BRIAN LAWRENCE AND AKSHAY VENKATESHmath.uchicago.edu/~brianrl/padicTorelli.pdf · BRIAN LAWRENCE AND AKSHAY VENKATESH ABSTRACT.We give a new proof of the ﬁniteness of solutions

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DIOPHANTINE PROBLEMS AND THE p-ADIC TORELLI MAP

BRIAN LAWRENCE AND AKSHAY VENKATESH

ABSTRACT. We give a new proof of the finiteness of solutions to the S-unit equation,and of finiteness of points on a genus 2 curve Y over a number field, under a genericitycondition on the 2-torsion of Jac(Y ).

Under a purely topological assumption – namely, that certain Kodaira–Parshin cover-ings have large monodromy group – the same technique proves finiteness of Y (K) for allcurves Y of genus greater than one.

The proof uses monodromy computations and properties of p-adic Galois representa-tions. We discuss a conjecture on period mappings that would allow the same technique tobe applied to other finiteness results of Shafarevich type.

1. INTRODUCTION

1.1. Let K be a number field. In this paper, we will give a new proof (§4) of finiteness ofsolutions to the S-unit equation x+ y = 1, where x, y are both S-units in K.

In fact, if Y is any projective curve over K of genus ≥ 2, the same technique alsoproves that Y (K) is finite, assuming that the monodromy of a certain family of abelianvarieties X → Y is large (see Definition 5.2 for a precise condition, and Theorem 5.6 forthe statement). This condition is purely topological.

We compute this monodromy in the simplest case. Namely, suppose that Y has genus2, K doesn’t contain a CM field, and the image of the Galois group acting on the 2-torsionof the Jacobian is as large as possible (i.e. Sp4(F2) ' S6). Then one can take the familyX → Y to have for each fiber a disjoint union of abelian varieties of dimension 3, and weverify large monodromy in Section 7. Therefore we get a complete proof in this case.

These proofs are related to Faltings’ proof [5], but are based on a closer study of thevariation of p-adic Galois representations in a family; they do not make any usage of tech-niques specific to abelian varieties.

1.2. Outline of the proof. Consider a smooth projective familyX → Y overK, where Yis itself a smooth K-variety; we suppose this extends to a family π : X → Y over the ringO of S-integers of K, for some finite set S of places of K (containing all the archimedeanplaces).

For y ∈ Y (K) callXy the fiber over y. We will show that Y(O) is finite, making use ofthe fact that, if y ∈ Y (K) extends to Y(O), then Xy admits a smooth proper model overO. That one can thus reduce Mordell’s conjecture to finiteness results for varieties withgood reduction was observed by Parshin [8] and then used by Faltings in his proof of theMordell conjecture [5].

Choosing a rational prime p that is unramified in K and not below any prime of S,write ρy for the Galois representation of GK = Gal(K/K) on H∗et(Xy ×K K,Qp). Asobserved by Faltings, one deduces from Hermite–Minkowski finiteness that there are onlyfinitely many possibilities for the semisimplification of ρy . In the contexts of interest, wewill complete the proof by establishing that:

1

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2 BRIAN LAWRENCE AND AKSHAY VENKATESH

(1) There exists a place v of K above p such that the map

y ∈ Y (Kv) −→ isomorphism class of ρy restricted to GKvhas finite fibers. (Here GKv is the absolute Galois group of Kv .)

(2) For all but finitely many y ∈ Y (K), the Galois representation ρy is semisimple.We will analyze these with p-adic Hodge theory:1 under the correspondence of p-

adic Hodge theory, the restricted representation ρy,v from (1) corresponds to a filteredφ-module, namely the de Rham cohomology of Xy over Kv equipped with its Hodge fil-tration and a semilinear Frobenius map. The variation of this filtration is governed by adifferential equation; using a monodromy computation, we will deduce that the isomor-phism class of the φ-module varies nontrivially, forcing (1) (see e.g. Proposition 3.2). Wediscuss this step in a little more detail in §1.3 below.

Although these p-adic Hodge theory arguments do not detect the semisimplicity of theglobal representation, we will be able to show (2) in a similar fashion, using the fact thatthe Hodge weights of a global representation are highly constrained by purity (Lemma2.6; for an example of how this is used, see Claim 1 and its proof in Section 6). Thispurity argument is also reminiscent of a more subtle argument in Faltings’ proof (the useof Raynaud’s results on [5, p. 364]).

Faltings proves stronger versions of (1) and (2) by a remarkable argument with heightsof abelian varieties. In particular, he shows that when the fibers Xy are curves or abelianvarieties, every ρy is semisimple. The advantage of our argument is that, since we do notrely on the theory of abelian varieties, our methods may generalize to families of higherrelative dimension.

1.3. p-adic Torelli theorems. Because of the method of proof and the parallel with theclassical Torelli theorem, we may regard results of type (1) as “p-adic Torelli theorems.”However, (1) does not follow directly from the Torelli theorem. Indeed, different filtrationson the underlying φ-module can give filtered φ-modules which are abstractly isomorphic,the isomorphism being given by a linear endomorphism commuting with φ. Hence, oneneeds to know not only that the period mapping

(1.1) residue disc in Y (Kv) −→ Kv-points of a flag variety,

is injective, but that its image has finite intersection with an orbit of the action of thecentralizer of φ on the period domain.

This can be reduced to a similar question for the complex period map, using an embed-dingK → C. In this way, we arrive at an issue of “unlikely intersections” for the complexperiod map: can the image of a complex period map intersect an algebraic subvariety of theperiod domain in an unexpectedly large set? In the case when the base is one-dimensional,this can be answered in the negative using monodromy computations.

However, this really uses one-dimensionality of Y : without this, we can only prove thatthe exceptional set is a properKv-analytic subvariety of Y (Kv) which a priori need not befinite. In §8 we formulate a “transcendence conjecture on period mappings” which wouldsurmount this issue.

1.4. Enlarging the base field. A major cause for concern, when analyzing (1.1), is thatthe centralizer of the Frobenius φ might be very large.

This already occurs in a basic example. When analyzing the S-unit equation, it isnatural to take Y = P1 − 0, 1,∞ and X → Y to be the Legendre family, so that Xt is

1We make no use of p-adic Hodge theory in families: we need only the statements over a local field.

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DIOPHANTINE PROBLEMS AND THE p-ADIC TORELLI MAP 3

the curve y2 = x(x − 1)(x − t). Unfortunately (1) fails: for t ∈ Zp, if we write ρt forthe representation of the Galois group GQp on the (rational) Tate module of Xt, then ρtbelongs to only finitely many isomorphism classes so long as the reduction t ∈ Fp is notequal to 0 or 1. 2 The issue is that, with reference to (1.1), both the Frobenius centralizerand the period domain are one-dimensional over Qp.

In general, Frobenius is a semilinear operator on a vector space over Kv , an unramifiedextension of Qp; semilinearity alone gives rise to a nontrivial bound (Lemma 2.1) on thesize of its centralizer, which, in effect, becomes stronger as [Kv : Qp] gets larger. To takeadvantage of this effect, we must find a way to enlarge the base field.

For instance, in our analysis of the S-unit equation in §4, we replace the Legendrefamily instead by the family with fiber

Xt =∐z2k=t

y2 = x(x− 1)(x− z),

for a suitable large integer k. In our situation, the corresponding map t 7→ [ρt] will nowonly have finite fibers, at least on residue discs where t is not a square – an example of theimportance of enlarging Kv .

Said differently, we have replaced the Legendre family X `→ P1 − 0, 1,∞ with afamily with the following composite structure:

X ′`′→ P1 − 0, µ2k ,∞ → P1 − 0, 1,∞

where the second map is given by u 7→ u2k , and `′ is simply the restriction of the Legendrefamily over P1 − 0, µ2k ,∞. The composite defines a family over P1 − 0, 1,∞ withgeometrically disconnected fibres, and this disconnectedness is, as we have just explained,to our advantage.

It turns out that the families introduced by Parshin (see [8, Proposition 9]), in his re-duction of Mordell’s conjecture to Shafarevich’s conjecture, automatically have a similarstructure. That is to say, if Y is a curve of genus 2, Parshin’s families factorize as

X → Y ′ → Y,

where Y ′ → Y is finite étale and X → Y ′ is a relative curve.There is in fact a lot of flexibility in this construction; in Parshin’s original construction

the covering Y ′ → Y is obtained by pulling back multiplication by 2 on the Jacobian,and as such each fiber is a torsor under H1(YK , µ2). Because we want to ensure that theGalois action on this fiber is large, we use a variant where each fiber is identified, instead,with H1(YK ,Z/qZ) (for a suitable auxiliary prime q). The Weil pairing alone implies thatthe Galois action on this is nontrivial, and this (although very weak) is enough to run ourargument.

1.5. Discussion. Our method of proof can, in principle, be made algorithmic in the samesense as Chabauty’s methods. For example, given a genus 2 curveC as above, it is possibleto “compute” a finite subset S ⊂ C(Kv) which contains C(K); “compute” means thatthere is an algorithm that will compute all the elements of S to a specified p-adic precisionin a finite time. However, the resulting method is completely impractical.

We hope that our methods may be pushed to proving (Shafarevich-type) finiteness re-sults for families of higher-dimensional varieties. For example, can one prove that there

2Near 0 or 1 – i.e. for curves with multiplicative reduction – the situation is better, as observed by Serre [10,IV.2.3] in his proof of the isogeny theorem for elliptic curves with nonintegral j-invariant. Unfortunately, it seemsto be difficult to use this kind of phenomenon.

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are only finitely many projective smooth hypersurfaces, of degree d in PN , over the baseSpec O? The only crucial inputs to our technique are general properties of v-adic Galoisrepresentations, and the classical Torelli theorem; but in higher dimension the “monodromyargument,” mentioned above, is not strong enough. For this reason we have formulated the“transcendence conjecture” of §8, which seems to be an interesting question to study in itsown right.

