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$: Title Torsion of abelian schemes and rational points on ...RIMS K^okyur^ oku Bessatsu B12 (2009), 7{29 Torsion of abelian schemes and rational points on moduli spaces By Anna CADORET∗$
Title Torsion of abelian schemes and rational points on modulispaces (Algebraic Number Theory and Related Topics 2007)

Author(s) CADORET, Anna; TAMAGAWA, Akio

Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2009), B12: 7-29

Issue Date 2009-08

URL http://hdl.handle.net/2433/176799

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

RIMS Kokyuroku Bessatsu

B12 (2009), 7–29

Torsion of abelian schemes andrational points on moduli spaces

By

Anna CADORET∗ and Akio TAMAGAWA∗∗

Abstract

In [CT], we proved, in characteristic 0, certain 1-dimensional base versions of the

uniform boundedness conjecture for p-primary torsion of abelian varieties and of Fried’s

modular tower conjecture related to the regular inverse Galois problem. In this paper,

we prove these results in arbitrary characteristics.

Contents

§0. Notation§1. Introduction§2. Main results§3. Preliminaries§4. Proof of Theorem A

§0. Notation.

Let S be a scheme and p a prime number. Given a commutative group scheme Aover S, we use the following notations:

　 Received June 2, 2008. Revised December 11, 2008.2000 Mathematics Subject Classification(s): Primary 14H25; Secondary 14H30, 14K15.

　*Institut de Mathematiques de Bordeaux, Universite Bordeaux 1, 351 Cours de la Liberation,

　 F-33405 Talence Cedex, France

　 e-mail: [email protected]

　**Research Institute for Mathematical Sciences, Kyoto University, KYOTO 606-8502, Japan.

　 e-mail: [email protected]

　　 c⃝ 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

8 ANNA CADORET AND AKIO TAMAGAWA

A[N ] def= Ker([N ] : A→ A),

where [N ] stands for the multiplication-by-N endomorphism (N ≥ 1),

Atorsdef= lim−→

N

A[N ] (i.e., Ators(T ) = lim−→N

A[N ](T ) for each S-scheme T ),

A[p∞] def= lim−→nA[pn] (i.e., A[p∞](T ) = lim−→

nA[pn](T ) for each S-scheme T ),

A[pn]∗ def= A[pn] rA[pn−1] (A[p0]∗ def= A[p0]).

If, moreover, S = Spec(k) for some field k of characteristic = p, we also use the followingnotations:

Tp(A) def= lim←−nA[pn](k) (the p-adic Tate module of A),

Tp(A)∗ def= Tp(A) r pTp(A) = lim←−nA[pn]∗(k),

Vp(A) def= Tp(A)⊗Zp Qp,

and

Zp(1) def= Tp(Gm),

where Gmdef= P1 r {0,∞} is the multiplicative group scheme.

§1. Introduction.

In this §, we shall explain two important problems in arithmetic geometry that havemotivated our study — the uniform boundedness conjecture for torsion of abelian vari-eties and the regular inverse Galois problem (especially, the modular tower conjecture).

⟨Uniform boundedness conjecture for torsion of abelian varieties ⟩

Let k be an algebraic number field (i.e., d def= [k : Q] <∞) and g an integer ≥ 0.

Uniform Boundedness Conjecture (UB).There exists a constant N = N(k, g) ≥ 0, such that for any g-dimensional abelianvariety A over k and any v ∈ Ators(k), the order of v is ≤ N .

(UB) is trivially valid for g = 0 and has been proved for g = 1 (by Mazur, Kamienny,Abramovich for smaller values of d and by Merel [Me] for d general) via intensive studyof geometry of modular curves. However, it is widely open for g > 1.

We also have the following variant. Let p be a (fixed) prime number.

TORSION OF ABELIAN SCHEMES AND RATIONAL POINTS ON MODULI SPACES 9

p-Uniform Boundedness Conjecture (pUB).There exists a constant N = N(k, g, p), such that for any g-dimensional abelian varietyA over k and any v ∈ A[p∞](k), the order of v is ≤ N .

(pUB) is trivially valid for g = 0 and has been proved for g = 1 by Manin [Ma2].Although (pUB) is clearly weaker than (UB), it is still widely open for g > 1.

⟨Inverse Galois problem ⟩Let G be a finite group and k an algebraic number field.

Inverse Galois Problem (IGP).Does there exist a Galois extension L/k, such that Gal(L/k) ≃ G?

The following variant of (IGP) is more closely related to our study in this paper.

Regular Inverse Galois Problem (RIGP).Does there exist a Galois extension L/k(T ) (T : indeterminate), such that L is regularover k (i.e. L ∩ k = k) and that Gal(L/k(T )) ≃ G?

It is well-known that (RIGP) implies (IGP) (by Hilbert’s irreducibility theorem).

While (IGP) is purely number-theoretic, (RIGP) is arithmetico-geometric in nature,as follows. First, we have the following one-to-one correspondences:

a Galois extension L/k(T ) with Gal(L/k(T )) ≃ G

1:1←→ a (branched) connected Galois cover Y → P1k with Aut(Y/P1

k) ≃ G

1:1←→ a surjection π1(P1k r S) � G, considered modulo Inn(G)

(where S runs over the finite sets of closed points of P1k).

Here, π1 stands for the etale fundamental group of scheme (with a suitable base point).Moreover, in these one-to-one correspondences, the condition for the first object

L/k(T ) that L is regular over k corresponds to the condition for the second objectf : Y → P1

k that Yk is connected (or, equivalently, f is “geometric”), and to thecondition for the third object π1(P1

k rS) � G that the induced map π1(P1k

rSk)→ G

is already surjective. Note that π1(P1k r S) fits into the following exact sequence of

profinite groups:

1→ π1(P1k

r Sk)→ π1(P1k r S)→ Gal(k/k)→ 1,

where π1(P1k rS) and π1(P1

krSk) are sometimes referred to as the arithmetic and the

geometric fundamental groups, respectively.It is well-known that (as k is of characteristic 0) the geometric fundamental group

π1(P1k

r Sk) is a free profinite group of rank r − 1, where r is the cardinality of thepoint set Sk. Thus, it is easy to construct a surjection π1(P1

kr Sk) � G. (Indeed, this

is possible if and only if G is generated by at most (r − 1) elements.) However, it is asubtle descent problem to extend a surjection π1(P1

kr Sk) � G to π1(P1

k r S).The above geometric interpretation of (RIGP) further leads to a modular interpreta-

tion of (RIGP), as follows. For this, let us introduce moduli spaces of covers of curves,i.e., Hurwitz spaces. (For details, see, e.g., [BR].)


Definition. Let G be a finite group and r an integer ≥ 0.(i) We denote by HG,P1,r the moduli stack (over Z[1/|G|]) of pairs (f, t), where f :Y → P1 is a geometric Galois cover equipped with Aut(Y/P1) ≃ G, and t ⊂ P1 is anetale divisor of degree r that contains the branch locus of f . It is known that HG,P1,r

is a Deligne-Mumford stack and that the associated coarse space HG,P1,r is a scheme.(ii) Let g be an integer ≥ 0 with the hyperbolicity condition 2− 2g − r < 0. Then wedenote by HG,g,r the moduli stack (over Z[1/|G|]) of pairs (f, t), where f : Y → X isa geometric Galois cover over a proper, smooth, geometrically connected curve X ofgenus g equipped with Aut(Y/X) ≃ G, and t ⊂ X is an etale divisor of degree r thatcontains the branch locus of f . It is known that HG,g,r is a Deligne-Mumford stack andthat the associated coarse space HG,g,r is a scheme.

By definition, each geometric Galois cover f : Y → P1k over k equipped with

Aut(Y/P1k) ≃ G, whose branch locus in P1

k is contained in an etale divisor t of de-gree r, defines a k-rational point in HG,P1,r(k) depending on the pair (f, t). (When thebranch locus coincides with t, this k-rational point depends only on f , since t is thendetermined by f .) If, moreover, HG,P1,r is representable (i.e., HG,P1,r = HG,P1,r), thenthis defines a one-to-one correspondence between the set of isomorphism classes of such(f, t) and HG,P1,r(k). Similarly, for a proper, smooth, geometrically connected curveX of genus g over k, each geometric Galois cover f : Y → X over k equipped withAut(Y/X) ≃ G, whose branch locus in X is contained in an etale divisor t of degreer, defines a k-rational point in HG,g,r(k) depending on the pair (X, f, t). (When thebranch locus coincides with t, this k-rational point depends only on (X, f), since t isthen determined by (X, f).) If, moreover, HG,g,r is representable (i.e., HG,g,r = HG,g,r),then this defines a one-to-one correspondence between the set of isomorphism classesof such (X, f, t) and HG,g,r(k).

