a Galoisorbit, so we have Cσ = (C2, C3, C

σN ). As discussed in Proposition 5, τ now preserves

Cσ, giving a semirigid situation. So the field of rationality is Fr = F2N , the fieldof weak rationality is again the subextension of Fr of degree ϕ(2N)/2a over Q, butnow what we can say about the minimal field of definition of the endomorphismsK is that Fr ⊂ K with index at most 2. This completes the proof of Theorem 1.

Note: In all the rigid cases we get that K contains Q(√

p∗), assuming the (rea-sonable) convention that Q(

√2∗) = Q. Thus it seems plausible that in the generic

case the field K is in fact F2N (√

p∗), so in particular is abelian over Q.

Example: We single out the (rigid) case p = 2, N = 7. Then G = PSL2(F8)is the Fricke-Macbeath curve, of genus 7, the second-smallest genus for a curve uni-formized by a subgroup of the “Hurwitz group” ∆(2, 3, 7). This curve has field ofmoduli Q and has a canonical model defined over the real cubic field F3 = Q(ζ7+ζ7),which is also the minimal field of definition for tbe action of its automorphisms.[Macbeath 2] finds a beautiful system of defining equations for the canonicallly em-bedded curve over Q(ζ7). In his model the automorphisms are given as explicit 7×7matrices with Q-rational coordinates. There is an analogous (but more amazing)isogeny decomposition of the Jacobian of J(7, 2): it is isogenous to E7, where Eis a non-CM elliptic curve with rational j-invariant. Indeed, as a Q(ζ7)-basis forthe global differentials on X(7, 2) Macbeath writes down 7 genus one curves (notin Weierstrass normal form), each defined over Q(ζ7): these curves have the sameQ-rational j-invariant.

20 PETE L. CLARK

We could equally well single out the curves X(11, 2) – with automorphism groupPSL2(F25) defined minimally over F11 – and X(19, 2) – with automorphism groupsPSL2(F29) defined minimally over F19 and both possessing noncanonical Q-rationalmodels, except that we know nothing about them. Indeed their genera are alreadygetting out of hand: XXX and XXX. Can one dare to hope that their Jacobianssplit as powers of a single elliptic curve with rational j-invariant?

5. The Shimura curves

Finitely many of the families X(N, p) correspond to Shimura curves associatedto quaternion algebras over totally real fields FN which are ramified at no finiteplace of FN : the values of N are

N = 7, 9, 11, 14, 18.

Note that the condition of a quaternion algebra B/F being unramified at all finiteplaces implies that the degree of F/Q is odd. This provides further justification forto the restriction to odd degree extensions of Fp, since there is a larger finite list ofquaternionic Shimura curves commensurable with ∆(2, 3, N) [Takeuchi], but in theother quaternionic cases there is a prime p dividing the quaternionic discriminantD (and dividing 6N), so that the Galois group of XD(p)/XD(1) is an irregularprojective group, i.e., of the form PGL2(k)..

We note also that the canonical models of Shimura curves XD(p) for D/FN aredefined over FN (ζp) – or at least, I think they are, but this needs to be checked.(For instance, if one puts the fields FN (pn) of definition of XD(pn) into a tower,one should get a Zp-extension of FN . But, being a totally real field, FN has noZp-extensions other than the cyclotomic one.)

6. Appendix: Some results of Belyi-Wolfart Uniformization Theory

In this section we will give a summary account of the arithmetic theory of Riemannsurfaces uniformized by subgroups of cocompact Fuchsian triangle groups. In thissection we work in the context of (a, b, c)-triangle groups, since most of the theoryworks in this level of generality; moreover, by getting a glimpse of the generalsetting we see the one thing that is gained by restricting to (a, b, c) = (2, 3, N) –namely, maximality is guaranteed – and also the features distinct to the case ofgcd(6, N) = 1.

6.1. Abstract and geometric triangle groups. Fix positive integers (a, b, c).Our basic construction will be symmetric in a, b and c, so we may and do assumethat a ≤ b ≤ c. Define f(a, b, c) = 1

a + 1b + 1

c . We say (a, b, c) is spherical, Eu-clidean or hyperbolic according to whether f(a, b, c) is, respectively, greater than1, equal to 1 or less than 1. As in the sequel we are interested almost exclusively inthe hyperbolic case, we record explicitly the list of spherical and Euclidean triples,namely:

spherical: (1, b, c), (2, 2, c), (2, 3, 3), (2, 3, 4), (2, 3, 5).