1.6. Acknowledgements. We thank Zeb Brady for help with the monodromy computa-tions and Brian Conrad for many helpful conversations and suggestions. A.V. would liketo thank Andrew Snowden for an interesting discussion.

CONTENTS

1. Introduction 12. Notation and preparatory results 43. Fibers with good reduction in a family 84. The S-unit equation 135. Mordell’s Conjecture 166. A variant of Proposition 3.2 207. A monodromy calculation 258. Higher dimensional varieties: a transcendence conjecture of period mappings 26References 27

2. NOTATION AND PREPARATORY RESULTS

We gather here some notation and some miscellaneous lemmas that we will use in thetext. We suggest that the reader refer to this section only as necessary when reading themain text.

Throughout the paper, K denotes a number field, and we will fix an algebraic closureK; we write GK = Gal(K/K) for the Galois group. By a GK-set we mean a set witha continuous action of GK ; here we regard all GK sets to be equipped with the discretetopology. The symbol S will denote a finite set of finite places of K containing all thearchimedean places, and OS will be the ring of S-integers; we will abridge this simply toO when S is understood. The (rational) prime number p will always be chosen so that noplace of S lies above p. Finally, for w a prime of O, we denote by Kw the completion,and Kw an algebraic closure; we write Fw for the residue field at w, with cardinality qw,and Fw for its algebraic closure, which we identify with the residue field of Kw. We alsodenote by O(w) the localization of O at w.

For a variety X over a field E, we denote by H∗dR(X/E) the de Rham cohomol-ogy of X → Spec(E). If E′ ⊃ E is a field extension, we denote by H∗dR(X/E′) =HdR(X/E)⊗E E′, i.e. the de Rham cohomology of the base-change XE′ .

For any scheme S, we shall use the phrase “family over S” to mean any S-schemeπ : Y → S. This is called a “curve over S” when π is smooth and proper of relativedimension 1 and each geometric fiber is connected. (Note that we will also make useof “open” curves, for example in §4, but we will avoid using the phrase “curve” in thatcontext.)

Let E/Qp be a finite unramified extension of Qp, and σ the unique automorphism of Einducing the p-th power map on the residue field. By φ-module (over E) we will mean a

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pair (V, φ), with V a finite-dimensional E-vector space and φ : V → V a map semilinearover σ. A filtered φ-module will be a triple (V, φ, F iV ) such that (V, φ) is a φ-module and(F iV )i is a descending filtration on V . We demand that each F iV be anE-linear subspaceof V but require no compatibility with φ.

2.1. Linear algebra.

Lemma 2.1. Suppose that σ : E → E is a field automorphism of finite order e, with fixedfield F . Let V be an E-vector space of dimension d, and ϕ : V → V a σ-semilinearautomorphism. Then the centralizer Z(ϕ) of ϕ in the ring of E-linear endomorphisms ofV satisfies

dimF Z(ϕ) = dimE Z(ϕe),

where ϕe : V → V is now E-linear. In particular, dimF Z(ϕ) ≤ (dimV )2.

Proof. Let E be an algebraic closure of E, and let Σ be the set of F -embeddings E → E.Then V = V ⊗F E is a E ⊗F E ' EΣ-module, and splitting by idempotents of E ⊗F Ewe get a decomposition

V =⊕τ∈Σ

V τ .

Moreover, ϕ extends to an E-linear endomorphism ϕ of V ; this endomorphism carries V τ

to V τσ. Fix τ0 ∈ ΣE ; then projection to the τ0 factor induces an isomorphism

Z(ϕ) ' centralizer of ϕe on V τ0 ,

whence the result.

Lemma 2.2. Let H 6 G be a finite index inclusion of groups, and let ρ : H → GLn(F )be a semisimple representation of the group H over the characteristic zero field F . Thenthe induction ρG = IndGHρ is also semisimple.

Proof. This follows readily from the fact that a representation ρ of G is semisimple if andonly if its restriction to a finite index normal subgroupG1 6 G is semisimple. For “if” onecan promote a splitting from G1 to G by averaging; for “only if” we take an irreducible G-representation V , an irreducible G1-subrepresentation W ⊂ V , and note that G-translatesof W must span V , exhibiting V |G1 as a quotient of a semisimple module.

2.2. Hurwitz spaces for curves. References need to be added for this material; for repre-sentability by a stack see [7, §3.22].

Let Y be a relative curve over the field K of characteristic zero.For a K-scheme S, a “G-cover of Y × S branched at a single point” consists of the

data:

(a) A morphism e : S → Y , and(b) A G-scheme Y and a finite flat surjective morphism Y → Y × S;

where we shall require that:

- Y → S is a relative curve, and- Y → Y × S is an étale G-torsor when restricted to the complement of the graph

of e.- The associated map π1(YS − graph(e), ∗) → G is surjective and doesn’t factor

through π1(YS), where ∗ is any geometric basepoint.

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The functorH, defined by

H(S) = isomorphism classes of G-covers of Y × S branched at a single point.

is representable as long as G has trivial center.If we let (Y ′, h ∈ H(Y ′)) be a representing object, the universal class h ∈ H(Y ′)

defines a map e′ : Y ′ → Y (the branch locus map, which is necessarily finite étale) andalso a universal branched cover

Z → Y × Y ′

which is a G-torsor away from the graph of e′.

2.3. Relative Picard and Prym schemes. Let C → S be a curve over S. Then we canform the relative Picard scheme Pic0

C/S . This is a relative abelian scheme over S and it isequipped with a symmetric, fiberwise ample line bundle – see [9, Chapter 9, Proposition4].

If G is a finite group acting on C → S (compatibly with the trivial action on S) then Gacts on the relative Picard scheme. If e ∈ Q[G] is an idempotent element, we may formthe e-component of the relative Picard scheme thus: let e′ = 1 − e ∈ Q[G], choose aninteger k such that ke′ ∈ Z[G], and set

(2.1) Pic0C/S [e] := relative identity component of the kernel of ke′.

(For the notion of “relative identity component,” see [4, Proposition 15.6.4].) This is anabelian scheme over S; it inherits a symmetric ample line bundle, and its degree is divisibleonly by primes dividing k.

2.4. Global Galois representations.

Lemma 2.3. (Faltings) Fix integers q, d ≥ 0. There are, up to conjugation, only finitelymany semisimple Galois representations ρ : GK → GLd(Qp) such that

(a) ρ is unramified outside S, and(b) The characteristic polynomial of every Frobenius ρ(Frobv), for v /∈ S, has coeffi-

cients in Q, and its roots are Weil numbers of weight q.

Proof. See the proof of [5, Satz 5]. We now give some lemmas which limit the reducibility of a global Galois representa-

tion.

Definition 2.4. (Friendly places). Let K be a number field. If K has a CM subfield, thenlet E be its maximal CM subfield [10, II.3.3] and E+ the maximal totally real subfield ofE. We say that a place ν ofK is friendly if it is unramified over Q, and it lies above a placeof E+ that is inert in E. If K has no CM subfield, any place ν of K which is unramifiedover Q will be understood to be friendly.

Lemma 2.5. Let ν be any friendly place ofK. For any continuous character η : A∗K/K∗ −→

Q∗p that is both pure of weight w and locally algebraic [10, Chapter III] at each primeabove p, one has

η2|K∗ν = χ ·NormwKν/Qp

,

where χ has finite order. In particular, w is even and the Hodge–Tate weight of η at theplace ν equals w/2.

The key point of the proof is the result, due to Artin and Weil, that an algebraic Heckecharacter factors through the norm map to the maximal CM subfield.

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Proof. Being locally algebraic, η gives rise to an algebraic character of ResK/QGm, whichis trivial on a finite index subgroup of O∗. Said differently, we obtain a Qp-rational char-acter S −→ Gm of the Serre torus S. Here S is the quotient of ResK/QGm by theZariski-closure of (a finite index subgroup of) the units.

The structure of this torus is computed by Serre [10, Chapter II]: Let E be the largestCM subfield of K, and let E+ be the totally real subfield of E. Then the norm mapS→ SE is an isogeny; in other words, a suitable power ηk factors through the norm fromK to E.

Therefore it is enough to prove the Claim for K = E, replacing v by the place ν of Ebelow it. In particular, by definition, ν lies above an inert prime of E/E+.

The map x 7→ x/x, from E∗ to E∗, is trivial on a finite index subgroup of units. Itarises from a map of algebraic groups

θ : S→ (ResE/QGm)1

where the superscript 1 denotes the kernel of the norm to E+. Together with the norm mapthis gives an isogeny S ∼ Gm × (ResE/QGm)1. Twisting by a power of the cyclotomiccharacter, it is enough to see that any Qp-rational character

(ResE/QGm)1 → Gm

has trivial pullback to E∗ν under

E∗ν ⊂ (E ⊗Qp)∗ → S(Qp)

θ→ (ResE/QGm)1(Qp).

(because it is easy to see all these characters have weight 0). But, because ν was above aninert prime of E+, we see that E∗ν maps into a Qp-anisotropic subtorus of (ResE/QGm)1

under x 7→ x/x. For a decreasing filtration F •V on a vector space V , we define the “weight” of the

filtration to equal

weightF (V ) =

∑p p dim grp(V )

dimV,

where grp(V ) = F p(V )/F p+1(V ) is the associated graded. For the other p-adic Hodgetheory terms that appear in the following result, see [3, §6].

Lemma 2.6. Let K be a number field and ν a friendly place. Let V be a Galois represen-tation of GK on a Qp-vector space which is de Rham at all primes above p, and pure ofweight w.

Let VdR = (V ⊗QpBdR)GKv be the filtered Kv-vector space that is associated to ρ|Kv

by p-adic Hodge theory.Then the weight of the Hodge filtration on VdR equals w/2.

Proof. Apply the previous Lemma to det(V ).

