Applications of projective geometryjessica2.msri.org/attachments/13550/13550.pdfApplications of...

Applications of projective geometry

January 2009

Euclid (Elements, Book I)

An old result

Dans certains pays d’Europe, dont la France, le theoreme deThales designe un theoreme de geometrie qui affirme que, dans unplan, une droite parallele a l’un des cotes d’un triangle sectionne cedernier en un triangle semblable. Dans d’autres langues,notamment en anglais, ce resultat est connu sous le nom detheoreme d’intersection.

En anglais et allemand, le theoreme de Thales designe un autretheoreme de geometrie qui affirme qu’un triangle inscrit dans uncercle et dont un cote est un diametre est un triangle rectangle.

Le “theoreme de Thales suisse” exprime par contre le carre de lahauteur dans un triangle rectangle.

An old result

Dans certains pays d’Europe, dont la France, le theoreme deThales designe un theoreme de geometrie qui affirme que, dans unplan, une droite parallele a l’un des cotes d’un triangle sectionne cedernier en un triangle semblable.

Dans d’autres langues,notamment en anglais, ce resultat est connu sous le nom detheoreme d’intersection.

An old result

Multiplication

More on projective geometry: axiomatization

Steiner 1832: Durch gehorige Aneignung der wenigen

Grundbeziehungen macht man sich zum Herrn des ganzen

Gegenstandes; es tritt Ordnung in das Chaos ein, und man sieht,

wie alle Theile naturgemass ineinander greifen, in schonster

Ordnung sich in Reihen stellen ...

Staudt 1847: Ich habe in dieser Schrift versucht, die Geometrie der

Lage zu einer selbststandigen Wissenschaft zu machen, welche des

Messens nicht bedarf.

Klein, Pasch, Pieri, ...

Schur: proof of the fundamental theorem of projectivegeometry from incidence axioms, Desargues and Pappus axiom

Hessenberg: Desargues follows from Pappus

Veblen 1910: What we call general projective geometry is,

analytically, the geometry associated with a general number field.

Hilbert ... Klein: When people run out of ideas they start

axiomatizing.

Fano plane

Universality theorems

Configuration spaces: moduli of finitely many points with specifiedalignments.

Mnev 1988

Any scheme over Z arises as a configuration space of points in P2.

Lafforgue 2002: singularities of certain strata in some modulispaces arising in the Geometric Langlands Program, e.g.,compactifications of PGLn+1

r /PGLr .

Vakil: Murphy’s law - badly behaved moduli spaces, e.g.,Hilbert schemes of smooth curves in projective space, surfacesin P4, etc.

Mnev 1988

r /PGLr .

Mnev 1988

r /PGLr .

Mnev 1988

r /PGLr .

Mnev 1988

r /PGLr .

Axioms

Definition

A projective structure is a pair (S ,L) where S is a (nonempty) set(of points) and L a collection of subsets l ⊂ S (lines) such that

P1 there exist an s ∈ S and an l ∈ L such that s /∈ l;

P2 for every l ∈ L there exist at least three distinct s, s ′, s ′′ ∈ l;

P3 for every pair of distinct s, s ′ ∈ S there exists exactly one

l = l(s, s ′) ∈ L

such that s, s ′ ∈ l;

P4 for every quadruple of pairwise distinct s, s ′, t, t ′ ∈ S one has

l(s, s ′) ∩ l(t, t ′) 6= ∅ ⇒ l(s, t) ∩ l(s ′, t ′) 6= ∅.

Axioms

A morphism of projective structures ρ : (S ,L)→(S ′,L′) is a mapof sets ρ : S → S ′ preserving lines, i.e., ρ(l) ∈ L′, for all l ∈ L.

A projective structure (S ,L) satisfies Pappus’ axiom if

PA for all 2-dimensional subspaces and every configuration of sixpoints and lines in these subspaces as below

the intersections are collinear.

Axioms

A morphism of projective structures ρ : (S ,L)→(S ′,L′) is a mapof sets ρ : S → S ′ preserving lines, i.e., ρ(l) ∈ L′, for all l ∈ L.

A projective structure (S ,L) satisfies Pappus’ axiom if

PA for all 2-dimensional subspaces and every configuration of sixpoints and lines in these subspaces as below

the intersections are collinear.