Facts. Here, dim stands for the relative dimension over the base.(i) dim(HG,P1,r) = r, unless HG,P1,r = ∅. HG,P1,r is representable if and only if eitherHG,P1,r = ∅ or the center of G is trivial.(ii) dim(HG,g,r) = 3g − 3 + r, unless HG,g,r = ∅. HG,g,r is representable if and only if,for any object f : Y → X classified by HG,g,r, the centralizer of G in Aut(Y ) is trivial.(iii) HG,P1,r/PGL2 = HG,0,r (for r ≥ 3).

⟨Fried’s modular tower conjecture ⟩

The modular tower conjecture is a conjecture arising from (RIGP) that was posedby M. Fried in the early 1990s. Here, we formulate some variants of this conjecture.For more details, see [F], [FK] and [D].

Let p be a prime number. Let G = {Gn+1 � Gn}n≥0 be a projective system of

finite groups, such that G def= lim←−Gn is p-obstructed in the following sense:

Definition. We say that a profinite group G is p-obstructed, if G contains an opensubgroup that admits a quotient isomorphic to Zp.

Modular Tower Conjecture (MT).Let k be an algebraic number field and r an integer ≥ 0.


(i) There exists a constant N = N1(p,G, r, k), such that, for any n ≥ N and for anygeometric Galois cover f : Y → P1

k with group Gn, the degree of the (reduced) branchdivisor of f in P1

k is > r.(ii) There exists a constant N = N2(p,G, r, k), such that HGn,P1,r(k) = ∅ for anyn ≥ N .(iii) Let g be an integer ≥ 0 with 2 − 2g − r < 0. Then there exists a constant N =N3(p,G, g, r, k), such that HGn,g,r(k) = ∅ for any n ≥ N .

Here, the projective systems {HGn,P1,r}n≥0 and {HGn,g,r}n≥0 of Hurwitz spacesare often referred to as “modular towers”. Note that the following implications areimmediate:

(MT)(iii) for g = 0 =⇒ (MT)(ii) =⇒ (MT)(i).

Finally, the following (weaker) variant of (MT)(iii) has been already proved in ar-bitrary characteristics. Here, for a profinite group G, denote by ΣG the set of primenumbers which divide the order of (some finite quotient of) G.

Theorem 1.1 ([C], Corollary 3.6. See also [BF], [K]). Let k be a field finitelygenerated over the prime field of characteristic q ∈ ΣG (hence, in particular, q = p),and g, r integers ≥ 0 with 2− 2g − r < 0. Then we have

lim←−HGn,g,r(k) = ∅.

§2. Main results.

The main results of this paper are solutions of certain 1-dimensional versions of (pUB)and (MT) over fields finitely generated over the prime field of arbitrary characteristic= p. To state the former in some more generality, we shall introduce the notion ofnon-Tate characters as in [CT].

Definition. Let p be a prime number and k a field of characteristic q = p. (Denote byΓk the absolute Galois group Gal(ksep/k) of k.) We say that a character χ : Γk → Z∗

p

is non-Tate, if it does not appear as a subrepresentation of the p-adic representationassociated with an abelian variety over k. Equivalently, χ is non-Tate if and only if, forany abelian variety A over k,

A[p∞](χ) def= {T ∈ A[p∞](k) | σT = χ(σ)T for all σ ∈ Γk}

is finite.

When k is finitely generated over the prime field, the trivial character and the p-adiccyclotomic character are typical examples of non-Tate characters. (See [CT].)

Let d be an integer ≥ 0. Now, we can formulate the d-dimensional version of (pUB),as follows.


Conjecture (pUBd). Let p be a prime number. Let k be a field finitely generated overthe prime field of characteristic q = p and χ : Γk → Z∗

p a non-Tate character. Let S bea scheme of finite type over k with dim(S) ≤ d and A an abelian scheme over S. Thenthere exists an integer N = N(p, k, χ, S,A), such that As[p∞](χ) ⊂ As[pN ](k) for anys ∈ S(k).

(pUB0) follows immediately from the definition of non-Tate characters. Now, thefirst main result of this paper is:

Theorem A. (pUB1) holds.

Next, we formulate the d-dimensional version of (MT) (in arbitrary characteristics).

Conjecture (MTd). Let p be a prime number and G = {Gn+1 � Gn}n≥0 a projective

system of finite groups, such that G def= lim←−Gn is p-obstructed. Let g, r be integers≥ 0 with 2 − 2g − r < 0. Let k be a field finitely generated over the prime field ofcharacteristic q ∈ ΣG (hence, in particular, q = p), S a scheme of finite type over kwith dim(S) ≤ d, and ξ : S → HG0,g,r a k-morphism. Then there exists an integer

N = N(p,G, g, r, k, S, ξ), such that Sn(k) = ∅ for any n ≥ N . Here, we set Sndef=

S ×HG0,g,rHGn,g,r.

Observe that (MTd) implies (MT)(iii) for (g, r) with 3g − 3 + r ≤ d. (MT0) (hence,(MT)(iii) for (g, r) = (0, 3) and (MT)(i)(ii) for r = 3) follows immediately from Theorem1.1. Now, the second main result of this paper is:

Theorem B. (MT1) holds. (In particular, (MT)(iii) for (g, r) = (0, 4), (1, 1) and(MT)(i)(ii) for r = 4 hold.)

Theorem A in characteristic 0 and the deduction Theorem A =⇒ Theorem B inarbitrary characteristics are main results of [CT]. In this paper, we shall give a proof ofTheorem A in arbitrary characteristics, eventually in §4. Before that, we shall collectsome preliminaries in the next §.

§3. Preliminaries.

In this §, we collect various arithmetico-geometric preliminaries for the proof ofTheorem A in the next §. They were not needed in [CT], but are needed here to coverarbitrary characteristics. Some of them may be of some interest independent of theproof of Theorem A.

⟨Non-Tate characters ⟩Fix a prime p and let k be a field of characteristic q = p. The following was proved

in [CT].

Lemma 3.1. For any finitely generated extension K of k, χ : Γk → Z∗p is non-Tate

if and only if χ|ΓK: ΓK → Z∗

p is non-Tate. Here, we set χ|ΓK

def= χ ◦ |ksep , where|ksep : ΓK → Γk stands for the restriction from Ksep to ksep (with respect to a fixedembedding ksep ↪→ Ksep over k).

Here, we shall prove two more lemmas on non-Tate characters.


Lemma 3.2. The following are all equivalent:(i) The p-adic cyclotomic character χcyc : Γk → Z∗

p has open image and is non-Tate.(ii) For any finitely generated extension K of k and any character χ : ΓK → Z∗

p, χ(ΓKk)is finite.(ii′) For any finitely generated extension K of k and any (additive) character ψ : ΓK →Zp, ψ(ΓKk) is trivial.

Proof. First, as Z∗p ≃ Zp ×M with M finite, the equivalence (ii)⇐⇒ (ii′) is clear.

Next, assume that (i) holds. We shall prove the assertion of (ii′) by induction on thetranscendence degree d of K over k. If d = 0, the assertion is trivial. If d > 0, take asubextension k′/k of K/k with transcendence degree d−1. Replacing k′ by its algebraicclosure in K, we may assume that k′ is algebraically closed in K, hence the naturalmap ΓK → Γk′ is surjective. Now, observe that the conditions of (i) are also satisfiedwhen k is replaced by k′. Indeed, as the natural map Γk′ → Γk has open image, thefirst condition is satisfied, and, by Lemma 3.1, the second condition is satisfied. Thus, ifwe assume the implication (i) =⇒ (ii′) for d = 1 (for k′), ψ : ΓK → Zp factors throughΓK � Γk′ , or, equivalently, induces a character ψ : Γk′ → Zp. Applying the assumptionof induction to the finitely generated extension k′/k and the character ψ : Γk′ → Zp,we are done.

Thus, up to replacing K/k by K/k′, it suffices to settle the case where d = 1 and kis algebraically closed in K. Let kperf denote the perfect closure of k. Then, Kkperf isregarded as the function field of a proper, smooth, geometrically connected curve Ckperf

over kperf. Further, Ckperf → Spec(kperf) descends to a proper, smooth, geometricallyconnected curve Ck over some finite subextension k of k in kperf, so that Kk is regardedas the function field of Ck. Now, by Lemma 3.1, we may replace K/k by Kk/k andassume that K is the function field of a proper, smooth, geometrically connected curveC over k.