Lemma 2.7. Let K be a number field, and L ⊃ K a finite extension. Let ρ : GL →GLn(Qp) be a representation of GL that is de Rham at all primes above p, and pure ofweight w; let au(ρ) be the weight of the associated Hodge filtration at each such prime u.Then, for any friendly prime v of K above p,∑

u|v

[Lu : Kv]au(ρ) = [L : K]w

2.

Proof. We apply Lemma 2.5 to the determinant of IndLKρ. Up to finite order characters, itsdeterminant coincides with the composition of det(ρ) with the transfer map, and the resultfollows easily.

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2.5. Group-theoretic lemmas.

Lemma 2.8. Let S be a nonabelian simple group and f : Sd → S a surjective grouphomomorphism. Then f factors as f = φ pi, where pi is one of the canonical projectionsSd → S, and φ is an automorphism of S.

Proof. Let Si denote the canonical copies of S sitting inside Sd. One sees that each f(Si)is a normal subgroup of S, hence either S or the trivial subgroup. The subgroups f(Si)commute pairwise, so at most one can be nontrivial. The result follows.

Lemma 2.9. (Goursat) Suppose S is a nonabelian simple group and H a subgroup of theCartesian product Sd. Suppose the induced map pi : H → S onto the i-th factor is asurjection for every i. Then H is the set of elements (g1, g2, . . . , gd) ∈ Sd satisfying anumber of relations of the form gi = e and gi = φij(gj), for some automorphisms φij ofS.

In particular, suppose we know additionally that the projection H → S × S onto anytwo factors contains elements (a, e) and (e, b) in its image, for nontrivial a, b ∈ S. Thenwe have that H = Sd.

Proof. The proof is by induction on d, the base case d = 1 being trivial. Assume the resultfor Sd and suppose given H ⊆ Sd+1.

LetK = k ∈ S|(e, . . . , e, k) ∈ H.

One sees easily that K is normal in S, so K is either S or the trivial subgroup. If K = Sthe result is trivial, so suppose K is trivial.

Let Hd ⊆ Sd denote the projection of H onto the first d factors. The projection H →Hd is one-to-one, hence an isomorphism; thus we obtain a map

f : Hd∼= H → S

by projection onto the (d + 1)st factor. By the inductive hypothesis, Hd is abstractlyisomorphic to Sr, with the isomorphism given by some r of the d canonical projections.By Lemma 2.8, the map f : Sr → S is given by projection onto one of the factors,followed by an automorphism of S, and the result is proved.

Lemma 2.10. Let q, r be primes with r ≡ 1 modulo q, and let Gq,r be the semidirectproduct (Z/rZ) o (Z/qZ). For any s ≥ 1 consider the map f : G2s

q,r −→ (Z/rZ) givenby

f : g = (g1, g′1, · · · , gs, g′s) 7→ [g1, g

′1] · [g2, g

′2] · · · · · [gs, g′s].

Fix a nonzero (2s)-tuple y1, y′1, . . . , ys, y

′s ∈ (Z/qZ) and let L be the set of lifts of this

(2s)-tuple to a generating set g ∈ G2sq,r. Then all fibers of the map

g ∈ L : f(g) 6= 0 −→ (Z/rZ)− 0have the same size.

Proof. Note that if f(g) 6= 0 then the gi, g′i automatically generateG2sq,r. Parameterizing L

by an affine space over (Z/rZ), the map f is given by a nontrivial affine-linear map, andso all its fibers have the same size.

3. FIBERS WITH GOOD REDUCTION IN A FAMILY

In this section we give a general criterion (Proposition 3.2) which controls, in a givenfamily of smooth proper varieties, the collection of fibers that have good reduction outsidea fixed set of primes.

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3.1. Basic notation. We use notation K,O,O(w), S,GK ,Fw as in §2.Let Y be a smooth K-variety, and π : X → Y a proper smooth morphism.We shall suppose that this admits a good model over O, i.e. it extends to a proper

smooth morphism π : X → Y of smooth O-schemes. We will suppose, moreover, thatall the cohomology sheaves Rqπ∗Ω

pX/Y are sheaves of locally free OY -modules, and the

same is true of the the relative de Rham cohomology H q = Rqπ∗Ω•X/Y . There is no

harm in these assumptions, because they can always be achieved by possibly enlarging theset S of primes.

The generic fiber of H q is equipped with a Gauss–Manin connection (by [6, Theorem1]) and, again by shrinking S if necessary, we may suppose that this extends to a morphism

(3.1) H q →H q ⊗ Ω1Y/O.

For any y ∈ Y (K), we shall denote by Xy = π−1(y) the fiber of π above y; it is asmooth proper variety over K. Our goal in this section will be to bound Y(O). We willdo this by studying the p-adic properties of the Galois representation attached to Xy , fory ∈ Y(O) → Y (K). Fixing a degree q ≥ 0, we denote by ρy the representation of theGalois group GK on the étale cohomology group of (Xy)K :

ρy : GK → Aut Hqet(Xy ×K K,Qp).

We fix an archimedean place ι : K → C and a finite v : K → Kv , where we assumethat:

• if p is the rational prime below v, then p > 2, and• Kv is unramified over Qp, and• no prime above p lies in S.

Fix y0 ∈ Y(O). In what follows, we will analyze the set

(3.2) U := y ∈ Y(O) : y ≡ y0 modulo v.

and give criteria for the finiteness of U in terms of the associated period map. Clearly if Uis finite for each choice of y0, then Y(O) is finite too.

3.2. The cohomology at the basepoint y0. For any K-variety Z, we shall denote by ZC

its base change to C via ι, and byZKv its base change toKv via v. IfZ is aK-vector space,then ZC and ZKv are vector spaces of the same dimension, over C and Kv , respectively.

Let X0 = π−1(y0) be the fiber above y0. Let

(3.3) V = HqdR(X0/K).

Let d = dimK V . We will also denote by Vv and VC theKv- and C-vector spaces obtainedby ⊗KKv or ⊗(K,ι)C. Then VC is naturally identified with the de Rham cohomology ofthe variety X0,C, which is also (by the comparison theorem) identified with the singularcohomology of X0,C with complex coefficients:

VC ' Hqsing(X0,C,C).

In particular, monodromy defines a representation µ : π1(YC(C), y0) −→ GL(VC),whose Zariski closure we denote by Γ:

(3.4) Γ = Zariski closure of image(µ),

an algebraic subgroup of GL(VC). Note that both VC and Γ depend on the choice ofarchimedean place ι, although this dependence is suppressed in our notation.

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3.3. The Gauss–Manin connection. This connection (3.1) allows us to identify the co-homology of nearby fibers. Specifically, if one fixes a local basis v1, . . . , vr for H q atsome point of the scheme Y , and write∇vi =

∑j Aijvj , where Aij are sections of Ω1

Y insome neighbourhood of the given point, then a local section

∑fivi is flat exactly when it

solves the equation

(3.5) d(fi) = −∑j

Ajifj .

In particular, if y0 ∈ Y(O) and the place v is as before, let y0 ∈ Y(Fv) be the reduction,and choose a system of parameters p, z1, . . . , zm ∈ OY,y0 for the local ring of Y at y0; wemay do this so that (z1, . . . , zm) generate the kernel of the morphism OY,y0 → O(v).corresponding to y0.

The completed local ring OY,y0 at y0 is therefore identified with Ov[[z1, . . . , zm]], andthe image of OY,y0 in it is contained in O(v)[[z1, . . . , zm]].

Fix a basis v1, . . . , vr for H q at y0, which we assume to be compatible with theHodge filtration, i.e. each step of the Hodge filtration F iH q at y0 is spanned by a subsetof vi. Then by lifting we obtain a similar basis v1, . . . , vr for H q over the local ringOY,y0 of Y at y0. With respect to such a basis vi, the coefficients Aij of (3.5) are of theform Aij =

∑rk=1 aij,kdzk, where aij,k ∈ OY,y0 . In particular, the coefficients of aij,k,

considered as formal power series in the zi, lie in O(v).We may write down a formal solution to (3.5), where the fis are given by formal power

series inK[[z1, . . . , zm]]. By direct computation we see that these are v-adically absolutelyconvergent for |zi| < |p|1/(p−1)

v (where p is the residue characterstic of Ov) and ι-adicallyabsolutely convergent for sufficiently small |zi|.

Now we are assuming that p > 2, and that v is unramified above p. Thus we obtain anidentification

(3.6) GM : HqdR(Xy0)

∼→ HqdR(Xy)

whenever y ∈ Y(Ov) satisfies y ≡ y0 modulo v, and

(3.7) GM : HqdR(Xy0,C)

∼→ HqdR(Xy,C),

when y ∈ YC(C) is sufficiently close to y0. In the coordinates of the basis vi fixed above,GM is given by an r × r matrix

Aij(z1, . . . , zm) ∈ O(v)[[z1, . . . , zm]],

convergent in the regions noted above.We will use topological computations of monodromy to understand (3.7). However, the

fact that the series Aij has coefficients in K allows us to transfer the results to the v-adicGauss-Manin connection (3.6); see, for example, Lemma 3.1.

The fiber over the O-point y0 of Y gives a smooth proper O-model X0 for X0. Fory ∈ Y (Ov) with y ≡ y0 modulo v, we have a commutative diagram

(3.8) HqdR(Xy/Kv)

GM

∼

))Hq

cris(X0)⊗Ov Kv.

Vv = HqdR(Xy0/Kv)

∼55

DRAFT


where GM denotes the map induced by the Gauss–Manin connection, Hqcris is the crys-

talline cohomology of X0 (as a reference for crystalline cohomology, see [1, 2]), the diag-onal arrows are the canonical identification [2, Corollary 7.4] of crystalline cohomologywith the de Rham cohomology of a lift, and the commutativity of the diagram can bededuced from the results of [1, Chapter V] (see Proposition 3.6.4 and prior discussion).