Fundamental theorem

Reconstruction

Let (S ,L) be a projective structure of dimension n ≥ 2 whichsatisfies Pappus’ axiom. Then there exists a vector space V over afield k and an isomorphism

σ : Pk(V )∼−→ S .

Moreover, for any two such triples (V , k , σ) and (V ′, k ′, σ′) thereis an isomorphism

V /k∼−→ V ′/k ′

compatible with σ, σ′ and unique up to homothety v 7→ λv ,λ ∈ k∗.

Main example

Let k be a field and Pn the usual projective space over k ofdimension n ≥ 2. Then Pn(k) carries a projective structure: linesare the usual projective lines P1(k) ⊂ Pn(k).

Let K/k be an extension of fields. Then

S := Pk(K ) = (K \ 0)/k∗

carries a natural (possibly, infinite-dimensional) projectivestructure. Multiplication in K ∗/k∗ preserves this structure.

Main example

S := Pk(K ) = (K \ 0)/k∗

carries a natural (possibly, infinite-dimensional) projectivestructure.

Multiplication in K ∗/k∗ preserves this structure.

Main example

S := Pk(K ) = (K \ 0)/k∗

carries a natural (possibly, infinite-dimensional) projectivestructure. Multiplication in K ∗/k∗ preserves this structure.

Main theorem

Reconstructing fields

Let K/k and K ′/k ′ be field extensions of degree ≥ 3 and

ψ : S = Pk(K )→Pk ′(K ′) = S ′

a bijection of sets which is an isomorphism of abelian groups andof projective structures. Then

k ' k ′ and K ' K ′.

Main theorem

Reconstructing field homomorphisms

Let K/k and K ′/k ′ be field extensions of degree ≥ 3 and

ψ : S = Pk(K )→Pk ′(K ′) = S ′

an injective homomorphism of abelian groups compatible withprojective structures. Then k ' k ′ and K is isomorphic to asubfield of K ′.

Pregeometries and geometries

A combinatorial pregeometry (finitary matroid) is a pair (P, cl)where P is a set and

cl : Subsets(P)→ Subsets(P),

such that for all a, b ∈ P and all Y ,Z ⊆ P one has:

Y ⊆ cl(Y ),

if Y ⊆ Z , then cl(Y ) ⊆ cl(Z ),

cl(cl(Y )) = cl(Y ),

if a ∈ cl(Y ), then there is a finite subset Y ′ ⊂ Y such thata ∈ cl(Y ′) (finite character),

(exchange condition) if a ∈ cl(Y ∪ {b}) \ cl(Y ), thenb ∈ cl(Y ∪ {a}).

A geometry is a pregeometry such that cl(a) = a, for all a ∈ P,and cl(∅) = ∅.

Y ⊆ cl(Y ),

Examples

1 P = V /k , a vector space over a field k and cl(Y ) the k-spanof Y ⊂ P

2 P = Pk(V ), the usual projective space over a k

3 P = Pk(K ), a field K containing an algebraically closedsubfield k and cl(Y ) - the normal closure of k(Y ) in K ; ageometry is obtained after factoring by x ∼ y iff cl(x) = cl(y).

Examples

3 P = Pk(K ), a field K containing an algebraically closedsubfield k and cl(Y ) - the normal closure of k(Y ) in K ;

ageometry is obtained after factoring by x ∼ y iff cl(x) = cl(y).

Examples

Combinatorial geometries of field extensions

Evans–Hrushovski 1991 / Gismatullin 2008

Let k and k ′ be algebraically closed fields, K/k and K ′/k ′ fieldextensions of transcendence degree ≥ 5 over k , resp. k ′. Then,every isomorphism of combinatorial geometries

Pk(K )→ Pk ′(K ′)

is induced by an isomorphism of purely inseparable closures

K → K ′.

K-theory

Let KMi (K ) be i-th Milnor K-group of a field K . Recall that

KM1 (K ) = K ∗

and that there is a canonical surjective homomorphism

σK : KM1 (K )⊗KM

1 (K )→KM2 (K )

whose kernel is generated by symbols (x , 1− x), for x ∈ K ∗ \ 1.Let

KMi (K ) := KM

i (K )/infinitely divisible, i = 1, 2,

be the component generated by nondivisible elements.