For each closed point x of C, denote the residue field at x by k(x) and let ΓK ⊃Dx ⊃ Ix be the decomposition and the inertia subgroups (defined up to conjugacy). Itis well-known that the maximal pro-p quotient Ip

x of Ix is isomorphic to Zp (as p = q)and that the natural action of Γk(x) on Ip

x ≃ Zp (induced by the conjugate action of Dx

on Ix) is via χcyc|Γk(x) . Since the p-adic cyclotomic character χcyc : Γk → Z∗p has open

image by assumption, so is χcyc|Γk(x) , hence (Ipx)Γk(x) is finite. (Here, for a (topological)

group G and a (topological) G-module M , MG denotes the coinvariant module of M ,i.e., the maximal quotient of M on which G acts trivially.) Now, first, since Zp is pro-p,ψ|Ix : Ix → Zp factors through Ip

x . Next, since Zp is abelian, ψ|Dx factors through Dabx ,

hence ψ|Ix factors through (Ipx)Γk(x) . Finally, since Zp is torsion-free, ψ|Ix is trivial.

In summary, the image under ψ of the inertia subgroup at any closed point of C istrivial. Thus, ψ : ΓK → Zp (resp. ψ|Γ

Kk: ΓKk → Zp ) factors through ΓK � π1(C)

(resp. ΓKk � π1(C ×k k)). In particular, ψ induces a Γk-equivariant homomorphismψJ : Tp(J)→ Zp, where J is the Jacobian variety of C. Suppose that ψJ is nontrivial.Then, by duality, we get a nontrivial Γk-equivariant homomorphism Zp(1) → Tp(J).This is absurd, since χcyc is assumed to be non-Tate. Therefore, ψJ is trivial, or,equivalently, ψ(ΓKk) is trivial.

Finally, assume that (i) does not hold, or, equivalently, that either χcyc(Γk) ⊂ Z∗p is


not open or χcyc is not non-Tate. In the first case, χcyc(Γk) is finite, hence there existsa finite extension k′ of k such that χcyc(Γk′) is trivial. This means that k′ contains all

p-power roots of unity of k. Now, set K def= k′(t), where t stands for an indeterminate.By Kummer theory, K(t1/p∞

) def= ∪n≥0K(t1/pn

) defines a Zp-extension of K, and thecorresponding additive character

ψ : ΓK � Gal(K(t1/p∞)/K) ≃ Zp

satisfies ψ(ΓKk) = Zp, which gives a counterexample for (ii′). In the second case, thereexists an abelian variety A over k that admits an injective, Γk-equivariant homomor-phism Zp(1) → Tp(A). By duality, we get a nontrivial, Γk-equivariant homomorphismTp(A∨) → Zp, where A∨ stands for the dual abelian variety of A. The latter Γk-equivariant homomorphism yields a homomorphism π1(A∨)→ Zp such that the image

of π1(A∨k) is nontrivial. So, K def= k(A∨) gives a counterexample for (ii′). This completes

the proof. �We say that k is p-arithmetic, if one (hence all) of the conditions in Lemma 3.2 is

satisfied. Note that, if k is finitely generated over the prime field (of characteristicq = p), then k is p-arithmetic. Moreover, if k is finitely generated over a p-arithmeticfield, then k is p-arithmetic.

Lemma 3.3. Assume that k is p-arithmetic. Let T be a normal, integral scheme offinite type over k and χT : π1(T ) → Z∗

p a character. For each (not necessarily closed)point t ∈ T , denote by χt : Γk(t) → Z∗

p the character obtained by taking the composite ofχT and the map Γk(t) → π1(T ) associated with the natural morphism Spec(k(t)) → Twith image t. Then, χt0 is non-Tate for some t0 ∈ T if and only if χt is non-Tate forall t ∈ T .

Proof. SetK def= k(T ), and denote by k the algebraic closure of k inK. Then, we see thatk is again p-arithmetic (as k is a finite extension of k), and that the structure morphismT → Spec(k) factors as the composite of a morphism T → Spec(k) and the naturalmorphism Spec(k)→ Spec(k) (as T is normal). Moreover, T is geometrically connectedover k by definition. So, replacing k by k, we may assume that T is geometricallyconnected over k.

Then the natural map ΓKk → π1(T ×k k) is surjective. Indeed, by the functorialityproperty of π1, we have the following commutative diagram:

ΓKk → π1(T ×k k)

↓ ↓

ΓKksep → π1(T ×k ksep),

in which both vertical arrows are isomorphisms (cf. [GR], Expose IX, Theoreme 6.1).As T is normal and ksep/k is separated, T×kk

sep is normal. Thus, the bottom horizontalarrow is surjective ([GR], Expose V, Proposition 8.2), hence so is the top horizontalarrow, as desired.


Now, since k is p-arithmetic and K is finitely generated over k, we see thatχT (π1(T ×k k)) is finite. Thus, there exists a connected finite etale cover T ′ → T suchthat χT (π1(T ′ ×k′ k)) is trivial (where k′ denotes the algebraic closure of k in k(T ′)),or, equivalently, that χT ′

def= χT |π1(T ′) factors through π1(T ′) � Γk′ . Let χk′ denotethe character of Γk′ induced by χT ′ .

Suppose that χt0 is non-Tate for some t0 ∈ T , and take a point t′0 ∈ T ′ above t0.Then k′(t′0) is a finite extension of k(t0), hence χt′0

is non-Tate. Here, χt′0is defined to

be the composite of the natural map Γk′(t′0)→ π1(T ′) and χT ′ : π1(T ′) → Z∗

p, hencecoincides with the composite of the natural map Γk′(t′0)

→ Γk′ and χk′ : Γk′ → Z∗p.

Now, since k′(t′0) is finitely generated over k′ and χt′0is non-Tate, we conclude that χk′

is non-Tate by Lemma 3.1.Finally, let t ∈ T be any point and take a point t′ ∈ T ′ above t. Since χk′ is non-Tate,

χt′ is non-Tate, hence χt is non-Tate. This completes the proof. �⟨A variant of the Serre-Tate criterion ⟩

Let p be a prime. Let R be a discrete valuation ring and K the field of fractions of R.Assume that the characteristic of the residue field of R is not p. Thus, in particular, thecharacteristic of K is not p. Let I be the inertia subgroup (determined up to conjugacy)for R in ΓK .

Let A be an abelian variety over K. Then ΓK acts naturally on the p-adic Tatemodule Tp(A) of A. The Serre-Tate criterion for good reduction of abelian varieties([ST]) tells that A has good reduction over R if and only if I acts trivially on Tp(A).

Now, we shall prove the following variant of this criterion under the assumption thatK is finitely generated over the prime field. We need this extra assumption to resortto the semisimplicity and the Tate conjecture, which were proved by Tate, Zarhin andMori in positive characteristic (cf. [MB], Chapitre XII) and by Faltings in characteristic0 (cf. [FW], Chapter VI).

Proposition 3.4. Assume moreover that A is nontrivial and K-simple and that K isfinitely generated over the prime field. Then A has good reduction over R if and onlyif there exists a nontrivial ΓK-submodule T of Tp(A) on which I acts trivially.

Proof. The ‘only if’ part immediately follows from the ‘only if’ part of the original Serre-Tate criterion. (Take, say, T = Tp(A).) To see the ‘if’ part, we resort to the semisim-plicity and the Tate conjecture for the ΓK-module Vp(A). As Vp(A) is a semisimpleΓK-module, there exist a finite number of simple ΓK-submodules W1, . . . ,Wr of Vp(A)which are mutually non-isomorphic as ΓK-modules and positive integers n1, . . . , nr,such that

Vp(A) = V1 ⊕ · · · ⊕ Vr, Vi ≃W⊕nii

as ΓK-modules. Then we have

EndΓK (Vp(A)) ≃Mn1(D1)× · · · ×Mnr (Dr),

where Didef= EndΓK

(Wi) is a division algebra, as Wi is a simple ΓK-module. On theother hand, as a consequence of the Tate conjecture, we have

EndΓK (Vp(A)) ≃ D ×Q Qp,


where D def= EndK(A) ⊗Z Q is a division algebra, as A is a K-simple abelian variety.Let Fi (resp. F ) denote the center of Di (resp. D), which is a finite extension field ofQp (resp. Q). As

D ×Q Qp ≃Mn1(D1)× · · · ×Mnr (Dr),

we haveF ×Q Qp ≃ F1 × · · · × Fr.