This crystalline cohomology is equipped with a Frobenius operator

ϕv : Vv −→ Vv,

which is semilinear with respect to the Frobenius on the unramified extension Kv/Qp. Bythe isomorphisms of (3.8), this ϕv acts on Hq

dR(Xy/Kv) and HqdR(Xy0/Kv) as well.

3.4. The period mappings in a neighbourhood of y. Now V = HqdR(X0/K) is equipped

with a Hodge filtration:

(3.9) V = F 0V ⊃ F 1V ⊃ . . .

LetH be the K-variety parameterizing flags in V with the same dimensional data as (3.9),and let h0 ∈ H(K) be the point corresponding to the Hodge filtration on V .

Base changing by means of v and ι, we get a Kv-variety Hv and a C-variety HC. Wedenote by Han

v and HanC the associated rigid-analytic space and the associated complex-

analytic space, respectively. We denote by hι0 ∈ HanC the image of h0.

Let ΩC be a contractible analytic neighbourhood of y0 ∈ Y anC . The Gauss-Manin

connection defines an isomorphism HdR(Xt/C) ' HdR(X0/C) for each t ∈ ΩC. Inparticular, the Hodge structure on the cohomology of Xt defines a point of H(C); thisgives rise to the complex period map

ΦC : ΩC −→ HanC .

Indeed, ΦC extends to a map from the universal cover of Y anC to Han

C , and this map isequivariant for the natural action of π1(Y an

C , y0) (which acts by monodromy on HanC ).

From this we see, in particular, that

(3.10) Γ · hι0 ⊂ the Zariski-closure of ΦC(ΩC) insideHC,

since the preimage under ΦC of any algebraic subvariety containing ΦC(ΩC) gives acomplex-analytic subvariety of Y an

C containing an open subset.We need a v-adic analogue. Again, if y ∈ Y (Ov) satisfies y ≡ y0 modulo v, the Gauss–

Manin connection (3.8) allows one to identify the Hodge filtration on HqdR(Xy/Kv) with

a filtration on Vv , and thus with a point ofH(Kv). This gives rise to a function

Φv : Ωv −→ HKvwhere Ωv = y ∈ Y (Ov) : y ≡ y0 modulo v.

Lemma 3.1. The dimension of the Zariski closure (in the Kv-variety HKv ) of Φv(Ωv) isat least the (complex) dimension of Γhι0.

In particular, ifHbadv ⊂ Hv is a Zariski-closed subset of dimension less than dimC(Γhι0),

then Φ−1v (Hbad

v ) is contained in a proper Kv-analytic subset of Ωv , by which we mean asubset cut out by v-adic power series converging absolutely on Ωv .

Proof. Fix an embedding H → PN−1 over K. Our previous discussion shows that thereis a system of coordinates z1, . . . , zm for Y near y0 and power series B1, . . . , BN ∈K[[z1, . . . , zm]] where:

DRAFT


- (B1, . . . , BN ) converges in a small complex neighbourhood of y0, and (shrinkingΩC if necessary) gives the composite

ΩCΦC−→ HC → PN−1

C

- (B1, . . . , BN ) converges absolutely on Ωv and coincides there with the composite

ΩvΦv−→ HKv → PN−1

Kv.

Let SpecA ⊆ H be an open affine containing h0. The formal power series (B1, . . . , BN )induce a map B : A → K[[z1, . . . , zm]], hence (for any field K ′ ⊃ K) also a mapBK′ : A ⊗K K ′ → K ′[[z1, . . . , zm]]. We define ZA,K′ to be the closed subscheme ofSpec(A⊗K K ′) cut out by the kernel of BK′ .

One verifies the following functoriality properties of ZA,K′ . For any f ∈ A not vanish-ing at h0, the schemes ZA[1/f ],K′ and ZA,K′ are related by

ZA[1/f ],K′ = ZA,K′ ∩ SpecA[1/f ],

and conversely ZA,K′ is the Zariski closure of ZA[1/f ],K′ in SpecA. Furthermore, ZA,K′is the base change of ZA,K to K ′.

Hence, if we take Z to be the Zariski closure in H of ZA,K (for any choice of A), thenwe recover all ZA,K′ by pullback to Spec(A ⊗K K ′). Informally, we may consider Z tobe (the Zariski closure of) the image of our formal power series map intoH.

In the analytic setting (p-adic or complex), an absolutely convergent power series van-ishes identically exactly when it is formally zero; it follows that the Zariski closure ofΦv(Ωv) is exactly ZKv , and the Zariski closure of ΦC(ΩC) is exactly ZC. The resultfollows.

3.5. Hodge structures. We use p-adic Hodge theory to relate Galois representations tocrystalline cohomology. A good reference is [3].

For each y ∈ U the representation ρy is crystalline upon restriction to Kv , because ofthe existence of the model Xy for Xy . By p-adic Hodge theory, there is [3, Proposition9.1.9] a fully faithful embedding of categories:

(3.11) crystalline representations of GalKv on Qp vector spaces → FL,where the objects of FL are triples (W,α, F ) of a Kv-vector space W , a Frobenius-semilinear automorphism α : W →W , and a descending filtration F of W .

By the crystalline comparison theorem of Faltings [5], the embedding (3.11) carries ρyto the triple (Hq

dR(Xy/Kv),Frv, Hodge filtration for Xy). But (3.8) induces an isomor-phism in FL:

(HqdR(Xy/Kv),Frv, Hodge filtration for Xy) ' (Vv, ϕv,Φv(y)),

We obtain the following:

Proposition 3.2. Notation as above: in particular X → Y is a smooth proper family overK, V is the de Rham cohomology of a given fiber X0 above y0 ∈ Y (K), H a space offlags in V ,

Φv : y ∈ Y(Ov) : y ≡ y0 −→ Hanv

is the v-adic period mapping, Γ is the Zariski closure of the monodromy group, and h0 isthe image of y0 under the period mapping.

Suppose that

(3.12) dimKv

(Z(ϕ[Kv :Qp]

v ))< dimC Γhι0

DRAFT


where the left-hand side Z(. . . ) denotes the centralizer, in AutKv (Vv), of the Kv-linearoperator ϕ[Kv:Qp]

v .Then the set

(3.13) y ∈ Y (O) : y ≡ y0 modulo v, ρy semisimple

is contained in a proper Kv-analytic subvariety of the residue disk of Y (Kv) at y0.

Proof. For any y as in (3.13) the Galois representation ρy belongs to a finite set of iso-morphism classes (Lemma 2.3). By our previous discussion the triple (Vv, ϕv,Φv(y))also belongs to a finite set of isomorphism classes (now in the category FL). Choosingrepresentatives (Vv, ϕv, hi) for these isomorphism classes, we must have

Φv(y) ∈ Z(ϕv) · hi,

where Z(ϕv) is the centralizer, in the Qp-algebraic group ResKvQpAutKv (Vv) of Kv-linear

automorphisms of Vv , of the crystalline Frobenius for X0.Now certainly Z(ϕv) ⊂ Z(ϕ

[Kv :Qp]v ), and the right-hand side is a Kv-algebraic sub-

group of GLKv (Vv). Therefore, any y as in (3.13) is contained in the preimage, under Φv ,of a proper Zariski-closed subset of Hv with dimension the left hand side of (3.12). Thisis obviously a Kv-analytic subvariety as asserted. It is proper because of Lemma 3.1.

4. THE S-UNIT EQUATION

As a first application, and a warm-up to the more complicated case of curves of highergenus, we will show finiteness of the set of solutions to the S-unit equation

U = t ∈ O∗S : 1− t ∈ O∗S.

We may freely enlarge both S and K. Thus, we may suppose that S contains all primesabove 2 and thatK contains the 8th roots of unity. Letm be the largest power of 2 dividingthe order of the group of roots of unity in K. By assumption m > 8.

First of all, it suffices to prove finiteness of the set

U1 = t ∈ O∗S : 1− t ∈ O∗S , t /∈ (K∗)2.

because U ⊂ U1 ∪ U21 ∪ U4

1 ∪ · · · ∪ Um1 . To see this, we take t ∈ U and try to repeatedlyextract its square root; observe that such a square root, if in K, also belongs to U . If wecannot extract an mth root of t, we are done; otherwise, write t = tm1 and adjust t1 by anmth root of unity to ensure that t1 is nonsquare.

If t ∈ U1 and t1/m is an mth root of t in K, there are (Hermite–Minkowski) onlyfinitely many possibilities for K(t1/m). Enumerate them; call them L1, . . . , Lr, say. Eachhas degree m over K. Thus it is sufficient to prove finiteness of the set

U1,L = t ∈ U1, t1/m ∈ L.

for a fixed field L ∈ L1, . . . , Lr; we understand t1/m ∈ L as meaning that t admits anmth root inside L.

Fixing L as above, we may choose a prime v of K, not in S, which remains inert in L;we require that the prime p of Q below v is unramified in K. Then any t ∈ U1,L is not asquare in Kv . Fixing t0 ∈ U1,L we shall show that

(4.1) t ∈ U1,L : t ≡ t0 modulo v

is finite.

DRAFT


4.1. A variant of the Legendre family. As discussed in the introduction, we apply Propo-sition 3.2 not to the Legendre family, but to a modification of it: Let Y = P1

O − 0, 1,∞(where 0, 1 denote the corresponding sections over SpecO) and letY ′ = P1

O−0, µm∞;let π : Y ′ → Y be the map u 7→ um.

Let X → Y ′ be the Legendre family, so that its fiber over t is the curve y2 = x(x −1)(x− t); and consider the composite

X −→ Y ′ π−→ Y.We will apply our prior results to the family X → Y; here, the fiber over t is the disjointunion of the curves y2 = x(x− 1)(x− t1/m) over all mth roots of t. Also, as before, wedenote by X and Y the fibers of X and Y over Spec(K).

We will show below thatClaim 1: The Zariski-closure Γ of monodromy for the family X → Y , considered as a

complex-analytic family by means of any embedding K → C, contains SL(2)m

(see Lemma 4.1 for precise statement).Claim 2: For all but finitely many y ∈ U1, the Galois representation ofGK onH1

et(Xy,Q`)is semisimple.