Euclidean: (2, 4, 4), (2, 3, 6), (3, 3, 3).

All others are hyperbolic. Note in particular that (2, 3, N) is hyperbolic if and


only if N ≥ 7.

Associated to any triple (a, b, c) we define the abstract triangle group

∆ = ∆(a, b, c) = 〈x, y, z | xa = yb = zc = xyz = 1〉

Notice that ∆ is actually generated by x and y, since z = (xy)−1; this leads to thealternate presentation

∆ = 〈x, y | xa = yb = (xy)c = 1〉,

which is also useful to keep in mind. I suppose the names “spherical,” “Euclidean”and “hyperbolic” are sufficiently suggestive to give away the geometric interpreta-tion, but just for a moment consider this trichotomy in terms of pure group theory.The three kinds of groups are indeed quite different from each other.

Indeed, the spherical groups are finite: ∆(1, b, c) ∼= Z/(gcd(b, c)), ∆(2, 2, n) ∼= Dn,the dihedral group of order 2n;11

∆(2, 3, 3) ∼= A4 (under x 7→ (234), y 7→ (214));∆(2, 3, 4) ∼= S4 (under x 7→ (1234), y 7→ (243));and ∆(2, 3, 5) ∼= A5 (under x 7→ (12345), y 7→ (153).)

The Euclidean groups are infinite and nonabelian, but solvable: indeed in eachof the three cases they fit into an exact sequence

1 7→ [∆,∆] ∼= Z2 → ∆ → ∆ab → 1.

We can at least compute the maximal abelian quotient ∆ab in general: it is thequotient of the finite abelian group Z/(a)×Z/(b) by the cyclic subgroup generatedby (c, c). This comes out to

∆(a, b, c)ab = Z/ gcd(c, lcm(a, b))Z× Z/ gcd(a, b)Z.

In particular, ∆ = [∆,∆] is perfect if and only if (a, b, c) are relatively primein pairs: we speak of a “perfect triple” in this case. In the perfect case, no fi-nite quotient of ∆ is solvable, or equivalently, the minimal finite quotients of ∆are all nonabelian simple groups. Note that this “explains” why the simple groupA5 = ∆(2, 3, 5) sneaks into the list of spherical groups. Note also that ∆(2, 3, N)is perfect if and only if N ∼= ±1(6), and in particular if N ≥ 7 is prime.

The hyperbolic groups are infinite and nonsolvable: indeed, every hyperbolic ∆admits a torsionfree finite index normal subgroup Γ, and also such groups Γ areisomorphic to Π(g) for some g, where

Π(g) = 〈x1, . . . , xg, y1, . . . , yg | [x1, y1] · · · [xg, yg] = 1〉,

i.e., to the fundamental group of a compact Riemann surface of genus g. In par-ticular in every case ∆ is residually finite (i.e., injects into its profinite completion)and virtually torsionfree. The existence of a finite index subgroup which is tor-sionfree is the crux of the matter (if such a group exists, then a normal one existsjust by intersecting all the conjugates of a given group; the fact that such groups

11When n = 2 this must be interpreted as the symmetry group of the “regular 2-gon,” i.e. ofthe line segment [−1, 1] in the plane: D2

∼= Z/2Z× Z/2Z

22 PETE L. CLARK

are isomorphic to Π(g) for some g also follows easily): we call this the uniformiza-tion theorem for triangle groups. It does not follow directly from the geometricinterpretation in terms of Fuchsian groups, and several of the standard textbookson Fuchsian groups do not give a proof. Fortunately, a wonderful discussion can befound in [Stillwell], where the uniformization theorem indeed appears as the climaxof the entire book.