K-theory

KM1 (K ) = K ∗

σK : KM1 (K )⊗KM

1 (K )→KM2 (K )

whose kernel is generated by symbols (x , 1− x), for x ∈ K ∗ \ 1.

i (K ) := KMi (K )/infinitely divisible, i = 1, 2,

K-theory

KM1 (K ) = K ∗

σK : KM1 (K )⊗KM

1 (K )→KM2 (K )

whose kernel is generated by symbols (x , 1− x), for x ∈ K ∗ \ 1.Let

KMi (K ) := KM

i (K )/infinitely divisible, i = 1, 2,

Let K and L be function fields of algebraic varieties of dimension≥ 2 over algebraically closed fields k and l , respectively. Assumethat there exist isomorphisms

ψi : KMi (K )→ KM

i (L), i = 1, 2,

of abelian groups with a commutative diagram

KM1 (K )⊗KM

1 (K )ψ1⊗ψ1 //

��

KM1 (L)⊗KM

��KM

2 (K )ψ2

// KM2 (L).

Bogomolov-T. 2008

Then there exists an isomorphism of fields

ψ : K → L,

compatible with ψ1.

Assume that there exist isomorphisms

i (L), i = 1, 2,

KM1 (K )⊗ KM

1 (K )ψ1⊗ψ1 //

��

KM1 (L)⊗ KM

��KM

2 (K )ψ2

// KM2 (L).

Then there exists a (compatible) isomorphism of fields

ψ : K → L.

Assume that there exist isomorphisms

i (L), i = 1, 2,

KM1 (K )⊗ KM

1 (K )ψ1⊗ψ1 //

��

KM1 (L)⊗ KM

��KM

2 (K )ψ2

// KM2 (L).

Then there exists a (compatible) isomorphism of fields

ψ : K → L.

K-groups of function fields

Let K and L be function fields of transcendence degree ≥ 2 overan algebraically closed field k , resp. l . Let

ψ1 : KM1 (K )→KM

be an injective homomorphism.

Assume that there is acommutative diagram

KM1 (K )⊗ KM

1 (K )ψ1⊗ψ1 //

��

KM1 (L)⊗ KM

��KM

2 (K )ψ2

// KM2 (L).

Assume that ψ1(K ∗/k∗) 6⊆ E ∗/l∗, for 1-dimensional E ⊂ L.

ψ1 : KM1 (K )→KM

be an injective homomorphism. Assume that there is acommutative diagram

KM1 (K )⊗ KM

1 (K )ψ1⊗ψ1 //

��

KM1 (L)⊗ KM

��KM

2 (K )ψ2

// KM2 (L).

ψ1 : KM1 (K )→KM

be an injective homomorphism. Assume that there is acommutative diagram

KM1 (K )⊗ KM

1 (K )ψ1⊗ψ1 //

��

KM1 (L)⊗ KM

��KM

2 (K )ψ2

// KM2 (L).

Theorem

Then there exist an r ∈ Q and a homomorphism of fields

ψ : K→L

such that the induced map on K ∗/k∗ coincides with ψr1.

Outlook:

existence of sections of fibrations X → B, e.g., uniqueness ofthe Brauer obstruction to the existence of points.

birational invariants of quotients V /G , where G is a finitegroup and V its representation

Theorem

ψ : K→L

Outlook:

Theorem

ψ : K→L

Outlook:

Sketch of proof

The ground field: Infinitely divisible elements

An element f ∈ K ∗ = KM1 (K ) is infinitely divisible if and only if

f ∈ k∗. In particular,

KM1 (K ) = K ∗/k∗.

Sketch of proof

1-dimensional subfields

Given a nonconstant f1 ∈ K ∗/k∗, we have

Ker2(f1) = E ∗/k∗,

where E = k(f1)K

is the normal closure in K of the 1-dimensionalfield generated by f1 and

Ker2(f ) := { g ∈ K ∗/k∗ = KM1 (K ) | (f , g) = 0 ∈ KM

2 (K ) }.

Sketch of proof

Reconstructing lines: Functional equations

Assume that x , y ∈ K ∗ are algebraically independent and that ifboth xb, yb ∈ K ∗ then b ∈ Z.