4.2. Proof of finiteness. Assuming these for the moment, we complete the proof usingProposition 3.2. Use the notation of the Proposition (specialized to the family above); inparticular we have fixed a complex embedding K → C, which we use to complexifyK-schemes and K-vector spaces.

The vector space V = H1dR(Xt0/K) has the structure of a 2-dimensional vector space

over K(t1/m).The splitting of Xt0,C into geometric components induces a splitting

(4.2) VC =

m⊕i=1

Vi,

where each Vi is a 2-dimensional complex vector space. Claim 1 shows that the algebraicmonodromy group Γ contains

∏SL(Vi). The pertinent flag variety H ' Gr(m,V ) is the

variety of m-dimensional subspaces in V ; the splitting (4.2) induces a natural inclusion∏mi=1 PVi → HC. Therefore the the orbit Γhι0 is all of

∏mi=1 PVi and, in particular, has

dimension m > 8.On the other hand, we may identify Vv = V ⊗K Kv with a 2-dimensional vector

space over Kv(t1/m). Let ϕv be the semilinear Frobenius operator on Vv , and consider

the Kv-linear operator ϕ[Kv :Qp]v . This operator is Kv(t

1/m)-semilinear, although it is Kv-linear; we can apply Lemma 2.1 to it, taking E = Kv(t

1/m) and σ to be a generator forGal(E/Kv). Therefore centralizer Z of Frobv , inside GLKv (Vv), has Kv-dimension atmost 4. We have now verified the input (3.12) for Proposition 3.2. In combination withClaim 2, it shows that the set described in (4.1) is finite.

Now we prove the claims:

Lemma 4.1. Consider the family of curves over C− 0, 1 whose fiber over t ∈ C is theunion of the elliptic curves Ez : y2 = x(x − 1)(x − z), over all mth roots zm = t. Thenthe action of monodromy

(4.3) π1(C− 0, 1, t0) −→ Aut

( ⊕zm=t0

H1(Ez,Q)

)has Zariski closure containing

∏z SL(H1(Ez,Q)).

DRAFT


Proof. Write Γ for the Zariski closure in question. Then:

- Γ transitively permutes the factors on the right-hand side of (4.3), by consideringthe action of local monodromy near t = 0;

- Γ∩ SL(2)m projects to SL(2) in each factor: indeed, this projection contains, as afinite index subgroup, the algebraic monodromy group of the Legendre family.

- Γ contains an element of the form

(1, 1, . . . , 1, u, 1, . . . , 1)

where u ∈ SL(2) is a nontrivial unipotent element, as we see by considering theaction of local monodromy near t = 1.

These facts, and the (Lie algebra) version of Lemma 2.9, imply that Γ ⊃ SL(2)m. For the second claim, we apply the following Lemma together with the fact that induc-

tion from GL to GK carries semisimples to semisimples (Lemma 2.2).

Lemma 4.2. Let L be a number field and p a rational prime, larger than 2, and unramifiedin L. There are only finitely many z ∈ L such that z, 1− z are both p-units, but for whichthe Galois representation of GL on the Tate module Tp(Ez) = H1

et(Ez ⊗L L,Qp) of theelliptic curve

Ez : y2 = x(x− 1)(x− z),

fails to be simple.

Of course much stronger results are known. The point here is that we prove this in a“soft” fashion, using the Torelli theorem as a substitute for more sophisticated arguments;although we use the specific feature of Hodge weights 0 and 1, the argument is robustenough to generalize (although with a little added complexity, see e.g. Lemma 6.2).

Proof. Fix z0 ∈ L with the quoted p-integrality properties; in particular, Ez0 has goodreduction at all primes of L above p.

It is enough to show the same finiteness when we restrict to the set

VL = z ∈ L : z ≡ z0 modulo v, for all v|p.

If Tp(Ez) is reducible there exists a one-dimensional subrepresentation Wz ⊂ Tp(Ez).By Lemma 2.7 (applied with K = Q) there is a place w of L above p such that Wz

corresponds to a weight 1 subspace on the de Rham side, i.e. F 1H1dR(Ez/Lw) is stable by

the semilinear Frobenius Frobw. Because the Newton and Hodge polygons have the sameendpoint, the slopes of the Frobenius at w must be unequal in this case; in particular, theLw-linear Frobenius Frob[Lw:Qp]

w is not scalar.As in the discussion around (3.8), Gauss–Manin induces an identification

H1dR(Ez0/Lw) ' H1

dR(Ez/Lw)

of Lw-vector spaces with semilinear Frobenius action. There are at most two lines in theleft-hand space that are stable by the semilinear Frobenius. But the position of the Hodgeline F 1H1

dR(Ez/Lw) varies w-adic analytically inside the disc VL, and the associated w-adic analytic function is nonconstant (by the – trivial – Torelli theorem for elliptic curves).It follows there are at most finitely many z ∈ VL for which F 1H1

dR(Ez/Lw) is Frobeniusstable. Taking the union over possible w we still see that the exceptional set is finite.

DRAFT


5. MORDELL’S CONJECTURE

Let Y be a curve of genus at least two over a number fieldK. In this and the subsequentsections, we will apply our method to analyze Y (K). In our analysis we will use thefollowing special type of family over Y :

Definition 5.1. An abelian-by-finite family over Y is, by definition, a sequence of mor-phisms

X −→ Y ′π−→ Y

where π is finite étale, and X → Y ′ is (equipped with the structure of) a polarized abelianscheme.

A “good model” for such a family, over an S-integer ring O ⊂ K, is a family X →Y ′ → Y of smooth, proper O-schemes, satisfying the same conditions, and recoveringX → Y ′ → Y on base change to K.

Of course the polarization on X → Y ′ is an additional structure but for brevity we donot explicitly include it in the notation. Recall that we have already used families with thisstructure, in the course of §4.

For any such abelian-by-finite family X → Y ′ → Y we may take a geometric pointy0 : Spec(K)→ Y , and consider the action of π1(Y, y0) on

H1(Xy0 ,Q`) '⊕

π(y)=y0

H1(Xy,Q`),

where the sum is taken over y ∈ Y ′(K) lying over y0.

Definition 5.2. We say that the family has full monodromy if the Zariski-closure of π1(Y, y0),in its action on the right-hand side, contains the product of symplecitc groups:

( image of π1(Y, y0)) ⊃∏

π(y)=y0

Sp(H1

et(Xy,Q`), ω),

where the symplectic group is with reference to the form defined by the polarization.

By standard comparison theorems, this is equivalent to the obvious analogous statementin singular cohomology, if we take y0 to be a complex point, and in particular this conditionis purely topological.

5.1. The Kodaira–Parshin family. If Y is a curve of genus 2 or greater, there exist manyinteresting (geometric) finite étale covers of Y ×Y branched exactly at the diagonal. Theseare the source of the “Kodaira–Parshin” construction. There is a lot of flexibility in thisconstruction, depending on what Galois group and branching one takes for the cover: wewill use a semidirect product of cyclic groups. To describe the precise family we use, weuse the theory of Hurwitz schemes, which has been briefly summarized in §2.2.

Let q, r be prime numbers, such that r ≡ 1 modulo q, and let Gq,r be the subgroup ofpermutations of Z/rZ of the form

x 7→ ax+ b,

where a ∈ (Z/rZ) satisfies aq = 1, and b ∈ Z/rZ. Thus Gq,r is isomorphic to thesemidirect product (Z/rZ) o (Z/qZ). Note that Gq,r has trivial center.

Definition 5.3. Let Y be a curve of genus at least 2 over a number field K, and let (q, r)be prime numbers with r ≡ 1 (q).

DRAFT


The Kodaira-Parshin curve family over Y with parameters (q, r) will be the sequenceof morphisms

(5.1) Zq,r −→ Y ′q,r × Yπpr1−→ Y,

where π : Y ′q,r → Y is the Hurwitz space parameterizing Gq,r-covers of Y ramified at asingle point,3 and Zq,r → Y ′q,r × Y is the universal curve over this Hurwitz space.

Therefore, in the construction above: π : Y ′q,r → Y is finite étale, Zq,r → Y ′q,r is arelative curve, and Zq,r → Y ′q,r × Y is finite étale away from the graph of π. Moreover,if we consider the composite morphism Zq,r → Y from (5.1), its fiber over a geometricpoint y of Y is the disjoint union of all curves that admit a degree-r map to Yy , with Galoisgroup Gq,r, and ramified at a single point.

What is more useful to work with is an associated Jacobian family, or more precisely a“Prym” version thereof. We can construct this using the representation theory of Gq,r.

To motivate the formula that follows, recall that a morphism C1 → C2 of curves overan algebraically closed field admits a Prym variety: by definition, this is the cokernel ofthe map Pic0(C2) → Pic0(C1). Now suppose that that the covering C1 → C2 is Galois,with Galois group Gq,r. In this case, we can form a smaller degree r covering C ′1 → C2

using the permutation action of Gq,r on Z/rZ; we will examine its Prym. The image ofPic0(C2) in Pic0(C1) is now the connected component of the Gq,r-invariants; similarlythe image of Pic0(C ′1) in Pic0(C1) is the connected component of the invariants by thesubgroup Hq,r = Z/qZ, which is a point stabilizer in the permutation action of Gq,r onZ/rZ. In summary, then, the Prym variety is the image of the idempotent

e :=1

#Hq,r

∑h∈H

h− 1

#Gq,r

∑g∈Gq,r

g ∈ Q[Gq,r]

acting on Pic0(C1).We may apply this discussion in a family. Namely, Zq,r → Y ′q,r is a relative curve over

Y ′q,r and it admits a Gq,r-action, where Gq,r acts trivially on the base; indeed, the mapZq,r → Y ′q,r × Y had the structure of a branched Gq,r-cover. We may therefore form

Xq,r = Pic0Zq,r→Y ′q,r [e],

which should be thought of as the relative Prym variety, not for the curve family Zq,r →Y ′q,r itself, but rather for the associated degree r family.

