Cent. Eur. J. Math. • 12(5) • 2014 • 659-674 DOI: 10.2478/s11533-013-0370-1

VERS ITA

Central European Journal of Mathematics

Abelian varieties over fields of finite characteristic

Research Article

Yuri G. Zarhin1,2*

1 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

2 Department of Mathematics, The Weizmann Institute of Science, P.O.B. 26, Rehovot 7610001, Israel

Received 3 February 2D 13; accepted 19 September 201 3

Abstract: The aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

MSC: 11G10, 14K15

Keywords: Abelian varieties • Isogenies • Points of finite order • Tate modules © Versita Sp. z o.o.

1. Introduction

Let K be a field, K its algebraic closure, Ks C K the separable algebraic closure of K, Gal(K) = Gal(Ks/K) = Aut(K/K) the absolute Galois group of K. If £ is a prime then we write F£ for the finite field Z/£Z.

Let X be an abelian variety over K. Then we write End(X) for its ring of K-endomorphisms and End0(X) for the finite-dimensional semisimple Q-algebra End(X)®Q. If n is a positive integer that is not divisible by char(K) then we write Xn for the kernel of multiplication by n in X(K); it is well known that Xn is free Z/nZ-module of rank 2 dimX [9], which is a Galois submodule of X(Ks). We write pnX for the corresponding (continuous) structure homomorphism

Pnx: Gal(K) ^ Autz/nz(Xn) = GL(2dim X, Z/nZ).

In particular, if n = £ is a prime different from char(K) then X£ is a 2 dimX-dimensional F£-vector space provided with

p£X: Gal(K) ^ AutF£(X£) = GL(2dimX, F£).

* E-mail: zarhin@math.psu.edu

Springer

We write Ge = Ge,XiK for the Image (subgroup) ptX(Gal(K)) C AutFe (Xe). By definition, GeXx,K is a finite subgroup of

AutFe (Xe) = GL(2 dim X, Fe).

If K(Xe) is the field of definition of all points of order e on X then it is a finite Galois extension of K and the corresponding Galois group Gal(K(Xe)/K) is canonically isomorphic to Gp_iXiK. If K'/K is a finite Galois extension of fields then Gal(K') is a normal open subgroup of finite index in Gal(K) while X' = X xKK' is a dim X-dimensional abelian variety over K' and the Gal(K')-modules Xe and X' are canonically isomorphic. Under this isomorphism, Ge,X',K' becomes isomorphic to a certain normal subgroup of GesXsK.

By functoriality, End(X) acts on Xn. This action gives rise to the embedding of free Z/nZ-modules

End(X)®Z/nZ ^ Endz/nz(Xn);

in addition, the image of End(X)®Z/nZ lies in the centralizer EndGal(K)(Xn) of Gal(K) in Endz/nz(Xn). Further we will identify End(X)®Z/nZ with its image in EndGal(K)(Xn) and write

End(X)®Z/nZ C EndGal(K)(Xn).

If e is a prime that is different from char(K) then we write Te(X) for the Ze-Tate module of X and Ve(X) for the corresponding Qe-vector space Te(X) ® ZeQe provided with the natural continuous Galois action [14]

pe,X: Gal(K) ^ Autzf(Te(X)) C AutQ (Ve(X)).

Recall [9] that Te(X) is a free Ze-module of rank 2dimX and Ve(X) is a Qe-vector space of dimension 2dimX. Notice that there are canonical isomorphisms of Gal(K)-modules Xei = Te(X)/'lTe(X) for all positive integers i. The natural embeddings

End(X)®Z/e'Z ^ EndZ/e,Z(Xe,)

are compatible and give rise to the embeddings of Ze-algebras End(X)®Ze ^ EndZe(Te(X)) and Qe-algebras End(X)®Qe EndQe(Ve(X)). Again, the images of End(X)®Ze in EndZe(Te(X)) and of End(X)®Qe in EndQe(Ve(X)) lie in the centralizers EndGal(K)(Te(X)) and EndGal(K)(Ve(X)) respectively. We will identify End(X)®Ze and End(X)®Qe with their respective images and write

End(X) ® Ze C EndCal,K)(Te (X)) C End Ze (Te (X)), End(X) ® Qe C EndCal,K)(Ve (X)) End Qe (Ve (X)).

Similarly, if Y is another abelian variety over K then we write Hom(X, Y) for the (free commutative) group of all K-homomorphisms from X to Y. Similarly, there are the natural embeddings

Hom(X, Y) ® Z/nZ C HomGal(K)(Xn, Yn) C HomZ/nZ(Xn, Yn), Hom(X, Y)®Z/e'Z C HomGal(K)(Xe., Ye,) C HomZ/e.Z(Xe., Ye,), Hom(X, Y)®Ze C HomGal,K)(Te(X),Te(Y)) C HomZe(Te(X),Te(Y)), Hom(X, Y)®Qe C HomGal,K)(Ve(X), Ve(Y)) C HomQe(Ve(X),Ve(Y)).

pe,X: Gal(K) ^ AutZe(Te(X)) C AutQe(Ve(X)) be the corresponding e-adic representation of Gal(K). The image

Ge,X,K = Pi,X (Gal(K)) C AutZe (Te (X)) C AutQe (Ve (X))

is a compact e-adic Lie (sub)group [11, 14].

Let d be a positive integer. We write Isog(X, K, d) for the set of K-isomorphism classes of abelian varieties Y over K that enjoy the following properties:

(i) Y admits a K-polarization of degree d.