Thus, in particular, giving an F×QQp-module M with dimQp(M) <∞ is equivalent

to giving an Fi-vector space Mi with didef= dimFi(Mi) < ∞ for each i ∈ {1, . . . , r}:

M ≃ M1 ⊕ · · · ⊕ Mr. Here, we shall refer to Mi as the Fi-component of M . (Forexample, Vi is the Fi-component of Vp(A).) Moreover, observe that M is free as anF ×Q Qp-module (i.e., M1 ⊕ · · · ⊕Mr is free as an (F1 × · · · × Fr)-module) if and onlyif d1 = · · · = dr.

We note that both Vp(A) and Vp(A)I are known to be free F ×Q Qp-modules (see,e.g., [T1], Lemmas (2.1) and (2.2)). Thus, the quotient Vp(A)/Vp(A)I is also a freeF ×Q Qp-module.

Now, suppose that I acts trivially on a nontrivial ΓK-submodule T of Tp(A), and setW = T ⊗Zp Qp ⊂ Vp(A). Take any simple ΓK-submodule of W , then it is necessarilyisomorphic to Wi0 for some i0 ∈ {1, . . . , r}. Thus, I acts trivially on Wi0 , hence onVi0 ≃ W

⊕ni0i0

. Namely, we have Vi0 ⊂ Vp(A)I , or, equivalently, the dimension of theFi0-component of Vp(A)/Vp(A)I is 0. Since Vp(A)/Vp(A)I is free as an F ×Q Qp-module, this implies that the dimension of the Fi-component of Vp(A)/Vp(A)I is 0 forall i ∈ {1, . . . , r}. Namely, Vp(A)/Vp(A)I is trivial, or, equivalently, I acts trivially onVp(A). Now, by the original Serre-Tate criterion, A has good reduction, as desired. �

⟨A consequence of Mordell’s conjecture over function fields ⟩

We mean by a curve a separated, normal, geometrically integral, 1-dimensionalscheme over a field, and by a proper curve a curve which is proper over the base field.(Observe that a curve in this sense is generically smooth over the base field in general,and smooth if the base field is perfect.) Mordell’s conjecture over function fields, provedby Manin and Grauert in characteristic 0 and by Samuel in positive characteristic, issummarized as follows. Here, for an extension k/F of fields and a k-scheme S, wesay that S is F -trivial, if there exists an F -scheme SF such that S is k-isomorphic toSF ×F k, and say that S is F -isotrivial, if the k-scheme S ×k k is F -trivial.

Theorem 3.5. Let F be an algebraically closed field of characteristic q ≥ 0, F theprime field of F , and k a field finitely generated over F . Let C be a proper curve overk, and assume that the normalization of C ×k k is of genus ≥ 2. Then at least one ofthe following holds:(i) C(k) is finite;(ii) there exists a curve CF over F , such that C is k-isomorphic to CF ×F k (i.e., C isF -trivial) and that, under the identification C = CF ×F k, C(k) r CF (F ) is finite;(iii) q > 0 and there exist a finite extension k′ of k, a finite subfield F′ of F , and acurve CF′ over F′, such that C ×k k

′ is k′-isomorphic to CF′ ×F′ k′. Moreover, given


such (k′,F′, CF′) and an identification C ×k k′ = CF′ ×F′ k′, there exists a finite subset

Ξ ⊂ C(k′), such that

C(k) ⊂ {ϕnF′(x) | x ∈ Ξ, n ≥ 0} ∪ CF′(F ) ⊂ CF′(k′) = C(k′),

where ϕF′ : CF′ → CF′ denotes the |F′|-th power Frobenius endomorphism.

Proof. For q = 0, see [Ma1], [Gra]. (See also [Sa], Theoreme in the introduction.) Forq > 0, this follows from [Sa]. More specifically, we may assume that C(k) is infinite.Then, by Theoreme 6, loc. cit., C is smooth over k, and, by Theoreme 5, loc. cit., wesee that either (ii) or (iii) holds. (More precisely, for case (iii), Theoreme 5, b), loc. cit.,ensures that the assertion holds if (k′,F′, CF′) is replaced by (k′′,F′′, CF′ ×F′ F′′),where k′′/k is a certain (Galois) subextension of k′/k and F′′/F′ is a (sufficiently large)finite subextension of F/F′. Now, observe that the assertion for (k′′,F′′, CF′ ×F′ F′′) isstronger than that for (k′,F′, CF′).) �

In the case of fields finitely generated over prime fields, we have the following strongerresult (due to Faltings for q = 0):

Theorem 3.6. Let F be the prime field of characteristic q ≥ 0, and k a field finitelygenerated over F. Let C be a proper curve over k, and assume that the normalizationof C ×k k is of genus ≥ 2. Then at least one of the following holds:(i) C(k) is finite;(ii) q > 0 and there exist a finite extension k′ of k, a finite subfield F′ of k′, and acurve CF′ over F′, such that C ×k k

′ is k′-isomorphic to CF′ ×F′ k′. Moreover, givensuch (k′,F′, CF′) and an identification C ×k k

′ = CF′ ×F′ k′, there exists a finite subsetΞ ⊂ C(k′), such that

C(k) ⊂ {ϕnF′(x) | x ∈ Ξ, n ≥ 0} ⊂ CF′(k′) = C(k′),

where ϕF′ : CF′ → CF′ denotes the |F′|-th power Frobenius endomorphism.

Proof. For q = 0, this is a theorem of Faltings ([FW], Chapter VI, Theorem 3). For q >0, suppose that C(k) is infinite. Then C(kF) is infinite, a fortiori. By applying Theorem3.5 to the curve C ×k kF over the field kF finitely generated over the algebraicallyclosed field F, we conclude that we are in the situation of either (ii) or (iii) of Theorem3.5. In fact, case (ii) cannot occur. Indeed, if case (ii) occurs, then there exists acurve CF over F, such that C ×k kF is kF-isomorphic to CF ×F kF and that (underthe identification C ×k kF = CF ×F kF) C(kF) r CF(F) is finite. Further, the kF-isomorphism C ×k kF ≃ CF ×F kF descends to kF′ for some finite extension F′ of F.More precisely, there exists a finite extension F′ of F and a curve CF′ over F′ such thatCF is F-isomorphic to CF′ ×F′ F and that C ×k kF′ is kF′-isomorphic to CF′ ×F′ kF′.Now, (under a suitable identification) we have

C(k) ∩ CF(F) ⊂ C(kF′) ∩ CF′(F) = CF′(kF′ ∩ F) = CF′(F′1),

where F′1 is the algebraic closure of F in kF′, which is a finite extension of F′. Thus,

C(k) = (C(k) r CF(F)) ∪ (C(k) ∩ CF(F)) ⊂ (C(kF) r CF(F)) ∪ CF′(F′1)


is finite, which contradicts the assumption.Finally, assume that case (iii) of Theorem 3.5 occurs. Then, first, there exist a finite

extension k′1 of kF, a finite subfield F′ of F, and a curve CF′ over F′, such that C×k k′1

is k′1-isomorphic to CF′×F′ k′1. This isomorphism descends to one over a finite extensionk′ of kF′ included in k′1, as desired. Second, suppose that we are given k′, F′, CF′ andC ×k k

′ = CF′ ×F′ k′ as in the second assertion of (ii). Then Theorem 3.5 ensures thatthere exists a finite subset Ξ1 ⊂ C(k′F), such that

C(kF) ⊂ {ϕnF′(x) | x ∈ Ξ1, n ≥ 0} ∪ CF′(F) ⊂ CF′(k′F) = C(k′F).

Let F′′ denote the algebraic closure of F′ in k′. Considering the action of the Galoisgroup Gal(k′F/k′) ≃ Gal(F/F′′) and taking the Galois-invariant parts, we obtain

C(k) ⊂ C(kF′′) ⊂ {ϕnF′(x) | x ∈ Ξ0, n ≥ 0} ∪ CF′(F′′) ⊂ CF′(k′) = C(k′),

where Ξ0def= Ξ1 ∩ C(k′). (Observe that x ∈ Ξ1 is Gal(k′F/k′)-invariant if and only if

so is ϕnF′(x).) Thus, the finite subset Ξ def= Ξ0 ∪ CF′(F′′) of CF′(k′) = C(k′) has the

desired properties. �The following consequence of Theorem 3.6 will be used in next §.

Proposition 3.7. Let F be the prime field of characteristic q ≥ 0, and k a field finitelygenerated over F. Let C be a proper curve over k, and assume that the normalizationof C×k k is of genus ≥ 2. Let S be a nonempty open subscheme of C (which is a curveover k). When S(k) is infinite, put the extra assumption that S is F-isotrivial. (Notethat C is automatically F-isotrivial by Theorem 3.6.) Then there exists an F-morphismf : S → T between separated, normal, integral schemes of finite type over F, such thatthe following hold: (a) the function field F(T ) of T is F-isomorphic to k; (b) under theidentification F(T ) = k, S is k-isomorphic to the generic fiber Sk of f ; and (c) underthe identification S = Sk, we have S(k) = S(T ), i.e., each element of S(k) = Sk(k)uniquely extends to an element of S(T ).