Definition 5.4. Notation as in the prior definition. The Kodaira–Parshin family of Jaco-bians over Y , with parameters (q, r), is by definition the abelian-by-finite family definedby the sequence

Xq,r −→ Y ′q,r → Y,

where Xq,r is the relative Prym, as defined above.

Proposition 5.5. Let Y be a curve over K of genus > 2.Let X → Y ′ → Y be an abelian-by-finite family over Y , with full monodromy (5.2);

and let g be the relative dimension of X → Y . Suppose that X → Y ′ → Y admits a goodmodel over the ring O of S-integers, and let v /∈ S be a friendly place of K (Definition2.4).

3This Hurwitz space is geometrically disconnected: the monodromy around the missing point gives an invari-ant. However, this does not matter.

DRAFT


Suppose given a finite GK-set Υ, where the action is unramified outside S, and a GK-equivariant map

geometric components of Y ′ −→ Υ;

for each o ∈ Υ let Y ′o be the corresponding union of components of Y ′, and deg(o) thedegree of the finite covering Y ′o → Y .

Suppose, finally, that the cycle structure (e1, . . . , ek) of Frobv on Υ satisfies

(5.2)

∑ei>d diei∑ei<d diei

> g,

where di is the constant value of o 7→ deg(o) on the orbit corresponding to ei, and d =max(5, 8g

g+1 ).Then Y (K) is finite.

This is in essence a variant of Proposition 3.2 but it requires some careful indexing. Wepostpone the proof to the next section §6.

Theorem 5.6. Let Y be a curve over the number field K with genus at least two, and letq, r be primes such that q > 8[K : Q], q is unramified in K, and r ≡ 1 modulo q.

If the Kodaira-Parshin family over Y with parameters (q, r) has full monodromy, thenY (K) is finite.

We emphasize that the condition of full monodromy is purely topological. It dependsonly on the genus of Y and the parameters (q, r). It seems to us very likely that this holdsfor “most” (q, r), perhaps even with only finitely many exceptions; we prove it holds forgenus(Y ) = 2, q = 2, r = 3 later.

Proof. We first observe that there exists a friendly place v of K (in the sense of Lemma2.5) with (qv, q) = 1 and such that the order of qv in (Z/q)∗ is at least d.

Indeed, the Galois closureK ′ is unramified at all primes above q. Since Q(ζq) is totallyramified at q, the fields Q(ζq) and K ′ are linearly disjoint over Q, and correspondinglyrestriction induces an isomorphism

Gal(K ′(ζq)/Q) −→ Gal(K ′/Q)×Gal(Q(ζq)/Q).

If K has no CM subfield, choose σ ∈ Gal(K ′/Q) arbitrarily. Otherwise let E be themaximal CM subfield of K, and let E+ the maximal totally real subfield; choose someσ ∈ Gal(K ′/E+) ⊆ Gal(K ′/Q) inducing the nontrivial automorphism of E over E+,and let σi be the least power of σ such that σi ∈ Gal(K ′/K); thus i ≤ [K : Q].

Let a be a primitive root modulo q. If q > d[K : Q] then ai has order at least d inside(Z/qZ)∗. By Chebotarev density, there is a place ℘ of K ′ such that the Frobenius Frob℘is the element (σ, a) of Gal(K ′(ζq)/Q) ' Gal(K ′/Q) × F∗q . Let p be the prime of Qbelow ℘; thus p ≡ amodulo q. The place v ofK below ℘ has residue field of size qv = pi,and therefore the order qv in (Z/q)∗ is at least d as desired. The place of E+ below ℘ isinert in E, by choice of σ. This shows that there indeed exists v as desired.

Now we consider the Kodaira–Parshin family of Jacobians

Xq,r −→ Y ′q,r −→ Yq,r,

and write g for the relative dimension of X → Y .The map Gq,r → Z/qZ induces a map of GK-sets

geometric components of Y ′q,r −→ H1et(YK ,Z/qZ)− 0.︸︷︷︸

Υ

DRAFT


Moreover, for o ∈ Υ, if we write Y ′o for the set of geometric components of Y ′ correspond-ing to o, the degree of Y ′o → Y is independent of o; this follows from Lemma 2.10.

This Υ has the structure of (2g)-dimensional vector space over Fq (remove its origin).It is equipped with a Galois-equivariant Weil pairing, valued in the q-th roots of unity. TheFrobenius at v induces, in particular, an automorphism T : Υ→ Υ that satisfies

〈Tv1, T v2〉 = qv〈v1, v2〉.The number of elements of Υ belonging to T -orbits of size less than d is at most 4qg: it iscontained in the union of the subspaces ker(T i − 1) for 5 ≤ i ≤ 8; these are all isotropicsince qiv is not equal to 1 modulo q for such i.

We find that of the q2g elements of Υ, at most 4qg lie in orbits (for the Frobenius action)of size 8 or smaller. Hence, writing (e1, . . . , ek) for the cycle structure of Frobv on Υ, wehave ∑

ei>8 ei∑ei<8 ei

>q2g − 4qg

4qg=qg

4− 1.

Noting that we necessarily have q ≥ 11, this quotient is guaranteed to exceed g.We may now apply Proposition 5.5 to Xq,r → Y ′q,r → Y , and to the set Υ, to get the

desired result.

Proposition 5.7. Let Y be a curve of genus 2. Then the Kodaira–Parshin family of Jaco-bians X2,3 → Y ′2,3 → Y , i.e. with (q, r) = (2, 3), has full monodromy.

Proof. See Section 7.

Theorem 5.8. Suppose that K does not contain a CM field, and Y is a genus 2 curve overK for which the GK action on Y [2] has image Sp4(F2). Then Y (K) is finite.

The condition that K does not have a CM field can be relaxed: see the discussion afterthe proof.

Proof. Such a curve Y is necessarily of the form y2 = f(x), where f is irreducible ofdegree 6 and the roots of f generate an S6-extension. The assumption on K means thatall places v are friendly (Definition 2.4). In particular we may choose v such that v isfriendly, Kv is an unramified extension of Qp, and the Frobenius conjugacy class attachedto v permutes cyclically the 6 roots of f .

Now apply the previous argument with parameters (q, r) = (2, 3); the relative dimen-sion of the Kodaira–Parshin family of Jacobians X2,3 → Y is then g = 3, and we mayidentify the set Υ ' H1(Y,Z/2Z) − 0 with unordered pairs of roots of f . The orbitsof Frobv on Υ are of size 6, 6, 3. The finiteness of Y (K) follows from Proposition 5.5,applied to X2,3 → Y ′2,3 → Y and with this Υ.

For generalK, the genericity condition on 2-torsion takes a slightly different shape. Wewill simply formulate it but will not write out the proof in this case: it is a straightforwardmodification of the foregoing argument.

Namely, let M be a number field. Fix an algebraic closure M over M and for any fieldextension M ′ ⊃ M we denote by ΣM ′ the set of embeddings M ′ → M that extend theinclusion on M . Thus GM acts on ΣM ′ ; we denote its image by GM ′/M 6 Sym(ΣM ′).Now let K ⊃ M be an extension of number fields. We say a number field L containingK has has “full Galois group over M” if the image of GM on ΣL is as large as possible –explicitly, we ask that GM induce on ΣL the full wreath product

σ ∈ Sym(ΣL) : σ preserves ΣL → ΣK , and induces on ΣK an element of GK/M ..Then the prior argument goes through in the following setting:

DRAFT


Let Y be the genus 2 curve y2 = f(x) (with f sextic). LetE be the largestCM subextension of K, and E+ its totally real subfield. Suppose that f isirreducible and the field K ′ generated by a root of f has full Galois groupover E+.

Then Y (K) is finite.

6. A VARIANT OF PROPOSITION 3.2

We prove Proposition 5.5, which we rewrite for the reader’s convenience.

Proposition 5.5. Let Y be a curve over K of genus ≥ 2.Let X → Y ′ → Y be an abelian-by-finite family over Y , with full monodromy (5.2);

and let g be the relative dimension of X → Y . Suppose that X → Y ′ → Y admits a goodmodel over the ring O of S-integers, and let v /∈ S be a friendly place of K (Definition2.4).

Suppose given a finite GK-set Υ, where the action is unramified outside S, and a GK-equivariant map

geometric components of Y ′ −→ Υ;

for each o ∈ Υ let Y ′o be the corresponding union of components of Y ′, and deg(o) thedegree of the finite covering Y ′o → Y .

Suppose, finally, that the cycle structure (e1, . . . , ek) of Frobv on Υ satisfies

(6.1)

∑ei≥d diei∑ei<d diei

> g,

where di is the constant value of o 7→ deg(o) on the orbit corresponding to ei, and d =max(5, 8g

g+1 ).Then Y (K) is finite.

Proof. Enlarging S further if necessary, we may assume that X → Y satisfies the assump-tions at the start of §3.1. Fix y0 ∈ Y (K). It is sufficient to show that there are only finitelymany points of Y (K) that lie in the residue disk

Ωv = y ∈ Y (Kv) : y ≡ y0 modulo v,which we are regarding as a Kv-analytic manifold.

Recall we have fixed an algebraic closure K with Galois group GK . Fix an extensionof v to that field; the completion of K gives an algebraic closure Kv of Kv .

For each y ∈ Y (K), write

Σy = preimage of y in Y ′(K).

Thus Σy is a finite set with GK-action, and it is identified with the set of geometric com-ponents of Xy . We write, for short, Σ0 instead of Σy0 . For y in the residue disk Ω of y0,reduction at the place of K above v gives an identification

Σy ' Σ0,

equivariantly for the action of Frobv .