(ii) There exists a K-isogeny Y — X whose degree is prime to char(K).

For example, if d = 1 then Isog(X, K, 1) consists of (K-isomorphism classes of) all principally polarized abelian varieties Y over K that admit a K-isogeny whose degree is prime to char(K).

We write Isog(X, K) for the set of K-isomorphism classes of abelian varieties Y over K such that there exists a K-isogeny Y — X whose degree is prime to char(K). Clearly, Isog(X, K) coincides with the union of all Isog(X, K, d). The following statement was proven under an additional assumption that p does not divide d by the author when p > 2 [18] and by Mori when p = 2 [8, Chapter XII, Corollary 2.4, p. 244]. (This is a strengthening of Tate finiteness conjecture for isogenies of abelian varieties [16, 29].)

Theorem 1.1 ([8, Corollary 2.4, p. 244]).

Assume that p = char(K) > 0 and K is finitely generated over the finite prime field Fp. Let d be a positive integer and X be an abelian variety over K. Then the set Isog(X, K, d) is finite.

The finiteness of \sog(X, K, d) combined with results of [17] implies the Tate conjecture on homomorphisms of abelian varieties and the semisimplicity of Tate modules over K (see [18] for p > 2 and [8, Chapter XII, Theorem 2.5, pp. 244-245]).

Theorem 1.2 ([8, Theorem 2.5, pp. 244-245]).

Assume that p = char(K) > 0 and K is finitely generated over the finite prime field Fp. Then for all abelian varieties A and B over K and every prime £ = char(K) the Galois module V£(A) is semisimple and the natural embedding of Z£ -modules

Hom(A, B)®Z£ ^ HomGai(K)(7>(A), T>(B))

is bijective.

Remark 1.3.

In fact, Theorem 1.2 follows even from a special case of Theorem 1.1 that deals only with principally polarized abelian varieties (i.e., when d = 1), see Remark 3.12.

By [16, Section 1, Lemma 1] the second assertion of Theorem 1.2 implies the following statement.

Theorem 1.4.

Assume that p = char(K) > 0 and K is finitely generated over the finite prime field Fp. Then for all abelian varieties A and B over K the natural embedding of Q£-vector spaces

Hom(A, B)®Q ^ HomGal(,q(V>(A),Ve(B))

is bijective.

Let K be a field that is finitely generated over the finite prime field Fp and A an abelian variety of positive dimension over K. Let £ be a prime different from p. By Theorem 1.4 (applied to B = A) and Theorem 1.2, the Gal(K)-module V£(A) is semisimple and

EndGai(K)(V£ (A)) = End (A) ® Q = End0(A) ® QQg. Since C£tAtK is the image of Gal(K) ^ AutZg(T£(A)) C AutQ£(Vg(A)), the C£AK-module V£(A) is semisimple and

End C£Ak (V£ (A)) = End (A) ® Q = End0(A) ® qQ£.

Let QeGetAtK be the Qe-subalgebra of EndQe(Ve(A)) spanned by Ge,A,K. It follows from the Jacobson density theorem [6, Chapter XVII, Section 3, Theorem 1] that QeGetA,tK coincides with the centralizer of End(A)®Qe in EndQe(Ve(A)). It follows easily that if ZeGe,A,K is the Ze-subalgebra of EndZe(Te(A)) spanned by GeiAiK then the centralizer of End(A)® Ze in EndZe(Te(A)) contains ZeGetAtK as a Ze-submodule of finite index.

2. Main results

The aim of this note is to prove variants of Theorem 1.2 where Tate modules are replaced by Galois modules An and Bn. Most of our results were already proven in [19, 28] or stated in [21] under an additional assumption p > 2. (See also [27] where the case of finite fields is discussed.) Throughout this section, K is a field that is finitely generated over the finite prime field Fp.

Theorem 2.1.

Let A be an abelian variety of positive dimension over K. Then the set Isog(A, K) is finite.

Remark 2.2.

A weaker version of Theorem 2.1 (where Isog(A, K) is replaced by its subset that consists of abelian varieties that admit a polarization of degree prime to p) is proven in [19, Theorem 6.1] under an additional assumption that p > 2.

Corollary 2.3.

Let A be an abelian variety of positive dimension over K. There exists a positive integer r = r(A) that is not divisible by p and enjoys the following properties.

(i) If C is an abelian variety over K that admits a K-isogeny C ^ A whose degree is not divisible by p then there exists a K-isogeny fi: A ^ C with ker fi C Ar.

(ii) If n is a positive integer that is not divisible by p and W is a Galois submodule in An then there exists u e End (A) such that rW C u(An) C W.

(iii) For all but finitely many primes e the Galois module Ae is semisimple.

Remark 2.4.

Corollary 2.3 (iii) is proven in [19, Theorem 1.1] under an additional assumption p > 2.

Theorem 2.5.

Let A be an abelian variety over K. Then there exists a positive integer r = r^A) such that, for any positive integer n that is not divisible by p and un an endomorphism of the Galois module An, it enjoys the following property: If we put m = n/(n, r-i) then there exists u e End (A) such that both u and un induce the same endomorphism of the Galois module Am.

If A and B are abelian varieties over K then applying Theorem 2.5 to their product X = A x B, we obtain the following statement.

Theorem 2.6.

Let A and B be abelian varieties over K. Then there exists a positive integer r2 = r2(A, B) such that, for any positive integer n that is not divisible by p and un : An ^ Bn a homomorphism of the Galois modules, it enjoys the following property: If we put m = n/(n,r-) then there exists u e Hom (A, B) such that both u and un induce the same homomorphism of the Galois modules Am ^ Bm.