Now for the geometric interpretation. For any ∆ = ∆(a, b, c) there is a trian-gle whose angles are π

a , πb , π

c : since the angle sum is π(f(a, b, c)), this triangle liveson the Riemann sphere, in the Euclidean plane or in the hyperbolic plane accordingas f(a, b, c) is positive, zero or negative. In any of the appropriate geometries, itmakes sense to consider the subgroup 4 of isometries generated by the reflectionsτ1, τ2, τ3 in the three sides of the triangle. 4 is a discrete group: indeed thetriangle itself is a fundamental domain for its action. But note that the reflectionsare orientation-reversing isometries, so it is natural to consider the subgroup oforientation-preserving isometries, denoted 4 = 4(p, q, r): indeed this group is gen-erated by the three elements R1 = τ1 ◦ τ2, R2 = τ2 ◦ τ3, R3 = τ3 ◦ τ1, which, as thecomposition of reflections with intersecting fixed lines, are rotations through twiceeach angle, i.e., through 2π

a , 2πb , 2π

c respectively. Clearly we have

Ra1 = Rb

2 = Rc3 = 1 = (τ1 ◦ τ2) ◦ (τ2 ◦ τ3) ◦ (τ3 ◦ τ1) = R1R2R3,

whence an epimorphism ϕ : ∆(a, b, c) → 4(a, b, c). It is not too hard to show –using, e.g., the method of Cayley graphs – that ϕ is indeed an isomorphism, so weare allowed to view our abstract triangle groups as a group of spherical, Euclideanor hyperbolic isometries.

6.2. The spherical triangle groups. The spherical groups are nothing else thanall the finite groups of orientation-preserving isometries of the Riemann sphere, i.e.,finite subgroups of SO(3), which were classified by Klein. The first two families∆(1, b, c) and ∆(2, 2, c) are not so exciting, corresponding to the groups whichare already subgroups of O(2)12: indeed in the first case we just draw gcd(p, q)meridians on the sphere (here one of the angles is π

1 , i.e., is not actually there atall!) and in the second case we draw r meridians and an equator. The remainingthree groups are the orientation-preserving symmetry groups of the platonic solids:A4 is the automorphism group of the tetrahedron, S4 is the common automorphismgroup of the octahedron and the cube, and A5 is the common automorphism groupof the dodecahedron and the icosahedron. Note that each group corresponds toa unique triangulated solid, and by inscribing this solid in the sphere and thenprojecting onto the boundary we get the three desired triangular tesselations of theRiemann sphere. Note that the one simple group, A5, really is the most interestingof the bunch. For instance, consider the three-manifold X = SO(3)/A5. Since thefundamental group of SO(3) (which is homeomorphic to RP3) is Z/2Z, we get anexact sequence

1 → Z/2Z → A5 := π1(X) → A5 → 1.

12Reflections in the plane can be viewed as 180 degree rotations in three-dimensional space,whence O(2) ↪→ SO(3).


Here A5 is a nontrivial central extension of A5 by Z/2Z which is, like A5, perfect[CITE?]. It follows that H1(X, Z) = A5

ab= 0, so it follows from Poincare duality

that all the homology groups of X are the same as those of S3: X is what is knownas a homology three-sphere. In fact X (and S3) are the only known homologythree-spheres with finite fundamental group.

6.3. The Euclidean triangle groups. Consider now the three Euclidean cases(3, 3, 3), (2, 4, 4) and (2, 3, 6). Here the associated triangular tilings are among themost familiar of all mathematical objects: the first is just the tiling of the planeby equilateral triangles, the second by isosceles right triangles and the third by30-60-90 right triangles. Note that the first and the third give tilings which arecommensurable: group-theoretically, this corresponds to an embedding of trianglegroups (3, 3, 3) ↪→ (2, 3, 6), of index 2. Let us examine the consequences of ourabove claim that the commutator subgroup Γ := [∆,∆] is isomorphic to Z2, i.e.,is torsionfree and hence gives a uniformizing subgroup. Now the only subgroups ofisometries of the Euclidean plane isomorphic to Z2 are the lattices in C, so thatE = Γ\C is an elliptic curve, and we have a canonical map E = Γ\C → ∆\C = P1

ramified at precisely three points and Galois. Now this is a very rare event: for allbut two elliptic curves E/C, the automorphism group is E(C) × | Z/2Z. SupposeG is a finite subgroup of Aut(E). Then G has a normal subgroup Gτ consist-ing of translations (by torsion elements of E), and E → E/G factors throughE → E/Gτ → (E/Gτ )/(G/Gτ ). Since E/Gτ is another elliptic curve (isogenousto E), we are reduced to the case of subgroups of automorphisms fixing the origin.But now for all but two elliptic curves, G/Gτ

∼= ±1, and E → E/±1 is ramifiedover E[2], which has order 4.