Proof. Since k is a finitely generated extension of F, there exists a separated (or evenaffine), normal, integral scheme T of finite type over F, such that k = F(T ). On theother hand, fix a finite k-morphism C → P1

k, and define C to be the normalization of P1T

in (the function field of) C. Then, as C is normal and finite over P1k, we have Ck = C.

Set D def= C r S, and define D to be the topological closure of D in C. Set S def= C rD,which is separated, normal, integral and of finite type over F. Then it is easy to seethat the F-morphism S → T thus constructed satisfies (a) and (b). Further, if weconsider the base change from T to a nonempty open subscheme U , all the conditionsare preserved. Thus, it suffices to find U such that fU : SU

def= S ×T U → U satisfies(c).

Assume first that S(k) is a finite set {x1, . . . , xr}. Each xi : Spec(k) → S = Sk

defines a rational map T 99K S, or, more precisely, there exists a nonempty opensubscheme Ui of T , such that xi (uniquely) extends to a morphism Ui → S over T .Now, it is easy to see that U def= U1 ∩ · · · ∩ Ur has the desired property.


Next, assume that S(k) is infinite. Then C(k) is infinite, a fortiori, hence, by The-orem 3.6, q > 0 and there exist a finite extension k′ of k, a finite subfield F′ of k′,a curve CF′ over F′, such that C ×k k

′ is k′-isomorphic to CF′ ×F′ k′. On the otherhand, by our extra assumption, S is also F-isotrivial, hence, by replacing k′ and F′ bysuitable finite extensions, we may also assume that there exists a curve SF′ over F′,such that S×k k

′ is k′-isomorphic to SF′ ×F′ k′. Further, define C1,F′ to be the smoothcompactification of SF′ (note that F′ is perfect). Since both CF′ ×F′ k = C ×k k andC1,F′ ×F′ k ⊃ SF′ ×F′ k = S ×k k are smooth compactifications of S ×k k, they arecanonically k-isomorphic to each other. This k-isomorphism descends uniquely to anF-isomorphism between CF′×F′ F and C1,F′×F′ F. Thus, up to replacing F′ by a finiteextension, we may assume that this F-isomorphism descends to an F′-isomorphism be-tween CF′ and C1,F′ . In particular, we obtain an open immersion SF′ ↪→ CF′ over F′

which is compatible (over k′) with the original open immersion S ↪→ C over k.Let T ′ be the normalization of T in k′ and denote by π the natural finite morphism

T ′ → T . Let S ′ be the normalization of S in the function field of S×k k′. On the other

hand, set S ′1def= SF′ ×F′ T ′. The generic fibers of the morphisms of finite type S ′ → T ′

and S ′1 → T ′ are S ×k k′ and SF′ ×F′ k′, respectively, which are identified with each

other. Accordingly, there exists a nonempty open subscheme V ′ of T ′ over which thesetwo families coincide with each other.

Now, by Theorem 3.6, there exists a finite subset Ξ ⊂ C(k′), such that, under theidentification C ×k k

′ = CF′ ×F′ k′, we have

C(k) ⊂ {ϕnF′(x) | x ∈ Ξ, n ≥ 0} ⊂ CF′(k′) = C(k′).

From this, we obtain

S(k) ⊂ {ϕnF′(x) | x ∈ ΞS , n ≥ 0} ⊂ SF′(k′) = S(k′),

where ΞSdef= Ξ ∩ S(k′). (Observe (ϕF′)−1(SF′) = SF′ .)

Write ΞS = {x1, . . . , xr}. Each xi : Spec(k′) → S ×k k′ = S ′V ′ ×V ′ k′ (where

S ′V ′def= S ′ ×T ′ V ′) defines a rational map V ′ 99K S ′V ′ , or, more precisely, there exists

a nonempty open subscheme V ′i of V ′, such that xi (uniquely) extends to a morphism

V ′i → S ′V ′ over V ′. Set U ′

1def= V ′

1 ∩ · · · ∩ V ′r , then xi extends to a morphism U ′

1 → S ′V ′

over V ′ for all i = 1, . . . , r. Moreover, as U ′1 ⊂ V ′, we have S ′V ′ = S ′1,V ′

def= SF′ ×F′ V ′,from which we conclude that ϕn

F′(xi) extends to a morphism U ′1 → S ′V ′ over V ′ for all

i = 1, . . . , r and all n ≥ 1. Thus, any element of S(k) ⊂ S(k′) extends (uniquely) to amorphism U ′

1 → S ′V ′ over V ′, or, equivalently, a morphism U ′1 → S ′U ′

1over U ′

1.

Finally, set U def= T r π(T ′ r U ′1) and U ′ def= π−1(U). Now, any element x ∈ S(k) ⊂

S(k′) extends to a morphism U ′ → S ′U ′ → SU . By the following Lemma 3.8, this impliesthat x extends to a morphism U → SU , as desired.

Lemma 3.8. Let U be a separated, normal, integral scheme, k the function field ofU , k′ an algebraic extension of k, and U ′ the normalization of U in k′. Then, in the


category of separated schemes, the following diagram is co-cartesian:

Spec(k′) → Spec(k)

↓ ↓

U ′ → U.

Namely, for any separated scheme Y , the natural map Y (U)→ Y (k)×Y (k′) Y (U ′) is abijection.

Proof. Let k′′ be the normal closure of k′ over k and U ′′ the normalization of U ′ in k′′.In the commutative diagram

Spec(k′′) → Spec(k′) → Spec(k)

↓ ↓ ↓

U ′′ → U ′ → U,

if the left square and the big rectangle are co-cartesian, then the right square is alsoco-cartesian. Thus, replacing k′ by k′′, we may assume that k′/k is a normal extension.Set G def= Aut(k′/k). Then G induces an action on U ′ over U , and the underlyingtopological space of U can be regarded as the quotient space of U ′ by this G-action.Indeed, this is well-known set-theoretically. As for the topology, note that the naturalmorphism π : U ′ → U is closed.

Let fk : Spec(k)→ Y and fU ′ : U ′ → Y be morphisms whose restrictions to Spec(k′)coincide with each other. For each σ ∈ G, the restriction of fU ′ ◦σ to Spec(k′) coincideswith that of fU ′ . Since Y is separated, this implies that fU ′ ◦ σ = fU ′ . Thus, by thepreceding argument, there exists a continuous map ϕ : U → Y , such that fU ′ = ϕ ◦ π.Now, considering suitable affine open neighborhoods of x ∈ U and ϕ(x) ∈ Y for eachx ∈ U , we may reduce the problem to the case that U (hence also U ′) and Y are affine.

So, write U = Spec(A), U ′ = Spec(A′) and Y = Spec(R). In this case, the assertionis equivalent to saying that the natural map

Hom(R,A)→ Hom(R, k)×Hom(R,k′) Hom(R,A′)

is a bijection. But this is a consequence of the equality A = k ∩A′ in k′, which followsfrom the fact that A′ is integral over A and that A is integrally closed. �

Thus, the proof of Proposition 3.7 is completed. �

Remark 3.9. In Proposition 3.7, the extra assumption that S is F-isotrivial, when S(k)is infinite, cannot be removed. Indeed, take a proper curve T of genus ≥ 2 over a finiteextension F′ of F, set C def= T ×F′ T and S def= C r ∆, where ∆ stands for the diagonal.Moreover, set k def= F(T ) = F′(T ), and define C and S to be the generic fibers of thenatural morphisms C → T and S → T obtained by the second projection pr. Then we


may identify C(k) = C(T ) = HomF′(T, T ) and S(k) = HomF′(T, T ) r {idT }. (Here,the natural identification C(k) = C(T ) is ensured essentially by the valuative criterionfor properness.) Under these identifications, consider the subset {ϕn | n > 0} ⊂ S(k),where ϕ = ϕF′ : T → T is the |F′|-th power Frobenius endomorphism. Let U be anopen subscheme of T . Then ϕn ∈ S(k) extends to an element of S(U) if and only ifU ⊂ T r pr(Γϕn ∩ ∆), where Γϕn stands for the graph of ϕn. However, pr(Γϕn ∩ ∆)coincides with the set of closed points of T that are F′

n-rational, where F′n denotes the

unique degree n extension of F′. Thus, there does not exist a nonempty open subschemeU of T such that ϕn ∈ S(k) extends to an element of S(U) for any n ≥ 0. (Such a Umust be contained in the one-point set consisting of the generic point!)