Theorem 2.6 implies readily the following assertion.

Corollary 2.7.

Let A and B be abelian varieties over K. Then for all but finitely many primes £ the natural injection

Hom(A, B)®Z/£Z ^ HomGal(K)(A£, B£)

is bijective.

Remark 2.8.

Theorem 2.6 was stated without proof in [21] under an additional condition that p > 2. Corollary 2.7 was proven in [19, Theorem 1.1] under an additional condition that p > 2.

Theorem 2.9.

Let A be an abelian variety of positive dimension over K. Then for all but finitely many primes £ the centralizer of End(A)®Z£ in EndZl(T£(A)) coincides with Z£C£iA,iK.

Remark 2.10.

When K is a field of characteristic zero that is finitely generated over Q, an analogue of Theorem 2.9 was proven by Faltings [3, Section 3, Theorem 1 (c)].

Recall that an old result of Grothendieck [10] asserts that in characteristic p an abelian variety of CM-type is isogenous to an abelian variety that is defined over a finite field. (The converse follows from a theorem of Tate [16].)

Theorem 2.11.

Let X be an abelian variety over K. Suppose that for infinitely many primes £ the group (j£x,K is commutative. Then X is an abelian variety of CM-type over K and therefore is isogenous over K to an abelian variety that is defined over a finite field.

Theorem 2.11 may be strengthened as follows.

Theorem 2.12.

Let X be an abelian variety over K. Suppose that for infinitely many primes £ the group G£!xsk is £ -solvable, i.e., its Jordan-Holder factors are either £-groups or groups whose order is not divisible by £. Then there is a finite Galois extension K'/K such that X x K K' is an abelian variety of CM-type over K' and therefore is isogenous over K to an abelian variety that is defined over a finite field.

Theorem 2.12 combined with the celebrated theorem of Feit-Thompson (about solvability of groups of odd order) implies readily the following statement.

Corollary 2.13.

Let X be an abelian variety of positive dimension over K that is not isogenous over K to an abelian variety that is defined over a finite field. Then for all but finitely many primes £ the group G£x!K is not solvable and its order is divisible by 2£.

Remark 2.14.

See [24-26] for plenty of explicit examples of abelian varieties in characteristic p without CM.

Theorem 2.11 was proven in [20] under an additional assumption that p > 2. Theorem 2.12 was stated without proof in [21] under an additional assumption that K is global and p > 2.

In order to state another (partial) strengthening of Theorem 2.11, we need to introduce the following notation. If A is a commutative group then we write TORS(A) for its subgroup of all periodic elements and TORS(A)(non-p) that consists of all elements of TORS(A), whose order is prime to p.

Theorem 2.15.

Let Kab C Ks be the maximal abelian extension of K. Let X be a simple abelian variety over K. If TORS(X(Kab))(non-p) is infinite then X is an abelian variety of CM-type over K and therefore is isogenous over K to an abelian variety that is defined over a finite field.

Remark 2.16.

In characteristic zero an analogue of Theorem 2.15 was proven in [23, Theorem 1.5]. Theorem 2.15 implies readily the following statement. (Compare with [23, Corollary, p. 132].)

Corollary 2.17.

Let Kab C Ks be the maximal abelian extension of K. Let X be an abelian variety of positive dimension over K. Let X1,... ,Xr be simple abelian varieties over K such that the product |~|= X. is K-isogenous to X. Then TORS(X(Kab))(non-p) is finite if and only if all the groups TORS(Xt(Kab))(non-p) are finite, i.e., all Xt are not of CM-type over K, 1 < i < r.

Now we discuss the torsion of abelian varieties in infinite Galois extensions of K with finite "field of constants".

Theorem 2.18.

Let X be an abelian variety of positive dimension over K such that the center of End0(X) is a direct sum of totally real number fields. Let K' C Ks be an infinite Galois extension of K. Let F' be the algebraic closure of Fp in K' and suppose that F' is a finite field. Then TORS(X(K'))(non-p) is finite.

Theorem 2.18 is an immediate corollary of the conjunction of following two assertions.

Theorem 2.19.

Let X be an abelian variety of positive dimension over K such that the center of End0(X) is a direct sum of totally real number fields. Let K' C Ks be an infinite Galois extension of K. If e = p is a prime such that the e-primary component of TORS(X(K')) is infinite then K' contains all e-power roots of unity. In particular, the algebraic closure of Fp in K' is infinite.

Theorem 2.20.

Let X be an abelian variety of positive dimension over K such that the center of End0(X) is a direct sum of totally real number fields. Let us choose a polarization A: X —> X* that is defined over K. Let e = p be a prime that enjoys the following properties:

(i) e is odd and prime to deg A;

(ii) the Gal(K)-module Xe is semisimple and EndGal(K)(Xe) = End(X)®Z/eZ.

If the e-primary component of TORS(X(K')) does not vanish then K' contains a primitive eth root of unity. In particular, if F' be the algebraic closure of Fp in K' then its order is strictly greater than e.

Remark 2.21.

Let S be the set of primes e that do not enjoy either property (i) or property (ii). Then S is finite.

The paper is organized as follows. In Section 3 we discuss isogenies of abelian varieties and their kernels (viewed as finite Galois modules). One of the goals of our approach is to stress the role of analogues of Tate's finiteness conjecture for isogeny classes of abelian varieties. In Section 4 we prove all the main results except Theorem 2.12, which will be proven in Section 5. Section 6 contains additional references to results that may be extended to characteristic 2 case.