Definition 6.1. Suppose s ∈ Σy . The stabilizer of s ∈ GK defines a field extensionK(s) ⊃ K, and the point s factors uniquely:

Spec(K)→ Spec(K(s))→ Y ′.

Accordingly the fiber X → Y ′ above s descends to K(s); we denote this (abelian) varietyby Xs.

DRAFT


We denote by ρs the representation ofGK(s) on the geometric étale cohomologyH1et(Xs×K(s)

K,Qp) of Xs. The polarization on X → Y ′ induces a symplectic form on the underlyingspace of ρs.

Similarly, by regarding s as a point inX (Kv) we obtain similarly an extensionKv(s) ⊂Kv , and a Kv(s)-variety Xsv ; by reducing s to a point s ∈ X (Fv) we obtain an extensionFv(s) ⊂ Fv and a Fv(s)-variety Xs.

Let E0 be the étaleK-algebra associated to theGK-set Σ0. Then the morphismXy0 →Spec(K) in fact factors through Spec(E0), andXy0 has the structure of a polarized abelianscheme over Spec(E0). Therefore, the space

V := H1dR(Xy0/K),

has the structure of a free E0-module, and the polarization induces a nondegenerate al-ternting form V × V → E0 (i.e. it remains nondegenerate under any extension of scalarsE0 → K).

Let us denote by

H = maximal isotropic E0-stable subspaces inside V .

Then the period map at y0 gives a Kv-analytic period mapping

Φv : Ωv −→ Hv.where the base changeHv toKv is naturally identified with the variety of maximal isotropic(E0 ⊗K Kv)-stable subspaces inside V ⊗K Kv . Now Lemma 3.1, and the assumption offull monodromy, imply that Φv(Ωv) is Zariski–dense inHv .

Now (E0 ⊗K Kv) splits into fields E(O)0 indexed by Frobv-orbits O on Σ0; explicitly,

choosing a representative ξ ∈ O, we have E(O)0 ' Kv(ξ). Correspondingly, Hv has a

decomposition indexed by orbits of Frobv on Σ0, and we shall denote by HO the factorassociated to an orbit O ⊂ Σ0. Thus

Hv =∏O

HO,

where, if ξ ∈ O is again any representative, we may identify

(6.2) HO = Lagrangian Kv(ξ)-subspaces of H1dR(Xξv ).

As in the discussion around (3.8), the identification of V ⊗ Kv with the crystallinecohomology of the reduction of Xy0 endows it with a semilinear Frobenius. This semilin-ear Frobenius induces a Kv-algebraic map Hv → Hv , which respects the decompositionabove.

For y ∈ Ωv we have

(6.3) projection toHO of Φv(y) = F 1H1dR(Xyv ).

where y ∈ Σy corresponds to ξ ∈ O under the bijection Σy ' Σ0, and we identifyH1

dR(Xyv ) with H1dR(Xξv ) (cf. (3.8)).

Before we come to the core of the proof a few remarks on indexing. For y ∈ Σy thereare bijections

embeddings K(y) → K ∼←− GK/GK(y) → Σy,

where the left arrow restricts an element σ ∈ GK to K(y), and the right arrow sends σ toσ(y). Considering Frobv orbits, we get

places of K(y) above v ∼←− Frobv orbits on GK/GK(y) → Frobv orbits on Σy .(6.4)

DRAFT


We shall denote by O(w, y) the Frobenius orbit on Σy corresponding to the place w ofK(y); thus [K(y)w : Kv] = #O(w, y).

Claim 1: There is a finite subset F ⊂ Ωv ∩ Y (K) such that, for y ∈(Ωv ∩ Y (K))−F , there exists y ∈ Σy such that the Frobv-orbit of y hassize at least d, and such that ρy is simple as a GK(y)-representation.

Recall that d = max(5, 8gg+1 ). Also recall that we denote by ρy theGK(y)-representation

H1et(Xy ×K(y) K,Qp); we shall denote by ρwy its restriction to a decomposition group at

the place w of K(y). We will prove the Claim in a moment. For the moment observe thatfor y ∈ (Ωv ∩ Y (K)) − F , and y ∈ Σy , there are but finitely many possibilities for thefield K(y). We will also show:

Claim 2: Let L be a field extension ofK inK, andw a place of L above vwith [Lw : Kv] ≥ d. Fix a GLw -module N0. Then there are only finitelymany y ∈ Y ′(L) such that all of the following conditions hold:

(i) the image y of y in Y (L) lies in Ωv ∩ Y (K);(ii) the Frobv-orbit of y has size at least d;

(iii) ρwy ' N0.Claim 1, Claim 2, and the finiteness result for global Galois representations (Lemma

2.3) complete the proof.Now, to prove Claim 1 and Claim 2 we shall analyze the period mapping more carefully.

Proof. (of Claim 2): Write y for the image of y in Y (K). We may as well fix a Frobv-orbit O ⊂ Σ0, with #O ≥ d, and consider only those y for which O(w, y)↔ O under thebijection Σy ' Σ0.

Under the correspondence of p-adic Hodge theory, ρwy corresponds to H1dR(Xy/Lw),

together with its natural semilinear Frobenius operator φ, and the flag defined byF 1HdR(Xy/Lw).Thus, for every y for which ρwy ' N0, the isomorphism class of the triple

(H1dR(Xy/Lw), F 1H1

dR, ϕv = semilinear Frobenius)

is the same.Now, in view of (6.3), this means that the projection of Φv(y) ∈ Hv toHO lies inside a

single orbit for the action of the Frobenius centralizer Z(ϕv) on HO, and so also a singleorbit of Z(ϕ

[Kv:Qp]v ). 4

Apply Lemma 2.1 with E = Lw, and σ a generator for Gal(Lw/Kv) to see that thisFrobenius centralizer has Kv-dimension at most 4g2. Since diagonal matrices act triviallyon the Grassmannian, the orbits of the Frobenius centralizer have dimension at most 4g2−1.

As noted earlier, the period map Φv has Zariski-dense image (in the Kv-variety Hv;therefore this remains true when projected to HO). Since dimHO = [Lw : Kv] · g(g+1)

2

and [Lw : Kv] >8gg+1 , we have finished the proof of Claim 2.

Proof. (of Claim 1): Let us call y ∈ Y (K) ∩ Ωv “bad” when, for every y ∈ Σy such that

(6.5) K(y) admits a place w above v of degree [K(y)w : Kv] > d,

the representation ρy fails to be simple. We must show there are only finitely many bady ∈ Y (K) ∩ Ωv .

4Strictly speaking, our phrasing is misleading, because the Frobenius centralizer does not act on HO . Rather,we regard HO as a subspace of an ambient Grassmannian, without any isotropicity imposed, and take the Frobe-nius orbit in this larger Grassmannian. The argument that follows works with this interpretation.

DRAFT


Suppose that y is bad and y ∈ Σy satisfies (6.5). Then there exists a GK(y)-invariantsubspace Wy of ρy . Passing from Wy to its orthogonal complement, if necessary, we maysuppose that dim(Wy) 6 g. Replacing Wy by W ∩W⊥ if it is nontrivial, we may assumethat either Wy = W⊥y or Wy ∩W⊥y = 0 – in other words, the symplectic form on W iseither trivial or nondegenerate.

Now, let w be a place of K(y) above v. Restricting Wy to the decomposition groupat w and applying p-adic Hodge theory, we obtain a filtered sub-φ-module Wy,w of theφ-module H1

dR(Xy/K(y)w). Because the comparison theorem between p-adic étale co-homology and de Rham cohomology respects cup products, we see thatWy,w is also eitherisotropic or nondegenerate, these notions now taken with reference to the symplectic formon H1

dR(Xy/K(y)) induced by the polarization.If y ∈ Σy does not satisfy (6.5), set Wy to be the full space of ρy .

Sublemma: If y ∈ Y (K) ∩ Ωv is bad, there exists:- y ∈ Σy- a place w of K(y) such that [K(y)w : Kv] > d- a subrepresentationWy ⊂ ρy such that dimF 1Wy,w ≥ dim(Wy)/2.

Proof of sublemma: Take a bad y ∈ Y (K) ∩ Ωv . For each y ∈ Σy defineWy as just described. Assume by way of contradiction that, for each suchy and w satisfying [K(y)w : Kv] > d we have

dimF 1Wy,w <1

2dimWy.

By Lemma 2.7, we have

(6.6)∑w|v

[K(y)w : Kv]dimF 1Wy,w

dimWy,w=

1

2[K(y) : K]

for any y ∈ Σy . Sum (6.8) over a set of representatives y for GK-orbitson Σy . Since dim(Wy,w) ≤ g whenever [K(y)w : Kv] ≥ d, we get

(6.7)∑

[K(y)w :Kv ]>d

[K(y)w : Kv]

(1

2− 1

2g

)+

∑[K(y)w :Kv ]<d

[K(y)w : Kv] >1

2#Σy.

Here all the summations are understood to be over a set of representativesy for GK-orbits, together with all the places w of K(y). Therefore,

(6.8)∑

[K(y)w :Kv ]<d

1

2[K(y)w : Kv] >

1

2g

∑[K(y)w :Kv ]>d

[K(y)w : Kv].

Now use the GK-equivariant map Σy → Υ to conclude∑[K(y)w:Kv ]>d

[K(y)w : Kv] >∑ei>d

diei,

and similarly with< d but with the inequality reversed. Therefore, we get

1

2

∑ei<d

diei >1

2g

∑ei≥d

diei,

which contradicts our numerical assumption. This contradiction provesthe sublemma.

We now return to the proof of Claim 1. Fix any orbit O ⊂ Σ0 of size ≥ d, and a basepointξ ∈ O. In view of the Sublemma and (6.3), it is enough to show that there are only finitely

DRAFT


many y ∈ Y (K)∩Ωv such the projection of Φv(y) toHO lies in the following subvarietyH′O:

H′O ⊂ HO is defined as all LagrangianKv(ξ)-subspacesF ⊂ H1dR(Xξv/Kv(ξv))

for which there exists a Frobenius-stableKv(ξ)-subspaceW ⊂ H1dR(Xξv/Kv(ξv)),

satisfying

(6.9) dim(W ) ≤ min(g, 2 dim(W ∩ F )), ω|W is nondegenerate or zero.

where ω is the symplectic form on H1dR(Xξv/Kv(ξv)).