§4. Proof of Theorem A.

Before starting the proof of Theorem A, we shall recall the main geometric resultof [CT]. Let k be a field of characteristic q = p, S a smooth, separated, geometricallyconnected curve (necessarily of finite type) over k, η the generic point of S, andK = k(η)the function field of S. Let A be an abelian scheme over S such that the generic fiberAη is of dimension d. Since A→ S is an abelian scheme and q = p, A[pn] = Ker([pn] :A → A) is finite etale over S. Thus, the natural action of the absolute Galois groupΓK of K on Aη[pn](K) (n ≥ 0) factors through π1(S). This, in turn, defines actionsof π1(S) on Aη[p∞](K) and on the Tate module Tp(Aη). We denote by ρ = ρA,p thecorresponding representation π1(S)→ AutZp(Tp(Aη)).

For each v ∈ Aη[p∞](K), write π1(S)v ⊂ π1(S) for the stabilizer of v. This is anopen subgroup of π1(S), and, by Galois theory, corresponds to a connected finite etalecover Sv → S (defined over a finite extension kv/k). We denote by gv the genus ofthe smooth compactification of Sv ×kv k. Finally, denote by (Aη)0 the largest abeliansubvariety of Aη which is isogenous to a k-isotrivial abelian variety.

Theorem 4.1. Assume that k is algebraically closed. Then, for any c ≥ 0, there existsan integer N = N(p, k, S,A, c) ≥ 0 such that, for all v ∈ Aη[p∞](K), either gv ≥ c orpNv ∈ (Aη)0.

Theorem 4.1 was proved in [CT] in arbitrary characteristics. There, roughly speak-ing, we estimate the genus by observing the Galois representation π1(S)→ GL(Tp(Aη))and using the Riemann-Hurwitz genus formula. Here, the main ingredient to estimatethe ramification terms in the genus formula is the Serre-Oesterle theorem ([Se][O]) onthe asymptotic behavior of the number of points on reduction modulo pn of p-adicanalytic subsets of Zm

p . For more details, see [CT].In [CT], we deduced Theorem A in characteristic 0 from Theorem 4.1. More specifi-

cally, we introduce a sequence of (disconnected) finite etale covers {Sn,χ}n≥0 of S withthe property that sn ∈ Sn,χ(k) lying above s ∈ S(k) corresponds to an element of orderexactly pn in As[p∞](χ). It is thus enough to prove that Sn,χ(k) = ∅ for n≫ 0. Accord-ing to Theorem 4.1, we may reduce this problem, roughly speaking, to either the casewhere Aη is isotrivial or the case where the genus of each component of Sn,χ is ≥ 2 forn≫ 0. The proof of [CT] for the first case works well in arbitrary characteristics, while


that for the second case, which resorts to Mordell’s conjecture (Faltings’ finiteness the-orem) and a compactness argument to produce a projective system of rational points,fails in positive characteristic as it is, since a curve of genus ≥ 2 over a field finitelygenerated over the (finite) prime field may admit infinitely many rational points. Toremedy this, we take a model T of the finitely generated base field k, consider models ofthe base scheme S and the abelian scheme A over T , and produce a projective systemof rational points on the fiber at a suitable closed point of T (whose residue field isfinite). Now, to make this argument work, we need various extra arguments resortingto the results of §3. See below for more details.

Proof of Theorem A. We divide the proof into several steps.

Step 1. First reductions. First, (pUB0) follows from the definition of non-Tate charac-ters. (Indeed, if S is of finite type over k and dim(S) = 0, S(k) is a finite set. So, wemay treat only finitely many abelian varieties As (s ∈ S(k)).)

Next, to prove (pUB1), assume that S is of finite type over k and dim(S) = 1. Byreplacing S by Sred, we may assume that S is reduced. By (pUB0), we may replace Sby an open dense subscheme freely. So, we may assume that S is regular and separated.Further, treating S componentwise, we may assume that S is connected. We may alsoassume that S(k) is nonempty, since otherwise there is nothing to do. Since S is regularand 1-dimensional, any point of S(k) is a smooth point, hence the smooth locus of Sis nonempty (and open). Thus, again by replacing S by an open dense subscheme, wemay assume that S is smooth and separated. Finally, since S is smooth, connectedwith S(k) = ∅, S is geometrically connected. Thus, in summary, we may assume thatS is a smooth, separated, geometrically connected curve over k.

For each n ≥ 0, set χndef= χ mod pn : Γk → (Z/pn)∗. Set i(p) = 1 for p = 2

and i(2) = 2. Then, up to replacing k by the fixed field of Ker(χi(p)) in ksep, onemay assume that χi(p) : Γk → (Z/pi(p))∗ is trivial. (Here, we have used the fact thatthe restriction of a non-Tate character to an open subgroup is non-Tate. See Lemma3.1.) This technical reduction ensures that Im(χn) ⊂ (Z/pn)∗ is contained in the orderpn−i(p) cyclic subgroup 1 + pi(p)Z/pnZ of (Z/pn)∗, when n ≥ i(p).

Step 2. Relation with rational points on various covers. For each vn ∈ Aη[pn]∗(K)(n ≥ 0), we shall define a connected finite etale cover Svn,χ of S. To do this, writeπ1(S)⟨vn⟩ for the stabilizer of ⟨vn⟩ = (Z/pn) · vn under π1(S) and S⟨vn⟩ → S for theresulting connected finite etale cover (defined over a finite extension k⟨vn⟩/k). Considerthe projection morphism pr⟨vn⟩ : π1(S⟨vn⟩) → Γk (whose image coincides with Γk⟨vn⟩)and the natural representation ρ⟨vn⟩ : π1(S⟨vn⟩)→ AutZ/pn(⟨vn⟩). These, together withχn = χ mod pn, define a representation

ρvn,χ : π1(S⟨vn⟩)→ AutZ/pn(⟨vn⟩), γ 7→ χn(pr⟨vn⟩(γ))−1ρ⟨vn⟩(γ).

Now, define Svn,χ → S⟨vn⟩ to be the connected finite etale Galois cover (defined over afinite extension kvn,χ/k⟨vn⟩) corresponding to the open normal subgroup Ker(ρvn,χ) ⊂π1(S⟨vn⟩), and denote by gvn,χ the genus of the smooth compactification of Svn,χ×kvn,χk.


Lemma 4.2. (i) Svn,χ ×kvn,χ k = Svn ×kvnk as covers of S ×k k. In particular,

gvn,χ = gvnis independent of χ.

(ii) For any k-rational point s : Spec(k)→ S, consider the specialization isomorphism

sps : Aη[p∞](K) ( ∼← A[p∞](K)) ∼→As[p∞](k).

Then sps(vn) ∈ As[p∞](χ) if and only if s : Spec(k) → S lifts to a k-rational pointsvn,χ : Spec(k)→ Svn,χ.

Proof. For (i), just observe that

π1(Svn,χ ×kvn,χ k) = Ker(ρvn,χ) ∩Ker(pr⟨vn⟩)

= Ker(ρ⟨vn⟩) ∩Ker(pr⟨vn⟩) = π1(Svn ×kvnk).

For (ii), denote again by s the section Γk ↪→ π1(S) of π1(S) � Γk induced (upto conjugacy) by s : Spec(k) → S, which identifies Γk with the decomposition groupat s. Then the existence of the lift svn,χ : Spec(k) → Svn,χ of s : Spec(k) → S isequivalent to the inclusion s(Γk) ⊂ π1(Svn,χ)(= Ker(ρvn,χ)), which can be rewritten ass(σ) ·vn = χ(σ)vn (σ ∈ Γk) or, applying the specialization isomorphism, as σ ·sps(vn) =χ(σ)sps(vn). �

Now, we shall introduce a projective system (Sn,χ)n≥0 of (disconnected) finite etalecovers of S. For each n ≥ 0, define

Sn,χdef=

⨿vn∈Aη[pn]∗(K)

Svn,χ.

Observe that (Sn,χ)n≥0 forms a projective system with transition maps induced by thecanonical morphisms Svn,χ → Spvn,χ over k. At the level of k-rational points, we have:

Claim 4.3. (i) lim←−Sn,χ(k) = ∅.(ii) The assertion of Theorem A is equivalent to saying that Sn,χ(k) = ∅ for any n≫ 0.(iii) Suppose that Sn,χ(k) = ∅ for any n ≥ 0. Then there exists an element (vn)n≥0 ∈lim←−Aη[pn]∗(K), such that Svn,χ(k) = ∅ for any n ≥ 0.