3. Isogenies and finite Galois modules

We write P for the set of all primes. Let K be a field. Let P C P be a nonempty set of primes that does not contain char(K). If X and Y are abelian varieties over K then a K-isogeny X —> Y is called a P-isogeny if all prime divisors of its degree are elements of P. For example, if P is a singleton {£} then a P-isogeny is nothing else but an £-power isogeny. We say that X and Y are P-isogenous over K if there is a P-isogeny X — Y that is defined over K. The property to be P-isogenous is an equivalence relation. Indeed, one has only to check that there exists a P-isogeny v: Y — X that is defined over K. Indeed, thanks to the Lagrange theorem, ker u C Xn where

n = deg u = #(ker u).

It follows that there is a K-isogeny v: Y — X such that the composition vu: X — Y — X coincides with multiplication by n in X. This implies that

n2dimX = deg (vu) = deg v deg u.

Since u is a P-isogeny, all prime divisors of n belong to P. This implies that all prime divisors of deg v also belong to P, i.e., v is a P-isogeny and we are done.

Let X* and Y* be the dual abelian varieties (over K) of X and Y respectively and

u*: Y* — X*, v*: X* — Y*

be the K-isogenies that are duals of u and v respectively. Since deg u* = deg u, deg v* = deg v, X* and Y* are also P-isogenous over K. (Warning: X and X* are not necessarily P-isogenous!) This implies that if X and Y are P-isogenous over K then (X x X*)4 and (Y x Y*)4 are also P-isogenous over K.

We write IsP(X, K) for the set of isomorphism classes of abelian varieties Y over K that are P-isogenous to X over K. We write IsP(X, K, 1) for the subset of IsP(X, K) that consists of all isomorphism classes of Y with principal polarization over K. For example, if P is P \ {char(K)} then

IsP(X, K) = Isog(X, K), IsogP(X, K, 1) = Isog(X, K, 1).

Now Theorem 2.1 becomes an immediate corollary of Theorem 1.1 and the following statement.

Theorem 3.1.

Let X be an abelian variety over a field K. Suppose that the set IsP((X x X*)4, K, 1) is finite. Then the set IsP(X, K) is also finite.

Proof. Let us fix an abelian variety X be over K. Let Y be an abelian variety over K that is P-isogenous to X over K. As we have seen, (Y x Y*)4 is P-isogenous to (X x X*)4 over K. Recall [8, 19, 22] (see also [27, Section 7]) that (Y x Y*)4 admits a principal polarization over K1. Since the set IsP((Y x Y*)4,K, 1) is finite, the set of K-isomorphism classes of all (Y x Y*)4 (with fixed X) is finite. On the other hand, each Y is isomorphic to a K-abelian subvariety of (Y x Y*)4 over K. But the set of isomorphism classes of abelian subvarieties of a given abelian variety is finite [7]. This implies that the set of K-isomorphism classes of all Y is finite. □

1 In [8, Chapter IX, Section 1] Deligne's proof is given.

Corollary 3.2.

Let X be an abelian variety of positive dimension over a field K. Suppose that the set Isp((X x X*)4, K, 1) is finite. Then there exists a positive integer r = r(X) that is not divisible by char(K) and enjoys the following properties.

(i) If Y is an abelian variety over K that is P-isogenous to X over K then there exists a P-isogeny fi: X — Y over K with ker fi C Xr.

(ii) If n is a positive integer, all whose prime divisors lie in P and W is a Galois submodule in Xn, then there exists u e End(X) such that rW C u(Xn) C W.

Proof. By Theorem 3.1, there are finitely many K-abelian varieties Y1,... ,Yd that are P-isogenous to X over K and such that every K-abelian variety Y that is P-isogenous to X over K is K-isomorphic to one of Yj. For each Y. pick a P-isogeny fi.: X — Y. that is defined over K. Clearly, ker fi. C Xmi where m. = deg fi.. Let us put r = |~| = m.. Clearly, for all Yj,

ker fi, C Xmt C Xr.

This implies that for every K-abelian variety Y that is P-isogenous to X over K there exists a P-isogeny fi: X — Y over K whose kernel lies in Xr. This proves (i), since every prime divisor of r is a prime divisor of one of m. = deg fi. and therefore lies in P.

(ii) The quotient Y = X/W is an abelian variety over K. The canonical map n: X — X/W = Y is a P-isogeny over K, because deg n = #(W) divides #(Xn) = n2dlmX. This implies that Y is P-isogenous to X over K. The rest of the proof goes literally (with the same notation) as in [27, Section 8, pp. 331-332] provided one replaces the reference to [27, Corollary 3.5 (i)] by the already proven case (i). (In [27], nX: X — X and nY: Y — Y denote the multiplication by n in X and Y respectively.) □

Theorem 3.3.

Suppose that P is infinite. Let X be an abelian variety of positive dimension over a field K. Suppose that the set IsP ((X x X*)4, K, 1) is finite. Then for all but finitely many primes e the Galois module Xe enjoys the following property: If W is a Galois submodule in Xe then there exists u e End(X)® Z/eZ such that u2 = u and u(Xn) = W. In particular, the Galois module Xn splits into a direct sum

Xn = u(Xn) ® (1 - u)(Xn) = W ® (1 - u)(Xn)

of its Galois submodules W and (1 — u)(Xn).

Theorem 3.3 implies immediately the following assertion.

Corollary 3.4.