Suppose on the other hand that E has complex multiplication by Z[ζ4] or by Z[ζ3],so the automorphism groups are cyclic of order 4 and 6, respectively. In the firstcase, put G = 〈ζ4〉, and E → E/G does the job: it is ramified over the fixed pointsof the elements of this group which are 0, 1/2 + 1/2ζ4 (for the generators) andE[2] for −1. But the point is that there are three – as opposed to four – orbitsof G on E[2]: the points 1/2 and 1/2ζ4 are identified, so there are indeed threebranch points. Note that this leads to ramification orders of 4, 4 and 2 over β(0),β(1/2 + 1/2i) and over β(1/2) = β(1/2i) respectively. Suppose now E has CM byZ[ζ3]. Put G = 〈ζ3〉. This time, the locus of points of E over which the quotientmap is not a local homeomorphism is contained in E[3], and by a similar analysiswe get that on the quotient curve there are only three branch points.

Note that in all cases (spherical, Euclidean or hyperbolic), since ∆ ⊂ ∆ with index2, a fundamental domain for ∆ consists of two of the basic triangles: indeed, it isenough to reflect the (a, b, c)-triangle through any of its sides: this quadrilateral isa fundamental domain. If its vertices are labelled counterclockwise as A,B,C, D,then the identified sides are AB ⇐⇒ AD and CB ⇐⇒ CD, so that the quotientspace is a compact Riemann surface of genus zero, i.e., is isomorphic to the complexprojective line.

6.4. Belyi’s Theorem and Wolfart’s Reinterpretation. Fix a triangle group∆ = ∆(a, b, c). From now on, unless otherwise specified, it will be assumed that ∆is hyperbolic.

24 PETE L. CLARK

Now let ϕ : ∆ → G be an epimorphism from ∆(a, b, c) to a finite group. Wesay that ϕ is nondegenerate if the images of x, y, and z have orders a, b andc repsectively (and not any smaller orders). Note that when (a, b, c) is perfect, ϕis degenerate only if G is trivial group. Let Γ be the kernel of ϕ, a finite-indexnormal subgroup of ∆. The nondegeneracy of ϕ implies that Γ is torsionfree, so asa cocompact Fuchsian group necessarily consists entirely of hyperbolic matrices. Itfollows that H → X = Γ\H is an unramified covering map, so exhibits H as theuniversal cover and Γ as the fundamental group of the compact Riemann surfaceX. Moreover, the natural map

X = Γ\H → ∆\H = P1

exhibits X as a finite covering of the projective line, branched over precisely threepoints (the images of the vertices of the triangle) which, owing to the triple transi-tivity of the automorphism group of P1, we may take to be 0, 1 and ∞.

These are very classical facts. But note that any Riemann surface branched over0, 1,∞ has algebraic moduli, i.e., can be defined over Q. More generally, any Rie-mann surface branched over P1 with branch points P1, . . . , Pn ∈ P1(Q) can bedefined over Q. This follows easily from a basic finiteness result which is the geo-metric analogue of Hermite’s theorem that there exist only finitely many numberfields of given absolute degree and unramified outside a fixed finite set of primenumbers, and the geometric analogue is arguably easier: it is enough to observethat a degree d cover of P1 unramified outside of P1, . . . , Pn corresponds to an in-dex d subgroup of π1(P1 \ {P1, . . . , Pn}), which is a free group on n− 1 generators,and indeed there are only finitely many such subgroups (this fact can be swiftlyreduced to the fact that there exist up to isomorphism only finitely many graphswith d vertices and dn edges, which is indeed obvious: see e.g. [Massey] for the fullargument). The moduli space M(0, n) of n marked points on the Riemann sphereup to isomorphism has dimension n − 3, so every such covering is algebraic whenn ≤ 3. Note that the cases of n = 1 and n = 2 are almost trivial: indeed since theaffine line A1 = P1 − {P1} remains simply connected13 every nontrivial branchedcover of P1 has at least two branched points. Similarly, the fundamental group ofGm = P1\{P1, P2} is Z, so finite covers with two branched points have cyclic Galoisgroups (and in particular they are necessarily Galois, since every subgroup of Z isnormal) so are given by Kummer theory as C(f(z)1/n) for some rational functionf ∈ C(z). Moreover, we may take the two branch points to be 0 and∞, so that f(z)is necessarily of the form czk and hence the field extension is of the form C(z1/m)for some m and is still a rational function field – i.e., only P1 can cover itself withtwo branch points (this can also be seen from the Riemann-Hurwitz formula).