Indeed, for (i), suppose that lim←−Sn,χ(k) = ∅ and take (sn)n≥0 ∈ lim←−Sn,χ(k). Then,by the definition of Sn,χ, there exists an element (vn)n≥0 ∈ lim←−Aη[pn]∗(K), such that

(sn)n≥0 ∈ lim←−Svn,χ(k). Set s def= s0. Then, by Lemma 4.2(ii), sps(vn) ∈ As[pn](χ)for all n ≥ 0. Thus, Γk acts on (sps(vn))n≥0 ∈ Tp(As)∗ via χ, which contradictsthe assumption that χ is non-Tate. For (ii), again by Lemma 4.2(ii), the assertion ofTheorem A is equivalent to saying that there exists an N ≥ 0 such that Svn,χ(k) =∅ =⇒ n ≤ N , hence also to saying that Sn,χ(k) = ∅ for any n ≫ 0. (iii) follows fromthe fact that Aη[pn]∗(K) is finite for each n ≥ 0.

Step 3. Second reductions. Now, suppose that the assertion of Theorem A fails forour abelian scheme A → S. Then, by Claim 4.3(ii)(iii), there exists an element v =(vn)n≥0 ∈ lim←−Aη[pn]∗(K), such that Svn,χ(k) = ∅ for any n ≥ 0. We fix such a


v = (vn)n≥0. Let Kn be the function field of Svn,χ and K = KA,v the union of Kn:

Kdef= lim−→Kn.To execute the proof of Theorem A in arbitrary characteristics, we need some more

reductions. First, we may replace A → S by A ×S Svm,χ → Svm,χ for any m ≥ 0.Indeed, vn can be regarded as an element of (A×S Svm,χ)[pn]∗(Km), and we have

(Svm,χ)vn,χ ={Svm,χ, n < m,

Svn,χ, n ≥ m.

Thus, (Svm,χ)vn,χ(k) = ∅ for any n ≥ 0. In particular, we may assume thatEndK(Aη ×K K) = EndK Aη, by replacing S by Svm,χ for m ≫ 0. (Indeed, sinceEndK(Aη×K K) ⊂ EndK(Aη×KK) is a finitely generated (abelian) group, all elementsof EndK(Aη ×K K) are already defined over Km for some m ≥ 0.)

Second, the generic fiber Aη of A → S is decomposed up to isogeny into a directproduct of K-simple abelian varieties. Namely, we have a K-isogeny Aη → A

(1)η ×

· · · × A(r)η , where A(i)

η is a K-simple abelian variety for i = 1, . . . , r. We have more:as a consequence of the above first reduction step, A(i)

η ×K K is K-simple. Let v(i) =(v(i)

n )n≥0 be the image of v in Tp(A(i)η ) = lim←−A

(i)η [pn](K). As the natural map Tp(Aη)→

Tp(A(1)η ) × · · · × Tp(A

(r)η ) is injective, there exists an i0 = 1, . . . , r such that v(i0) = 0.

Since the natural map Aη[p∞](K) → A(i0)η [p∞](K) is surjective, the action of ΓK on

A(i0)η [p∞](K) factors through π1(S), hence A(i0)

η has good reduction everywhere on Sby the (original) Serre-Tate criterion (cf. §3), or, equivalently, can be regarded as thegeneric fiber of a (unique) abelian scheme A(i0) over S. Now, since the natural mapAη[p∞](K) → A

(i0)η [p∞](K) is π1(S)-equivariant, we obtain a k-morphism Svn,χ →

Sv(i0)n ,χ

naturally. This implies that Sv(i0)n ,χ

(k) = ∅ for any n ≥ 0 and that KA(i0),v(i0) ⊂KA,v. Now, replacing (A, v) by (A(i0), p−av(i0)), where a ≥ 0 is defined to satisfyv(i0) ∈ pa(Tp(A

(i0)η )∗), we may assume that Aη is K-simple, and, a fortiori, K-simple.

Then, in particular, either (Aη)0 = Aη (Case 1) or (Aη)0 = 0 (Case 2).

Step 4. Case 1: (Aη)0 = Aη. For each n ≥ 0, there exists a k-rational point sn ∈Svn,χ(k), which yields a splitting sn : Γk ↪→ π1(Svn,χ) ⊂ π1(S) of the restrictionepimorphism π1(S) � Γk. Let ∆, Γ and Σsn denote the images in AutZp(Tp(Aη)) ofπ1(S×k k), π1(S) and sn(Γk), respectively, under ρ. Since π1(S) = sn(Γk) ·π1(S×k k),we have Γ = Σsn ·∆.

As Aη = (Aη)0 is isogenous to an isotrivial abelian variety, ∆ is finite. Now, Γ ⊂AutZp(Tp(Aη)) ≃ GL2d(Zp) (d def= dim(Aη)) is a compact p-adic Lie group, hence, inparticular, it is finitely generated. Since a finitely generated profinite group admits onlyfinitely many open subgroups of given bounded index and since [Γ : Σsn ] ≤ |∆| < ∞,there are only finitely many possibilities for the Σsn ⊂ Γ, n ≥ 0. Thus, there existss ∈ S(k) such that Σsn = Σs for infinitely many n ≥ 0. Write |∆| = pam with p |m.Then we have:

Claim 4.4. Let s, t ∈ S(k) with Σs = Σt. If spt(vn) ∈ At[pn](χ), then sps(pavn) ∈As[pn](χ).


Indeed, the statement is trivial for n ≤ a, so assume that n > a and write δ(σ) def=ρ(s(σ)t(σ)−1) ∈ ∆. Also, since ρ(s(σ)) ∈ Σs = Σt, there exists τ = τσ ∈ Γk such thatρ(s(σ)) = ρ(t(τ)). As a result, one obtains δ(σ)vn = χn(τ)χn(σ−1)vn. In particular, theorder of χn(τ)χn(σ−1) ∈ (Z/pn)∗ divides the order of δ(σ) ∈ ∆, hence divides the orderpam of ∆. On the other hand, by the assumption on χ put in Step 1, χn(τ)χn(σ−1) liesin the order pn−i(p) cyclic subgroup 1 + pi(p)Z/pnZ of (Z/pn)∗, when n ≥ i(p). Now, itfollows that χn(τ)χn(σ−1) ∈ 1 + pn−aZ/pn. Thus, we have

ρ(s(σ))pavn = χn(τ)pavn = χn(σ)pavn,

which completes the proof of Claim 4.4.

It follows from Claim 4.4 that, up to replacing vn by pavn+a, one may assume thats : Spec(k) → S lifts to a k-rational point svn : Spec(k) → Svn,χ for infinitely manyn ≥ 0, hence lim←−Svn,χ(k) = ∅. This contradicts Claim 4.3(i).

Step 5. Case 2: (Aη)0 = 0 — reductions. In this case, by Theorem 4.1 and Lemma4.2(i), there exists an integer N ≥ 0, such that gvn,χ ≥ 2 for n > N . Replacing A→ S

by A×S SvN+1,χ → SvN+1,χ, we may assume that the genus g of the smooth compact-ification of S ×k k is ≥ 2. Indeed, as we have already seen, KA,v = KA×SSvN+1,χ,v, sothat after this reduction Aη is still K-simple (hence, in particular, K-simple). Now, wemay make one more reduction. Let C be the normal compactification of S and AC theNeron model of Aη over C. Note that A is naturally identified with AC ×C S. Now,define S∼ to be the subset of points of C at which the fiber of AC → C is an abelianvariety. Thus, S ⊂ S∼ ⊂ C, and we may regard S∼ as an open subscheme of C. Thenwe have Svn,χ ⊂ (S∼)vn,χ for each n ≥ 0. So, replacing A → S by AC ×C S∼ → S∼,we may assume that S coincides with the set of points of C at which Aη has goodreduction.

If Svn,χ(k) = ∅ is finite for n ≫ 0, we have lim←−Svn,χ(k) = ∅, which contradictsClaim 4.3(i). (In particular, this, together with Faltings’ theorem (cf. Theorem 3.6),already completes the proof in characteristic 0, as in [CT].) So, we may assume thatSvn,χ(k) = ∅ is infinite for all n ≥ 0.

Step 6. Case 2: (Aη)0 = 0 — isotriviality of S. Our strategy is to apply Proposition3.7, to extend all the objects in question over k to ones over a suitable model T of kover the prime field F, to consider the fibers at a (fixed) closed point of T , and applythe above projective limit argument to the finite base field case. To do this, however,we have to check the extra assumption in Proposition 3.7 that (not only C but also) Sis F-isotrivial.