Let p ∈ k(x)∗, q ∈ k(y)

∗be such

that x , y , p, q are multiplicatively independent in K ∗/k∗. Assumethat there is a nonconstant

Π ∈ k(x/y)∗ · y ∩ k(p/q)

∗ · q.

Assume moreover that this Π arises from infinitely many, moduloscalars, elements p, q as above. Then, modulo k∗,

Π = Πκ,δ(x , y) := (xδ − κy δ)δ, (1)

with κ ∈ k∗ and δ = ±1.

Sketch of proof

Assume that x , y ∈ K ∗ are algebraically independent and that ifboth xb, yb ∈ K ∗ then b ∈ Z. Let p ∈ k(x)

∗, q ∈ k(y)

∗be such

that x , y , p, q are multiplicatively independent in K ∗/k∗.

Assumethat there is a nonconstant

Π ∈ k(x/y)∗ · y ∩ k(p/q)

∗ · q.

Π = Πκ,δ(x , y) := (xδ − κy δ)δ, (1)

Sketch of proof

∗, q ∈ k(y)

∗be such

Π ∈ k(x/y)∗ · y ∩ k(p/q)

∗ · q.

Assume moreover that this Π arises from infinitely many, moduloscalars, elements p, q as above.

Then, modulo k∗,

Π = Πκ,δ(x , y) := (xδ − κy δ)δ, (1)

Sketch of proof

∗, q ∈ k(y)

∗be such

Π ∈ k(x/y)∗ · y ∩ k(p/q)

∗ · q.

Π = Πκ,δ(x , y) := (xδ − κy δ)δ, (1)

Sketch of proof

The corresponding p and q are given by

pκx ,1(x) = x + κx , qκy ,1(y) = y + κy

pκx ,−1(x) = (x−1 + κx)−1, qκx ,−1(y) = (y−1 + κy )−1

withκxκy = κ.

Anabelian geometry

Grothendieck’s Anabelian program

The Galois group of a function field determines the field.

Two group operations, + and ·, are encoded in one group.

Let K be a field with absolute Galois group GK := Gal(K/K ).

Let GK be the pro-`-completion of GK , for ` 6= char(K ) a prime.

Uchida, Tamagawa, Mochizuki, Pop, Konigsmann, Zaidi ...:reconstruction of function fields from the full GK or GK .

Anabelian geometry

Almost abelian anabelian geometry

K := GK/[GK ,GK ], GcK := GK/[GK , [GK ,GK ]]

be the abelianization, resp. its canonical central extension.

Thegroup Ga

K is a torsion-free Z`-module of infinite rank.

Let ΣK be the set of all topologically noncyclic subgroups of GaK

that lift to abelian subgroups of GcK .

Bogomolov’s program

The pair (GaK ,ΣK ) determines K .

be the abelianization, resp. its canonical central extension. Thegroup Ga

Anabelian geometry of surfaces

Theorem (Bogomolov-T. 2004)

Let K and L be function fields over algebraic closures of finite fieldsk , l of characteristic 6= `. Assume that K = k(X ) is a functionfield of a surface X/k and that there exists an isomorphism

ψ : GaK ' Ga

inducing a bijection of sets

ΣK = ΣL.

Then, for some c ∈ Z∗` , cψ is induced by an isomorphism of purelyinseparable closures of K and L.

Sketch of proof: Kummer theory

The abelianized Galois group GaK is dual to K ∗, the

pro-`-completion of K ∗, and one obtains an isomorphism

K ∗ ' L∗.

In our setup, we can interpret GaK as homomorphisms

K ∗/k∗→Z`(1),

arising from

n 3 γn 7→(

f 7→ γ(`n√

f )/`n√

For a subfield E ⊂ K , the map GaK→Ga

E is simply restriction to E .

K ∗ ' L∗.

K ∗/k∗→Z`(1),

arising from

n 3 γn 7→(

f 7→ γ(`n√

f )/`n√

K ∗ ' L∗.

K ∗/k∗→Z`(1),

arising from

n 3 γn 7→(

f 7→ γ(`n√

f )/`n√

Valuations

A value group, Γ, is a totally ordered (torsion-free) abelian group.