Given this, one would expect that only a very small subset of all algebraic curvesX/Q can be uniformized by subgroups of hyperbolic triangle groups. Thus amaze-ment and admiration greeted Belyi’s announcement at the 1979 ICM that everyalgebraic curve defined over a number field is branched over the projec-tive line with three branch points. What is yet more amazing is that proof fitseasily onto two pages (the portion of Grothendieck’s Esquisse where he describes

13in characteristic zero!!


his own interaction with this result – daring to make the “crazy” conjecture, beingtold by Deligne that the result was “crazy indeed” but without any known coun-terexample, and then receiving a slim letter from Deligne reporting from the ICMand giving Belyi’s complete proof – makes compelling reading).

6.5. . . . and the rest, more briefly. Unfortunately our expository zeal has tem-porarily waned just as we get to the more interesting material. We state the resultsthat we need to use more concisely.

Theorem 14. Let ∆(a, b, c) be any triangle group, Γ any Fuchsian group, andsuppose ∆ ⊂ Γ.a) Then Γ is itself (conjugate to) an arithmetic triangle group ∆′ = ∆(a′, b′, c′).b) [Greenberg 1963] Moreover ∆(a, b, c) = Γ unless (a, b, c) is of one of the followingforms, said to be non-maximal:

(2, b, 2b), (3, b, 3b), (a, a, c).

A triangle group ∆(a, b, c) with values not in the above list is called maximal, andthe theorem says that such groups are indeed maximal among all Fuchsian groups.This is important in the computation of automorphism groups of curves, as follows.If X is a compact Riemann surface of genus at least two with uniformizing subgroupΓ, then Aut(X) = NΓ/Γ, where NΓ is the normalizer in all of PSL2(R): it is itselfa Fuchsian group. So if Γ ⊂ ∆(a, b, c) is a finite-index subgroup of an arithmetictriangle group, the theorem implies that the full automorphism group of X is ofthe form ∆(a′, b′, c′)/Γ, so that X → X/Aut(X) is a Belyi map, as claimed inthe introduction. Moreover, except for the special nonmaximal values of (a, b, c)listed in the theorem, a nondegenerate map ϕ : ∆(a, b, c) → G gives rise to aBelyi=Wolfart curve X whose full automorphism group is G. Note especially thatfor N ≥ 7, ∆(2, 3, N) is a maximal Fuchsian group.

Theorem 15. ([Wolfart 1997], [Wolfart 2000]) Let X/C be a compact Riemannsurface of genus at least 2. The following conditions are equivalent:a) There exists H ⊂ Aut(X) such that X → X/H is a Belyi map.b) The canonical map X → X/Aut(X) is a Belyi map.c) There exists an analytically open neighborhood U of X in the moduli space Mg

of genus g curves such that for all Y ∈ U , # Aut(Y ) < # Aut(X).d) There is no nontrivial deformation of the pair (X, Aut(X)).e) The uniformizing subgroup Γ of X is a finite-index normal subgroup of an arith-metic triangle group ∆(a, b, c).Under any of these conditions, X can be defined over Q and is called a Belyi-Wolfartcurve.

Corollary 16. Fix any positive integer g ≥ 2 and any finite group G. There are atmost finitely many Belyi-Wolfart curves with Galois group G (of any genus), andat most finitely many Belyi-Wolfart curves of genus g (with any Galois group).

Proof: First we observe that the two statements are equivalent. Indeed, let Xbe a Belyi-Wolfart curve of genus g, with Galois group G of order N and whosecanonical Belyi map has ramification indices (a, b, c). Then the Riemann-Hurwitzformula gives

g = 1 +N

2(1− 1

a− 1

b− 1

c).