Let Cn denote the normal compactification of Svn,χ (so, C0 = C), which is a propercurve over k. Then {Cn}n≥0 naturally forms a projective system. By Theorem 3.6, Cn

is F-isotrivial, or, more explicitly, there exists a curve Cn,F over F such that Cn ×k k

is k-isomorphic to Cn,F×F k. Moreover, under the identification Cn×k k = Cn,F×F k,the finite k-morphism Cn+1 ×k k → Cn ×k k uniquely descends to a finite F-morphismCn+1,F → Cn,F. (See [T2], Lemma (1.32).) We define Sn,F to be the image of Svn,χ×kkin Cn,F.


Claim 4.5. For each n ≥ 0, Sn,F is open in Cn,F, hence is regarded as an opensubscheme of Cn,F. Moreover, the finite F-morphism Cn+1,F → Cn,F restricts to afinite, etale F-morphism Sn+1,F → Sn,F.

Indeed, first, as the projection ϖn : Cn ×k k = Cn,F ×F k → Cn,F is an openmap ([Gro], Corollaire (2.4.10)), Sn,F is open. Next, consider the following cartesiandiagram:

Cn+1 ×k kϖn+1→ Cn+1,F

fk ↓ ↓ fF

Cn ×k kϖn→ Cn,F

From this, we first see that

fF(Sn+1,F) = fF(ϖn+1(Svn+1,χ ×k k))

= ϖn(fk(Svn+1,χ ×k k))

= ϖn(Svn,χ ×k k) = Sn,F.

Namely, fF : Cn+1,F → Cn,F restricts to a surjective F-morphism Sn+1,F → Sn,F.Moreover, as

f−1

k(Svn,χ ×k k) = Svn+1,χ ×k k

⊂ (Svn,χ ×k k)×Sn,F

Sn+1,F

⊂ f−1

k(Svn,χ ×k k),

we must have Svn+1,χ ×k k = (Svn,χ ×k k)×Sn,F

Sn+1,F. Namely, the diagram

Svn+1,χ ×k kϖn+1→ Sn+1,F

fk ↓ ↓ fF

Svn,χ ×k kϖn→ Sn,F

is cartesian. As ϖn : Cn ×k k → Cn,F is regarded as a base change of the mor-phism Spec(k) → Spec(F), it is affine (hence quasi-compact) and flat. Since the openimmersion Svn,χ ×k k ↪→ Cn ×k k is also quasi-compact and flat, we conclude thatϖn : Svn,χ ×k k → Sn,F is quasi-compact and flat. Moreover, it is surjective by def-inition. In summary, it is “fpqc”, hence, by descent theory, the finite-etaleness offk : Svn+1,χ ×k k → Svn,χ ×k k implies that of fF : Sn+1,F → Sn,F. Thus, the proof ofClaim 4.5 is completed.

A reformulation of Claim 4.5 in terms of fundamental groups is as follows: for eachn ≥ 0, there exists a subgroup Hn ⊂ π1(SF) (where SF

def= S0,F) such that the stabilizer


subgroup π1(S ×k k)vn at vn is the inverse image of Hn under π1(ϖ0) : π1(S ×k k) �π1(SF). Moreover, for each g ∈ π1(S ×k k), we have

π1(S ×k k)gvn = gπ1(S ×k k)vng−1 = gπ1(ϖ0)−1(Hn)g−1 = π1(ϖ0)−1(gHng

−1),

where g def= π1(ϖ0)(g). From this, we conclude that the action of π1(S×kk) on the subsetπ1(S ×k k)vn ⊂ Aη[pn](K) factors through π1(S ×k k) � π1(SF). This further impliesthat the actions of π1(S ×k k) on the submodules ⟨π1(S ×k k)vn⟩ ⊂ Aη[pn](K) and

Tdef= ⟨π1(S ×k k)v⟩ ⊂ Tp(Aη) also factor through π1(S ×k k) � π1(SF). In particular,

these actions factor through π1(S ×k k) � π1(SF ×F k). Namely, for each point x ofSF ×F k, the inertia subgroup I = Ix acts trivially on T . Now, by Proposition 3.4,Aη has good reduction at any such x. Recall that, in Step 5, we put the assumptionthat S coincides with the set of points of C at which Aη has good reduction. Thus, weconclude S ×k k = SF ×F k. In particular, S is F-isotrivial, as desired.

Step 7. Case 2: (Aη)0 = 0 — application of Proposition 3.7 and end of proof. Now,we may apply Proposition 3.7 to obtain an F-morphism f : S → T between separated,normal, integral schemes of finite type over F, such that the following hold: (a) thefunction field F(T ) of T is F-isomorphic to k; (b) under the identification F(T ) = k, Sis k-isomorphic to the generic fiber Sk of f ; and (c) under the identification S = Sk, wehave S(k) = S(T ), i.e., each element of S(k) = Sk(k) uniquely extends to an elementof S(T ). Moreover, the abelian scheme A over S = Sk extends to one over an opensubscheme of S. More precisely, there exists an open subscheme U of S containingS = Sk and an abelian scheme AU over U , such that AU ×U S is S-isomorphic to A (asabelian schemes). By definition, f(SrU) does not contain the generic point η of T . Asf(SrU) is constructible by Chevalley’s theorem, the topological closure Z def= f(S r U)does not contain η. Now, replacing T by T rZ and S by S×T (T rZ), and consideringAU ×U (S×T (T rZ)), we may assume (keeping the validity of (a)-(c)) that there existsan abelian scheme AS over S, such that AS ×S S is S-isomorphic to A (as abelianschemes). In particular, the action of π1(S) on Aη[p∞](K) factors through the naturalsurjection π1(S) � π1(S).

As T is normal, the natural map Γk → π1(T ) is surjective ([GR], Expose V, Proposi-tion 8.2). By Lemma 3.2, (i) =⇒ (ii′), together with the assumption (put at the begin-ning of the proof of Theorem A) that the image of χ is contained in 1 + pi(p)Zp ≃ Zp,χ : Γk → Z∗

p factors through Γk � ΓF′ , where F′ denotes the algebraic closure of F ink, hence, in particular, factors as Γk � π1(T )

χT→ Z∗p.

As in Step 2, we obtain a connected finite etale cover Svn,χTof S for each n ≥ 0,

such that Svn,χT ×S S = Svn,χ. More precisely, write π1(S)⟨vn⟩ for the stabilizer of⟨vn⟩ under π1(S) and S⟨vn⟩ → S for the resulting connected finite etale cover. Considerthe projection morphism pr⟨vn⟩ : π1(S⟨vn⟩) → π1(T ) and the natural representationρ⟨vn⟩ : π1(S⟨vn⟩) → AutZ/pn(⟨vn⟩). These, together with χT,n = χT mod pn, define arepresentation

ρvn,χT: π1(S⟨vn⟩)→ AutZ/pn(⟨vn⟩), γ 7→ χT,n(pr⟨vn⟩(γ))

−1ρ⟨vn⟩(γ).


Now, define Svn,χ → S⟨vn⟩ to be the connected finite etale Galois cover corresponding tothe open normal subgroup Ker(ρvn,χT

) ⊂ π1(S⟨vn⟩). Observe that Svn,χ(k) = Svn,χT(T )

for each n ≥ 0. Indeed, this follows from condition (c) (i.e., S(k) = S(T )), togetherwith the fact that T is normal.

Finally, fix a closed point t of T . Note that the residue field k(t) at t is a finitefield. Consider the fiber ASt → St → Spec(k(t)) of AS → S → T at t ∈ T andthe specialization isomorphism spt : Aη[p∞](K) ∼→ASt [p

∞](Kt), where Kt denote thefunction field of St. Then it is easy to see that Svn,χT

×T Spec(k(t)) ≃ (St)spt(vn),χtover

k(t). Here, Spec(k(t))→ T is the natural morphism with image t, and χt : Γk(t) → Z∗p

denotes the character obtained by taking the composite of χT : π1(T ) → Z∗p and the

natural map Γk(t) → π1(T ) associated with Spec(k(t)) → T . By Lemma 3.3, χt isnon-Tate. As Svn,χT (T ) = Svn,χ(k) is nonempty for any n ≥ 0, so is Svn,χT (k(t)) =(St)spt(vn),χt

(k(t)). Now, as k(t) is a finite field, (St)spt(vn),χt(k(t)) is finite for any

n ≥ 0, hence we conclude lim←−(St)spt(vn),χt(k(t)) = ∅, which contradicts Claim 4.3(i).

Thus, the proof of Theorem A is completed. �

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title torsion of abelian schemes and rational points on ...rims k^okyur^ oku bessatsu b12 (2009),...

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