By the lemmas that follow, H′O is contained in a proper Kv-subvariety of HO; weconclude as in the proof of Claim 2.

Lemma 6.2. Suppose Lw is a finite unramified extension ofKv of degree d ≥ 5. Let (V, ω)be a symplectic Lw-vector space, with dimLw V = 2g; let φ : V → V be semilinear forthe Frobenius of Lw/Kv and bijective.

Then there is a Zariski-open

A ⊆ ResLwKv LGr(V, ω)

(where LGr(V, ω) is the Lagrangian Grassmannian, and ResLwKv denotes Weil restrictionof scalars from Lw to Kv) with the following property:

if F ⊂ V is a Lagrangian Lw-subspace, corresponding to a point of A(Kv), there isno φ-invariant Lw-subspace W of V satisfying (6.9).

Proof. Passing from Kv to Kv , in a similar way to Lemma 2.1, we see that it suffices toprove the following statement:

Let (V, ω) be a symplectic vector space over an infinite field with dim(V ) =2g; write LGr(V, ω) for the Grassmannian of Lagrangian subspaces. LetE be the set of d-tuples of Lagrangian subspaces

(F1, . . . , Fd) ∈ LGr(V, ω)d

for which there exists a subspace W ⊂ V such that (6.9) holds (now forevery Fi). If d ≥ 5 then E is contained in a proper, Zariski-closed subsetof LGr(V, ω)d.

This is proved by a dimension count. It is sufficient to check that the moduli space of tuples(W,F1, F2, . . . , Fd) satisfying the properties described has dimension less than d · g(g+1)

2 .This moduli space is the union of two subsets, on whichW ∩W⊥ = 0 andW ∩W⊥ = W ,respectively. We analyze these two sets separately; write r = dim(W ).

(i) ω|W nondegenerate: the dimension of the space of possible W s has dimension

MW (r) = r(2g − r)︸︷︷︸dim Gr(2g,r)

.

Now consider Lagrangian F with dim(F ) = g. Put u = dim(W ∩ F ). Note thatu 6 r/2 because F ∩W is isotropic in the nondegenerate symplectic space W ;on the other hand, u ≥ r/2 by assumption. Therefore u = r/2, and the modulispace of such F has dimension at most

MF (r) =r

2

( r2

+ 1)

︸︷︷︸dimLGr(r,r/2)

+ (g − r

2)(g − r

2+ 1)︸︷︷︸

dimLGr(2g−r,g− r2)

=1

4r2 − 1

2gr +

g2 + g

2.

DRAFT


(ii) ωW = 0: Here the moduli space of such W has dimension

MW′(r) = r(2g − r)− 1

2r(r − 1),

the dimension of the isotropic Grassmannian of r-planes in 2g-dimensional sym-plectic space. Given W , the moduli space of F satisfying the required conditionshas dimension at most

MF′(r, u) = u(r − u)︸︷︷︸

dim Gr(r,u)

+ (g − u)(g − u+ 1)/2︸︷︷︸dim LGr(2g−2u,g−u)

.

Hence, the moduli space of tuples (W,F1, F2, . . . , Fd) has dimension the larger of

max0≤r≤g

MW (r) + dMF (r) and max0≤u≤r≤g

M ′W (r, u) + dM ′F (r, u)).

By a calculation we verify that this is less than d · g(g+1)2 so long as d > 4.

7. A MONODROMY CALCULATION

In this section we outline the calculation used to verify Proposition 5.7. We need toshow that the Kodaira-Parshin family of Jacobians X → Y ′ → Y with (q, r) = (2, 3)over a genus-2 curve has full monodromy. As mentioned above, this condition can bechecked topologically: throughout this section, X , Y ′ and Y will be topological spaces,the analytification of schemes over C.

A calculation involving Riemann-Roch shows that Z2,3 is a genus-9 curve. An order-2element of G induces a fixed-point-free endomorphism of Z2,3, giving a genus-5 curveZ ′2,3 of which Z2,3 is an étale double cover. The abelian variety X2,3, which is isogenousto the Prym of Z ′2,3 → Y , is an abelian scheme of relative dimension 3, and Y ′2,3 → Y isa finite map of degree d = 810.

Let R be a torsion-free ring. The Betti cohomology H1B(Xy, R) comes equipped with

an alternating form; we use Sp(H1B(Xy, R)) to denote the automorphisms of H1

B(Xy, R)preserving the form. If 3 is a unit in R then this is isomorphic to the usual symplecticgroup.

Choose a point y ∈ Y , and consider the monodromy action

π1(Y, y)→ Sp(H1B(Xy,Z)) ⊂ Sp(H1

B(Xy,Q)),

where Xy is the fiber of X over Y . We need to show that the Zariski closure of its imagecontains Sp6(Q)d. We perform a series of reductions to make this amenable to computa-tion.

Let y′1 and y′2 be any two distinct points of Y ′ above y ∈ Y , and let π′ ⊆ π1(Y, y)denote the subset fixing both y′1 and y′2 under the permutation action of π1(Y, y) on thefiber of Y ′ above y. If Xi denotes the fiber of X over y′i, then the monodromy actionabove restricts to a map

π′ →2∏i=1

Sp(HiB(Xy,Z)).

By Proposition 2.9 it is enough to prove that this map has Zariski dense image. Indeed,the quotient Sp6(Q)/±1 is simple, and applying Proposition 2.9 to the Zariski closureof the image of monodromy shows that if the result is true for d = 2, then the image ofmonodromy is a subgroup of Sp6(Q)d which surjects onto (Sp6(Q)/±1)d, as well asonto each individual factor Sp6(Q). But the only such subgroup is Sp6(Q)d itself.

DRAFT


The next proposition allows us to test surjectivity on a finite group.

Proposition 7.1. Suppose G is a subgroup of Sp6(Z)d that surjects onto Sp6(F5)2 underthe natural projection. Then G is Zariski dense in Sp6(Q)d.

Proof. We prove by induction that G surjects onto Hk = Sp6(Z/5kZ)2 for every k. As-suming the result for k, it is enough to show that the image of the mapG→ Hk+1 containsall matrices of the form I + 5kA. Suppose N is any 6-by-6 matrix with coefficients in F5

for which N2 = 0 and I +N is symplectic. Choose M ∈ G mapping to I +N under thenatural map; then M5k is congruent to I + 5kN modulo 5k+1. These matrices I + 5kNgenerate the kernel of Hk+1 → Hk, and the induction is complete.

Hence G is dense for the 5-adic topology on Sp6(Z5)2. The result follows.

Hence it is sufficient to test, for every pair (y1, y2), the surjectivity of the monodromymap

π′ →2∏i=1

Sp6(F5).

This is achieved by a calculation in SAGE, as follows: we produce four elements ofπ′ and let G be the subgroup of

∏2i=1 Sp6(F5) generated by their images under mon-

odromy. SAGE computes the order of G, and we verify that this is equal to the order of∏2i=1 Sp6(F5).

8. HIGHER DIMENSIONAL VARIETIES: A TRANSCENDENCE CONJECTURE OF PERIODMAPPINGS

(THIS SECTION IS INCOMPLETE, AND IS INDEPENDENT OF THE MAIN RE-SULTS OF THE PAPER. WE FORMULATE THE CONJECTURE ONLY; LATER ONWE’LL ADD A SAMPLE APPLICATION OF THE CONJECTURE.)

It is desirable to extend the method to settings where the base Y is higher-dimensional,thus feasibly leading to finiteness results for integral points on Y . As a typical example,we may wish to take X → Y to be the moduli space of smooth hypersurfaces in Pm; thenintegral points on Y correspond to integral homogeneous polynomials P (x0, . . . , xm+1)of degree d whose discriminant (disc P ) ∈ O∗.Conjecture 8.1. (Transcendence conjecture for period mappings): Suppose that we aregiven an algebraic family X → Y over the complex numbers; let H be the associatedperiod domain, so we have an analytic period map

Φ : Y −→ H

where Y is the universal cover of Y (C).Let H∗ be the Zariski-closure of Φ inside the ambient flag variety containing H; thus

H∗ is a flag variety for the algebraic monodromy group.Suppose that Z ⊂ H∗ is a complex-analytic subvariety, and

codim(Z) ≥ dim(image Φ).

Then the preimage of Z under Φ is contained inside the preimage, in Y , of the complexpoints of a proper subvariety of Y .

To put this conjecture in context, let us suppose that X → Y is defined over Q, andthat y ∈ M(Q). Grothendieck then conjectures that the transcendence degree of Φ(y) isas large as possible:

tr.deg. (Φ(y) ∈ H(C)) = dim h∗.

DRAFT


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[3] Olivier Brinon and Brian Conrad. Cmi summer school notes on p-adic Hodge theory.[4] Alexandre Grothendieck; Jean Dieudonné. éléments de géométrie algébrique: Iv. étude locale des schémas

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and number theory (Baltimore, MD, 1988), pages 25–80. Johns Hopkins Univ. Press, Baltimore, MD, 1989.[6] Nicholas M. Katz and Tadao Oda. On the differentiation of de Rham cohomology classes with respect to

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Sci, 1995.[8] A. N. Paršin. Algebraic curves over function fields. I. Izv. Akad. Nauk SSSR Ser. Mat., 32:1191–1219, 1968.[9] Siegfried Bosch; Werner Lütkebohmert; Michel Raynaud. Néron Models. Number Folge 3, 21 in Ergebnisse

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Wesley Publishing Company, Advanced Book Program, Redwood City, CA, second edition, 1989. With thecollaboration of Willem Kuyk and John Labute.

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