26 PETE L. CLARK

Since among hyperbolic triples the parenthesized expression is minimized at (a, b, c) =(2, 3, 7), we get the Hurwitz bound g ≥ 1 + N

84 . On the other hand, certainlymax a, b, c ≤ N , so N−1

2 ≥ g. Together with the observation that there are onlyfinitely many finite groups of any given order N , this gives the equivalence. Nowthe finiteness of the number of Riemann surfaces branched only over {0, 1,∞} withgiven Galois group (or even given order of Galois group) has been noted earlier(“geometric Hermite theorem”), and the result follows.

Remark: In [Wolfart 2000] one can find a complete list of all Belyi-Wolfart curvesof genus g where 2 ≤ g ≤ 4. For each curve the following information is given:defining equations, a list of subgroups H such that X → X/H is a Belyi map,and it is recorded whether or not the curve is hyperelliptic and whether or not theJacobian is of CM type. Examining this list is a highly educational experience, andit would be worthwhile to extend it to higher genus.

Theorem 17. Any Belyi-Wolfart curve can be defined over its field of moduli.

This theorem is to an extent folklore: it was remarked in [Debes-Emsalem] that itfollows easily from results of [Coombes-Harbater]. The proof, which is quite simple,seeems to have been written down for the first time in [Kock].

6.6. From Belyi maps to Belyi-Wolfart curves. Despite having said that Belyipairs β : X → P1 are ubiquitous and Belyi-Wolfart curves quite rare, there is anobvious way to go from one to the other: indeed, if β : X → P1 is a Belyi map(defined over Q), it induces an extension of function fields Q(X) → Q(t), andwe may simply take the Galois closure. As the compositum of unramified localextensions remains unramified, this Galois closure will still be a Belyi map, i.e.,every Belyi map gives rise to a Galois covering of an algebraic curve by a Belyi-Wolfart curve:

Y → Xβ→ P1,

where P1 = Y/Aut(Y ) and X = Y/H for some subgroup H ⊂ Aut(Y ). From thisit follows immediately that there are Belyi-Wolfart curves of arbitrary arithmeticcomplexity. Indeed, let K be any number field, and let E/K be an elliptic curvewithout complex multiplication whose j-invariant generates K. (E.g., take anygenerator α of the number field K, and divide by a sufficiently large integer n toensure that α/n is not integral.) Let β : E → P1 be a Belyi map (whose existence isguaranteed by Belyi’s theorem). Since the only two Belyi-Wolfart curves of genusone are the CM elliptic curves with j = 0 and j = 1728, and since every coveringof a genus one curve by a genus one curve is an unramified covering which, uponsuitable choice of basepoints, can in the geometric setting be viewed as an isogenyof elliptic curves, it is clearly impossible that the curve Y we obtain as the Galoisclosure of β has genus one. Moreover, although it is possible that the field of defi-nition of Y is smaller than K, if H is the subgroup of Aut(Y ) such that Y/H = E,then if X and all elements of H were defined over a field L, then so would E bedefined over L. Therefore the arbitrary field K is contained in the minimal field ofdefinition of (Y,Aut(Y )).

As remarked in the introduction, this construction gives us no clue on what theautomorphism group G might be. The main result of this paper says, morally, thatif K is sufficiently complicated (not given as the quadratic extension of an abelian


number field), then PSL2(Fq) will never arise via this construction. We are leftto speculate what groups do arise, and whether only a small fraction of all Belyi-Wolfart curves are obtained in this way.

Finally, even the question of which finite groups G can be the automorphism groupsof Belyi-Wolfart curves seems to be an interesting one. Note that if we asked whichgroups can be (sub)groups of automorphisms of Belyi-Wolfart curves, the answerwould be immediate: a necessary and sufficient condition is that G can be generatedby two elements, so e.g. (Z/NZ)r never arises for r ≥ 3. The abelian examplesare quite misleading: many, many finite groups which are “sufficiently nonabelian”can be generated by two elements: in particular every finite simple group has thisproperty. On the other hand, if we ask which groups can be the full automorphismgroup of a Belyi-Wolfart curve, the question seems to be more subtle, owing to theexistence of non-maximal triples.

Consider the simplest possible examples: when can Z/pZ (p a prime) be the fullautomorphism group of a Belyi-Wolfart curve (of genus at least 2)? The ramifica-tion degrees are forced to be (p, p, p), and applying Riemann-Hurwitz we get that ifX → P1 is a cyclic Galois cover of order p, fully ramified over three branch points,then g(X) = p−1

2 . Thus p = 2 and p = 3 are excluded by definition, and at p = 5we would get a curve of genus 2, so necessarily possessing a hyperelliptic involutiongiving an extra automorphism. Therefore p is at least 7.