Suppose that P is infinite. Let X be an abelian variety of positive dimension over a field K. Suppose that the set IsP ((X x X*)4, K, 1) is finite. Then for all but finitely many primes e e P the Galois module Xe is semisimple.

Proof of Theorem 3.3. It is well known that for all but finitely many primes e the finite-dimensional Fe-algebra End(X)®Z/eZ is semisimple. (See, e.g., [19, Lemma 3.2].) Let r be as in Corollary 3.2. Now let e e P be a prime that does not divide r and such that End(X) ® Z/eZ is semisimple. Let W be a Galois submodule in Xe. By Corollary 3.2, there exists u e End(X) such that

rW C u(Xe) C W.

Since e does not divide r, we have rW = W and therefore u(Xe) = W. Let ue be the image of u in End(X)®Z/eZ. Clearly, ue(Xe) = u(Xe) = W. Let I be the right ideal in semisimple End(X) ® Z/eZ generated by ue. The semisimplicity implies that there exists an idempotent u that generates I. It follows that

W = ue (Xe) = u(Xe).

We will need the following lemma [27, Lemma 9.2, p. 333]. Lemma 3.5.

Let Y be an abelian variety of positive dimension over an arbitrary field K. Then there exists a positive integer h = h(Y, K) that enjoys the following property: If n is a positive integer that is not divisible by char(K), u,v e End(Y) are endomorphisms of Y such that

{ker u n Yn} C {ker v n Yn}

then there exists a K-isogeny w: Y — Y such that hv — wu e n • End(Y). In particular, the images of hv and wu in

End(Y) ® Z/nZ C End Gal(K)( Yn) C End z/nz(Yn)

coincide.

Theorem 3.6.

Let X be an abelian variety of positive dimension over a field K. Suppose that the set Isog P( (X x X*)8, K, 1) is finite. Then there exists a positive integer r1 = ri(X, K) such that, for any positive integer n, all whose prime divisors lie in P and m = n/(n,r1), if enjoys the following property: If un e EndGal(K)(Xn) then there exists u e End(X) such that the images of un and u in EndGal(K)(Xm) coincide.

Proof. Let us put Y = X x X. Then (Y x Y*)4 = (X x X*)8. Let r(Y) be as in Corollary 3.2 and h(Y) as in Lemma 3.5. Let us put r1 = r1 (X, K) = r(Y, K)h(Y, K). Now the proof goes literally as the the proof of [27, Section 10, Theorem 4.1], provided one replaces the references to [27, Corollary 3.5, Lemma 9.2] by references to Corollary 3.2 and Lemma 3.5 respectively. □

Let A and B be abelian varieties over K. Applying Theorem 3.6 to X = A x B and using the obvious compatible decompositions

End(X) = End (A) ® End(B) ® Hom(A, B) ® Hom(B,A),

End Gal(K) (Xn) = End Gal(K )(An) 8 EndGal(K )(Bn) e HomGal(K )(An,Bn) e HomGal(K )(Bn,An),

we obtain the following statement.

Theorem 3.7.

Let A and B be abelian varieties of positive dimension over a field K. Suppose that the set IsP ((A x B x A* x B* )8, K, 1) is finite. Then there exists a positive integer r2 = r2(A, B, K) = r1(A x B, K) such that, for any positive integer n, all whose prime divisors lie in P and m = n/(n,r2), it that enjoys the following property: If un e HomGal(K)(An, Bn) then there exists u e Hom(A, B) such that the images of un and u in HomGal(K)(Am, Bm) coincide.

Corollary 3.8.

Let A and B be abelian varieties of positive dimension over a field K. Suppose that the set IsP ((A x B x A* x B* )8, K, 1) is finite. Then for all primes £ e P the natural injection

Hom(A, B)®Z£ ^ HomGal(K)(T£(A),Te(B))

is bijective.

Proof. Let r2 = r2(A, B) be as in Theorem 3.7. Let £'° be the exact power of £ that divides r2. Let v e HomGal(K)(T£(A),T£(B)). For each i > i0, v induces a homomorphism vi e HomGal(K)(A£i, B£i). By Theorem 3.7, there exists ui e Hom(A, B) such that the images of ui and vi in Hom(A£—i0, B£—i0) coincide. This means that ui — v sends T£(A) into £i—i0 T£(B). It follows that v coincides with the limit of the sequence {ui}°>^ in HomZ£(T£(A),T£(B)) with respect to £-adic topology. Since Hom(A, B)®Z£ is a compact and therefore a closed subset of HomZf(T£(A), T£(B)), the limit v also lies in Hom(A, B)®Z£. □

The following lemma will be proven at the end of this section.

Lemma 3.9.

Let X be an abelian variety of positive dimension over a field K. Suppose that the set Isog P( (X x X*)4, K, 1) is finite. Let r = r(X, K) be as in Corollary 3.2. Then every e e P enjoys the following property. Let S be a Galois-invariant Ze-submodule in Te(X) such that the quotient Te(X)/S is torsion-free. Then there exists u e End(X) ® Ze such that

r• S C u(Te(X)) C S.

Theorem 3.10.

Let X be an abelian variety of positive dimension over a field K. Suppose that the set Isog P( (X x X*)8, K, 1) is finite. Then every e e P enjoys the following property. If W is a Ga1(K)-invariant Qe-vector subspace in Ve(X) then there exists u e End(X)® Qe such that u2 = u and u(Ve(X)) = W. In particular, Ve(X) splits into a direct sum

Ve(X) = u(Ve(X)) ® (1 — u)(Ve(X)) = W ® (1 — u)(Ve(X))

of its Galois submodules W and (1 — u)(Ve(X)) and the Ga1(K)-module Ve(X) is semisimple.