A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν)consisting of a value group Γν and a map

ν : K→Γν,∞ = Γν ∪∞

such that

ν : K ∗→Γν is a surjective homomorphism;

ν(κ+ κ′) ≥ min(ν(κ), ν(κ′)) for all κ, κ′ ∈ K ;

ν(0) =∞.

Note that Fp admits only the trivial valuation. A valuation is a flagmap on K : every finite-dimensional Fp-subspace V ⊂ K has a flagV = V1 ⊃ V2 . . . such that ν is constant on Vj \ Vj+1. Conversely,every flag map gives rise to a valuation.

Valuations

A value group, Γ, is a totally ordered (torsion-free) abelian group.A (nonarchimedean) valuation on a field K is a pair ν = (ν, Γν)consisting of a value group Γν and a map

ν : K→Γν,∞ = Γν ∪∞

such that

ν(0) =∞.

Valuations

ν : K→Γν,∞ = Γν ∪∞

such that

ν(0) =∞.

Note that Fp admits only the trivial valuation.

A valuation is a flagmap on K : every finite-dimensional Fp-subspace V ⊂ K has a flagV = V1 ⊃ V2 . . . such that ν is constant on Vj \ Vj+1. Conversely,every flag map gives rise to a valuation.

Valuations

ν : K→Γν,∞ = Γν ∪∞

such that

ν(0) =∞.

Note that Fp admits only the trivial valuation. A valuation is a flagmap on K : every finite-dimensional Fp-subspace V ⊂ K has a flagV = V1 ⊃ V2 . . . such that ν is constant on Vj \ Vj+1.

Conversely,every flag map gives rise to a valuation.

Valuations

ν : K→Γν,∞ = Γν ∪∞

such that

ν(0) =∞.

Valuations

Denote by Kν , oν ,mν and Kν := oν/mν the completion of K withrespect to ν, the ring of ν-integers in K , the maximal ideal of oνand the residue field.

Keep in mind the exact sequences:

1→o∗ν→K ∗ → Γν→1

1→(1 + mν)→o∗ν→K∗ν→1.

Valuations

Denote by Kν , oν ,mν and Kν := oν/mν the completion of K withrespect to ν, the ring of ν-integers in K , the maximal ideal of oνand the residue field. Keep in mind the exact sequences:

1→o∗ν→K ∗ → Γν→1

1→(1 + mν)→o∗ν→K∗ν→1.

Valuations

A homomorphism χ : Γν → Z`(1) gives rise to a homomorphism

χ ◦ ν : K ∗ → Z`(1),

thus to an element of GaK , an inertia element of ν. These form the

inertia subgroup Iaν ⊂ Ga

The decomposition group Daν is the image of Ga

Kνin Ga

K . We havean embedding Ga

Kν↪→ Ga

K and an isomorphism

Daν/Ia

ν ' GaKν .

Valuations

A homomorphism χ : Γν → Z`(1) gives rise to a homomorphism

χ ◦ ν : K ∗ → Z`(1),

thus to an element of GaK , an inertia element of ν. These form the

inertia subgroup Iaν ⊂ Ga

The decomposition group Daν is the image of Ga

Kνin Ga

K . We havean embedding Ga

Kν↪→ Ga

K and an isomorphism

Daν/Ia

ν ' GaKν .

A dictionary

Let K be a function field over k = Fp. We have

GaK = {homomorphisms γ : K ∗→Z`(1)}Daν = {µ ∈ Ga

K |µ trivial on (1 + mν)},Iaν = {ι ∈ Ga

K | ι trivial on o∗ν}.

Inertia elements define flag maps on K .

A dictionary

Let K be a function field over k = Fp. We have

GaK = {homomorphisms γ : K ∗→Z`(1)}Daν = {µ ∈ Ga

K |µ trivial on (1 + mν)},Iaν = {ι ∈ Ga

K | ι trivial on o∗ν}.

Inertia elements define flag maps on K .

Projective geometry of the Galois group

Key fact

Let γ, γ′ ∈ GaK ' Z∞` be two nonproportional elements lifting to

commuting elements in GcK . Then, for any nonconstant f ∈ K ∗ the

restrictions of γ, γ′ to the projective line PFp (Fp ⊕ f Fp) areproportional (modulo addition of constants).