On the other hand, Kummer theory applies to this situation, and any such curveX as above is necessarily superelliptic, of the form yp = xa(x− 1)b, with 1 ≤ a, b ≤p− 1. Indeed, moving the third branch point from infinity to some finite point, weget the more symmetrical equation

yp = (x− x1)a1(x− x2)a2(x− x3)a3

with arbitrary numbers x1, x2, x3 ∈ A1/Q and a1 + a2 + a3 = p. Sometimes it isclear that there are extra automorphisms: e.g. if a1 = a2, then there is an involu-tion switching x1 and x2 and preserving the defining equation. This is actually ahyperelliptic involution, and leads to the family of curves

y2 = xp − 1,

whose full automorphism group is Z/2Z × Z/pZ. In general, it is not clear to mewhether there will be extra automorphisms. For instance, taking p = 7, then theonly triple (a1, a2, a3) distinct entries, up to the action of Aut(Z/pZ) = (Z/pZ)×,is (a1, a2, a3) = (1, 2, 4), giving the superelliptic curve y7 = x2(x− 1). But this is amodel for Klein’s quartic curve! – see [Elkies] – and the full automorphism groupis PSL2(F7). Thus Z/7Z is not the automorphism group of a Belyi-Wolfart curve.(This can also be deduced from Wolfart’s list of all Belyi-Wolfart curves of genus3.) But I don’t know how it goes for higher p (although probably someone does):e.g. does

y11 = (x− x1)(x− x2)3(x− x3)7

have automorphism group exceeding Z/11Z?

28 PETE L. CLARK

References

[Belyi] Belyi. Galois extensions of a maximal cyclotomic field, 1980.

[Coombes-Harbater] Coombes-Harbater. Hurwitz families and arithmetic Galois groups, 1985.

[Darmon-Mestre] Darmon and Mestre. Courbes hyperelliptiques a multiplications reelles etune construction de Shih, 2000.

[Debes-Emsalem] Debes-Emsalem. On fields of moduli of curves, 1999.

[Deligne 1971] Deligne. Travaux de Shimura, Springer Lecture Notes 244, 1971.

[Deligne 1989] Deligne. Le groupe fondamental de la droite projective moins trois points,

MSRI Pub. 16, 1989.

[Elkies] Elkies. The Klein quartic in number theory, 1999.

[Greenberg] Greenberg. Maximal Fuchsian groups, 1963.

[Kock] Kock. Belyi’s theorem revisited, http://arxiv.org/pdf/math.AG/0108222.pdf.

[Macbeath1965] Macbeath. On a curve of genus 7, 1965.

[Macbeath1969] Macbeath. Generators of the linear fractional groups, 1969.

[Malle-Matzat] Malle and Matzat. Inverse Galois Theory, SMM, 1999.

[Massey] Massey. A Basic Course in Algebraic Topology, GTM 127.

[Mazur] Mazur. Open problems regarding rational points on curves and varieties,

LMS 254, 1996.

[Schmidt-Smith] Schmidt and Smith. Galois orbits of principal congruence Hecke curves,2003.

[Shih] Shih. On the construction of Galois extensions of function fields and num-

ber fields, 1974.[Streit] Streit. Fields of definition and Galois orbits of the Macbath-Hurwitz

curves, 2000.


[Suzuki] Suzuki. Group theory. II, Grundelehren 248, 1986.

[Takeuchi] Takeuchi. Commensurability classes of arithmetic triangle groups, 1977.

[Volklein] Volklein. Groups as Galois groups. An introduction., Cambridge Studies53, 1996.

[Wolfart 1997] Wolfart. Wolfart. The “obvious” part of Belyi’s theorem and Riemann

surfaces with many automorphisms, 1997.

[Wolfart 2000] Triangle groups and Jacobians of CM type, available athttp : //www.math.uni− frankfurt.de/ wolfart/wolfart.html

ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE …math.uga.edu/~pete/psl2four.pdf · 2008-12-10 ·...

Documents

Transcript of ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE …math.uga.edu/~pete/psl2four.pdf · 2008-12-10 ·...