Proof. Let us put S = W n Te(X). Clearly, S Is a Galols-lnvarlant free Ze-submodule In Te(X) and W = QeS. In addition, the quotient Te(X)/S is torsion-free.

By Lemma 3.9, there exists u G End(X) ® Ze such that r • S C u(Te(X)) C S. It follows that r • W C u(Ve(X)) C W. Since r • W = W, we have u(Ve(X)) = W. Notice that

End(X)®Ze C End(X)®Qe = End0(X)®QQe

and End0(X) is a finite-dimensional semisimple Q-algebra. It follows that End (X) ® Qe is a finite-dimensional semisimple Qe-algebra. Let I the left ideal in semisimple End(X) ® Qe generated by u; there is an idempotent u that generates I. Clearly,

U(Ve (X )) = u(Ve (X)) = W. □

Proof of Lemma 3.9. If S = {0} then we just put u =0. If S = Te(X) then we take as u the identity automorphism 1X of X.

So, further we assume that S is a proper free Ze-module in Te(X) of positive rank say, d and let {e1,. . . , eJ, . . . , ed} be its basis. Since S is pure in Te(X), for all positive integers l the natural homomorphism of Galois modules

Sf = S/tS ^ Te(X)/eiTe(X) = Xe>.

is an injection of free Z/i"Z-modules. Further, we will identify Si with its image in Xei. We write ej for the image of ej in Si C Xei; clearly, the d-element set {ej}d=1 is a basis of the free Z/£iZ-module Si. By Corollary 3.2 applied to n = £i and W = Si there exists ui G End(X) such that rSi C ui(Xei) C Si. In particular, ui(Xei) contains re' for all j. It is also clear that for each z G Te(X), u(z) G S + e'Te(X). For each ej pick an element

zj G Te(X)/ejTe(X) = Xei

such that ui(zJl) = rej. Let us pick zj G Te(X) such that its image in Xei coincides with zj. Clearly, the image of ui(zjI) in Te(X)/elTe(X) = Xei equals rej. Using the compactness of End(X)®Ze and Te(X), let us choose an infinite increasing sequence of positive integers i1 < i2 < ... < im < ... such that {ulm converges in End(X) ® Ze to some u and {zJrri }T=1 converges in Te(X) to some zJ for all J with 1 < J < d. It follows that

u(zJ) = Um uim(z'im) = rej.

This implies that u(Te (X)) D r • S. On the other hand, for each z G Te (X), ulw (z) G S + eimTt (X). Since {im}™=1 is an increasing set of positive integers, {S + tmTe(X)}^=1 is a decreasing set of compact sets whose intersection is compact S. It follows that u(z) = Um ulw (z) lies in S. □

Lemma 3.11.

Let X be an abelian variety of positive dimension over K. Let £ be a prime that is different from char(K) and such that the Gal(K)-module X£ is semisimple and

EndGal(K)(X£) = End(X) ® Z/£ Z. Then the centralizer of End(X)®Z£ in EndZl(T£(X)) coincides with Z£G£iXiK.

Proof. Clearly, GX!esK is the image of Gal(K) ^ G£sXsK — AutF£(X£). It follows that the GXs£sK-module X£ is semisimple and

End (Xt) = End(X) ® Z/£ Z.

By the Jacobson density theorem, F£GX,£,K coincides with the centralizer of End(X)® Z/£Z in EndF£(X£). (Here F£GX,£,K is the F£-subalgebra of EndF£(X£) spanned by GXt£tK.)

Let M be the centralizer of End(X)®Z£ in EndZt(T£(X)). Clearly, M is a saturated Z£-submodule of EndZ£(T£(X)) (i.e., the quotient EndZ£(T£(X))/M is torsion-free);in addition, M contains Z£G£sXsK. We have

M/£M C Endz£ (T£(X)) ® Ze/£Ze = Endf£ (X£).

Clearly, M/£M lies in the centralizer of

End(X) ® Z£ ®it Ze/£Ze = End(X) ® Z/£Z.

This implies that M/£M C F£GX,£,K C EndF£(X£). On the other hand, the image of Z£G£tXtK in

End Z£ (T£ (X)) ® Ze/£Ze = End f£ (X,)

obviously coincides with F£GX,£,K. Since this image lies in M/£M, we conclude that M/£ = F£GX,£,K and M = Z£G£tXtK + £ • M. It follows from Nakayama's lemma that the Z£-module M coincides with its submodule Z£G£tXtK. □

Remark 3.12.

Let P be a singleton {£} and d = 1. Now Theorem 1.1 combined with Corollary 3.8 and Theorem 3.10 implies readily Theorem 1.2.

4. Proof of main results

Throughout this section, K is a field that is finitely generated over Fp. Let us put P = P \ {p}.

Proof of Corollary 2.3 and Theorem 2.5. Corollary 2.3 follows readily from Corollary 3.2 combined with Theorem 1.1. Theorem 2.5 follows readily from Theorem 3.6 combined with Theorem 1.1. □

Proof of Theorems 2.1 and 2.6. One has only to combine Theorem 1.1 with Theorems 3.1 and 3.7 respectively.