Consider the map

K ∗/k∗ = Pk(K ) → A2(Z`)

f 7→ (γ(f ), γ′(f ))

This maps every projective line into an affine line, a collineation.

Key fact

Consider the map

K ∗/k∗ = Pk(K ) → A2(Z`)

f 7→ (γ(f ), γ′(f ))

Key fact

Consider the map

K ∗/k∗ = Pk(K ) → A2(Z`)

f 7→ (γ(f ), γ′(f ))

A map α : P2(Fp)→Z/2 is a flag map iff the restiction to everyP1(Fp) ⊂ P2(Fp) is a flag map, i.e., constant on the complementof one point.

Counterexample: the Fano plane

(0:1:0)

(1:0:0)(1:0:1)(0:0:1)

(0:1:1) (1:1:0)

A map α : P2(Fp)→Z/2 is a flag map iff the restiction to everyP1(Fp) ⊂ P2(Fp) is a flag map, i.e., constant on the complementof one point.

Counterexample: the Fano plane

(0:1:0)

(1:0:0)(1:0:1)(0:0:1)

(0:1:1) (1:1:0)

Projective/affine geometry considerations produce a flag map inthe Z`-linear span of γ, γ′.

Every noncyclic subgroup of GaK lifting to an abelian subgroup of

GcK contains an inertia element ι = ιν for some valuation ν of K .

The elements “commuting” with ι form Daν .

The combinatorial structure of the fan ΣK allows to reconstructthe projective structure of Pk(K ).

What about curves?

Let k = Fp and K = k(C ). Let GK be the absolute Galois groupof K . Let

IK := {Iaν},

the set of inertia subgroups Iaν ⊂ G a

K of nontrivial divisorialvaluations of K (i.e., points of C ).

Bogomolov-T. 2008

Assume that g(C ) > 2 and that

(G aK , IK ) ' (G a

K, IK ).

ThenJ ∼ J.

Curves and their Jacobians

Let k be any field and C/k a smooth curve over k of genusg(C ) ≥ 2, with C (k) 6= ∅. For each n ∈ N, we have

(c1, . . . , cn) // (c1 + · · ·+ cn)

Cn // Symn(C )

��Jn

Choosing c0 ∈ C (k), we may identify Jn ' J.

Image(λg−1) = Θ ⊂ J, the Theta divisor

Torelli: the pair (J,Θ) determines C , up to isomorphism

for n ≥ 2g − 1, λn is a Pn−g-bundle

(c1, . . . , cn) // (c1 + · · ·+ cn)

Cn // Symn(C )

��Jn

(c1, . . . , cn) // (c1 + · · ·+ cn)

Cn // Symn(C )

��Jn

(c1, . . . , cn) // (c1 + · · ·+ cn)

Cn // Symn(C )

��Jn

(c1, . . . , cn) // (c1 + · · ·+ cn)

Cn // Symn(C )

��Jn

Abelian varieties over finite fields

Let A be an abelian variety of dimension g over a finite field k .Recall that

A(k) = p-part⊕⊕6=p

(Q`/Z`)2g.

Hom(A, A)⊗ Z` = HomZ`[Fr](T`(A),T`(A)).

In particular, A and A are isogenous iff the characteristicpolynomials of the Frobenius coincide.

(Q`/Z`)2g.

Divisibilities

Bogomolov-T. 2008

Let A and A be abelian varieties of dimension g over finite fields k ,resp. k . Let kn/k , resp. kn/k , be the unique extensions of degreen. Assume that

#A(kn) | #A(kn)

for infinitely many n ∈ N.

Then char(k) = char(k) and A and Aare isogenous over k .

Divisibilities

Bogomolov-T. 2008

Let A and A be abelian varieties of dimension g over finite fields k ,resp. k . Let kn/k , resp. kn/k , be the unique extensions of degreen. Assume that

#A(kn) | #A(kn)

for infinitely many n ∈ N. Then char(k) = char(k) and A and Aare isogenous over k .

Sketch of proof

Let A be an abelian variety over k1 := Fq. Let {αj}j=1,...,2g be theset of eigenvalues of Frobenius on H1

et(A,Q`), for ` 6= p, andΓA ⊂ C∗ the multiplicative subgroup spanned by α.