Proof of Theorem 2.9. The assertion follows readily from Lemma 3.11 combined with Corollary 2.3 and Theorem 2.5. □

Proof of Theorem 2.11. The proof of [20, Theorem 4.7.4] works literally provided one replaces the reference to [19, Theorem 1.1.1] by references to Corollaries 2.3 and 2.7. □

Proof of Theorem 2.15. Theorem 2.15 is an immediate corollary of the conjunction of two following statements. (Compare with [23, Theorems 2 and 3, p. 133].)

Theorem 4.1.

Let X be a simple abelian variety over K that is not of CM-type. Let £ = p be a prime, W a nonzero Galois-invariant Q£-vector space in V£(X) and GW the image of Gal(K) in AutQ£(W). Then the group GW is not commutative.

Theorem 4.2.

Let X be a simple abelian variety over K that is not of CM-type. Then for all but finitely many primes £ = p the following condition holds: Let W a nonzero Galois-invariant F£-vector space in X£ and GW the image of Gal(K) in AutF£(W), then the group GW is not commutative.

Proof of Theorems 4.1 and 4.2. The proof of [23, Section 3, Theorems 2 and 3] works literally provided one replaces the reference to [23, p. 139, Statements 1 and 2] by references to Theorem 1.2 (instead of Statement 1) and to Corollaries 2.3 and 2.7 (instead of Statement 2). □

This ends the proof of Theorem 2.15. □

Proof of Theorems 2.19 and 2.20. The proofs of [23, Section 4, Theorems 7 and 8] work literally in our case for Theorems 2.19 and 2.20 respectively. (As in the proof of Theorems 4.1 and 4.2 one should replace the reference to [23, Statements 1 and 2, p.139] by references to Theorem 1.2 and to Corollaries 2.3 and 2.7.) □

5. Torsion and ramification in solvable extensions

Let K be an arbitrary field and O C K a discrete valuation ring whose field of fractions coincides with K. We write p for the maximal ideal of O and p for the characteristic of the residue field O/p. Let L/K be a finite Galois field extension with Galois group Gal(L/K) and OL the integral closure of O in L. The following assertion is well known (see, e.g., [30, Chapter 5, Sections 7-10], [12, Chapter 1, Section 7]).

(i) OL is a principal ideal domain, the set of its maximal ideals is finite and consists of (say) g maximal ideals q1, . . . , qg such that

I * \ e(L/K) pO = (□ q)

where e(L/K) is the (weak) ramification index at p. The degree f(L/K) = [OL/qi: O/p] of the field extension (OL/qi)/(O/p) equals the product f0ps where f0 is the degree of the separable closure of O/p in OL/qi and s is a nonnegative integer that vanishes if and only if OL/qi is separable over O/p. The integers f0 and s do not depend on i. The product

e(L/K) • foPsg = e(L/K) • f • g = [L : K] = #(Gal(L/K)).

In particular, e(L/K) • ps divides [L: K]. The field extension L/K is tamely ramified at p if and only if #(/(qj)) is not divisible by p, i.e., e(L/K) is not divisible by p and OL/qj is separable over O/p. Here /(qf) C Gal(L/K) is the inertia subgroup attached to qf.

(ii) The Galois group Gal(L/K) acts transitively on the set {qf : 1 < i < g}. The corresponding inertia subgroups /(q£) C Gal(L/K) are conjugate subgroups of order e(L/K)ps in Gal(L/K). (See [30, Chapter 5, Section 10, Theorem 24 and its proof].)

(iii) Let L0/K be a Galois subextension of L/K, i.e., L0/K is a Galois field extension and L0 C L. Then q0 = qj n OL0 is a maximal ideal in OL0 that lies above p. The image of /(qj) under the surjection Gal(L/K) ^ Gal(L0/K) coincides with the inertia subgroup

/(q0) C Gal(L0/K)

attached to q0 [12, Chapter 1, Section 7, Proposition 22 (b)]; in particular, #(/(q0)) divides #(/(qt)). On the other hand, #(/(q0)) divides #(Gal(L0/K)) = [L0: K]. This implies that if [L0: K] and #(/(q,)) are relatively prime then #(/(q0)) = 1, i.e., L0/K is unramified at p.

Let Y be an abelian variety of positive dimension over K. Let n > 3 be an integer that is not divisible by p. Assume that Yn C Y(K), i.e., all points of order n on X are defined over K.

Let Y ^ Spec(O) be the Néron model of Y [1 ]; it is a smooth group scheme whose generic fiber coincides with Y. Since Yn C Y(K), the Raynaud criterion [5, Proposition 4.7] tells us that Y has semistable reduction at p, i.e., Y is a semiabelian group scheme.

Let ? be a prime different from p. For all positive integers j the field K(Ymj) with mj = is a finite Galois extension. The following assertion was inspired by [8, Chapter XII, Section 2.0, p.242].

Lemma 5.2.

Let n > 3 be an integer that is not divisible by p. Assume that Yn C Y(K). Let L = K(Y?) and q be a maximal ideal in OL, which lies above p. Then:

(i) The inertia group /(q) is a finite commutative ?-group. In particular, L/K is tamely ramified at p.

(ii) Let L0/K be a Galois subextension of L/K. If [L0: K] is not divisible by £ then L0/K is unramified at p.

Proof. The assertion (ii) follows readily from (i). So, let us prove (i).

For all positive integers j let us put Lj = K(Ymj) and Oj = OLj. We have L1 = L, O1 = OL. We have a tower of Galois extensions

K C Li C L2 C ... C Lj C ...