The sequence

R(n) := #A(kn) =

2g∏j=1

(αnj − 1).

is a simple linear recurrence with roots in Γ = ΓA.

There is an isomorphism of rings

{ Recurrences with roots in Γ} ⇔ C[Γ].

Sketch of proof

et(A,Q`), for ` 6= p, andΓA ⊂ C∗ the multiplicative subgroup spanned by α.The sequence

R(n) := #A(kn) =

2g∏j=1

(αnj − 1).

Sketch of proof

et(A,Q`), for ` 6= p, andΓA ⊂ C∗ the multiplicative subgroup spanned by α.The sequence

R(n) := #A(kn) =

2g∏j=1

(αnj − 1).

Sketch of proof: Recurrence sequences

Corvaja-Zannier 2002

Let R and R be simple linear recurrences such that

1 R(n), R(n) 6= 0, for all n, n� 0;

2 the subgroup Γ ⊂ C∗ generated by the roots of R and R istorsion-free;

3 there is a finitely-generated subring A ⊂ C withR(n)/R(n) ∈ A, for infinitely many n ∈ N.

ThenQ : N → C

n 7→ R(n)/R(n)

is a simple linear recurrence. In particular, the FQ ∈ C[Γ] and

FQ · FR = FR .

Sketch of proof: Recurrence sequences

Corvaja-Zannier 2002

Let R and R be simple linear recurrences such that

1 R(n), R(n) 6= 0, for all n, n� 0;

2 the subgroup Γ ⊂ C∗ generated by the roots of R and R istorsion-free;

3 there is a finitely-generated subring A ⊂ C withR(n)/R(n) ∈ A, for infinitely many n ∈ N.

ThenQ : N → C

n 7→ R(n)/R(n)

is a simple linear recurrence. In particular, the FQ ∈ C[Γ] and

FQ · FR = FR .

Let C be another smooth projective curve and J its Jacobian.Isomorphism of pairs:

φ : (C , J)→(C , J)

��

C (k)j1oo

��J(k) J1(k) C (k)

φ0: isomorphism of abstract abelian groups;

φ1: isomorphism of homogeneous spaces, compatible with φ0;

the restriction φs : C (k)→C (k) of φ1 is a bijection of sets.

For all #k � 0 the group J(k) is generated by C (k).

k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .

be the tower of degree 2 extensions. To characterize J(kn) itsuffices to characterize C (kn). Let C be a nonhyperelliptic curve ofgenus g(C ) ≥ 3.

Inductive characterization of J(kn), n ∈ N

J(kn) is generated by c ∈ C (k) such that there exists a pointc ′ ∈ C (k) with

c + c ′ ∈ J(kn−1).

For all #k � 0 the group J(k) is generated by C (k). Let

k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .

be the tower of degree 2 extensions.

To characterize J(kn) itsuffices to characterize C (kn). Let C be a nonhyperelliptic curve ofgenus g(C ) ≥ 3.

c + c ′ ∈ J(kn−1).

k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .

be the tower of degree 2 extensions. To characterize J(kn) itsuffices to characterize C (kn).

Let C be a nonhyperelliptic curve ofgenus g(C ) ≥ 3.

c + c ′ ∈ J(kn−1).

k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .

c + c ′ ∈ J(kn−1).

k = k1 ⊂ k2 ⊂ . . . ⊂ kn ⊂ . . .

c + c ′ ∈ J(kn−1).

Curves and their Jacobians: Torelli

Theorem

Let (C , J)→ (C , J) be an isomorphism of pairs. Then J isisogenous to J.

Proof.

1 Choose k1, k1 (sufficiently large) such that φ(J(k1)) ⊂ J(k1)

2 Define C (kn), resp. C (kn), intrinsically, as above.

3 Have φ(J(kn)) ⊂ J(kn), for all n ∈ N.

4 #J(kn) | #J(kn)

5 Apply the result about divisibility of recurrence sequences.

Theorem

Proof.

4 #J(kn) | #J(kn)

Theorem

Proof.

4 #J(kn) | #J(kn)

Theorem

Proof.

4 #J(kn) | #J(kn)

Theorem

Proof.

4 #J(kn) | #J(kn)

Theorem

Proof.

4 #J(kn) | #J(kn)

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