We write L^ for the union |JLt. For each j pick a maximal ideal q j in Oj in such a way that q(1) = q and q'j+1) lies above q(j). (Such a choice is possible, because the projective limit of nonempty finite sets is nonempty.) Then the Galois group Gal(L^/K) is the projective limit of finite groups Gal(Lj/K). It is also clear that

Gal(Lj/K) = Pmpy(Gal(K)) C Aut^^), Gal(L^/K) = plY(Gal(K)) C Autz?(T?(Y)) C AutQe(V?(Y)).

Recall that the natural homomorphisms /(q'j+1)) ^ /(q(j)) are surjective for all j. Let /^ be the projective limit of the corresponding inertia subgroups /(q^); clearly, /^ is a compact subgroup of Gal(L^/K) and for each j the natural group homomorphism /^ —> /(q(j)) is surjective, because the projective limit of nonempty finite sets is also nonempty. Therefore one may view /^ as a certain compact subgroup of

Autz?(T?(Y)) c AutQ?(V?(Y)).

Since Y has semistable reduction at q, there exists a Q?-vector subspace W C V£(Y) such that /^ acts trivially on W and V£(Y)/W [5, Proposition 3.5]. It gives us an injective continuous homomorphism of topological groups

/„-> HomQ?(V?(Y)/W, W), a ^{v + W ^ a(v) - v}, a e U v G V?(Y).

Since /^ is compact, there is a continuous isomorphism between /^ and its image, which is a compact subgroup of HomQ?(V?(Y)/W,W). Since the latter is a finite-dimensional Q?-vector space, all of its compact subgroups are commutative pro-?-groups (that are isomorphic either to a direct sum of several copies of Z? or to zero). It follows that /^ is also a commutative pro-?-group. Since there is a surjective continuous homomorphism

/^ /(q(j)),

every /(q(j)) is a finite commutative ?-group. This ends the proof, since q = q(1) and therefore /(q) = /(q(1)). □

Proof of Theorem 2.12. If K is finite then there Is nothing to prove. So, let us assume that K Is Infinite. Let d > 1 be the transcendence degree of K over Fp. Let us pick a positive integer n > 3 that is not divisible by p. Replacing K by K(Xn) and X by X xKK(Xn), we may assume that Xn C X(K).

Let P be an infinite set of primes f = p such that GftXtK is f-solvable. By deleting finitely many primes from P, we may and will assume that the Gal(K)-module Xf is semisimple for all f G P. Since GftXtK is the image of Gal(K) in AutFf (Xf) = GL(2g, Ff), the GfXK-module Xf is semisimple for all f G P.

Let C be the field of complex numbers. Let us put g = dim X. Recall that Xf is a 2g-dimensional Ff-vector space. Let G be a finite f-solvable subgroup of AutFf(Xf) such that the natural faithful representation of G in Xf is completely reducible. Let us split the semisimple Ff[G]-module X'e into a direct sum

Xf = 0 W

of simple Ff[G]-modules Wt. If dt = dimFf Wt then 2g = ^"=1 dt. By a theorem of Fong-Swan [13, Section 16.3, Theorem 38], each Wt lifts to characteristic zero;in particular, there is a group homomorphism pt : G ^ GL(dt, C) whose kernel lies in the kernel of G —> AutFf(Wt) (for all t). Clearly, the product-homomorphism

p = n p, : G n GL(dt, C) C GL(2g, C)

t=1 t=1

is an injective group homomorphism and therefore G is isomorphic to a finite subgroup of GL(2g, C). By a theorem of Jordan [2, Theorem 36.13], there is a positive integer N = N(2g) that depends only on 2g and such that G contains a normal abelian subgroup of index dividing N. By deleting from P all prime divisors of N, we may and will assume that f does not divide N for each f G P.

Let us apply this observation to G = GfiXiK. We obtain that for all f G P the group Ge!XsK contains an abelian normal subgroup Hf of index dividing N. For each f G P let us consider the corresponding subfield of Hf-invariants

Kf == (K(Xf))Hf c K(Xf).

Clearly, Kf/K is a Galois extension of degree dividing N while Gal(K(Xf)/Kf ) coincides with commutative (sub)group Hf. Since Kf C K(Xf) and [Kf:K] divides N and therefore is not divisible by f, Lemma 5.2 tells us that the Galois extension Kf/K is unramified with respect to every discrete valuation of K. (Since char(K) = p, its every residual characteristic is also p.)

Let k be the (finite) algebraic closure of Fp in K and let S be an absolutely irreducible normal d-dimensional projective variety over k whose field of rational functions k(S) coincides with K. Let V C S be a smooth open dense subset such that the codimension of S\ V in S is, at least, 2. Let Vf be the normalization of V in Kf/K. Now the Zariski-Nagata purity theorem tells us that the regular map Vf ^ V is an étale Galois cover;clearly, its degree equals [Kf : K] and therefore divides N. Let ^(V) be the fundamental group of V that classifies étale covers of V ([4], [8, Chapter XII, Section 1, pp.241-242]). This group is a (natural) topologically finitely generated (topological) quotient of Gal(K) [8, Chapter XII, Section 1, p.242] and the natural surjection

Gal(K) ^ Gal(Kf/K)

factors through ^(V), i.e., it is the composition of the canonical continuous surjection Gal(K) ^ ^(V) and a certain continuous surjective homomorphism

Yf : ^i(V) ^ Gal(Kf/K)

whose kernel is an open normal subgroup in ^(V) of index dividing N.

Since n1(V) is topologically finitely generated, it contains only finitely many open normal subgroups of index dividing N (because it admits only finitely many continuous homomorphisms to any finite group of order dividing N). If r is the intersection of all such subgroups then it is an open normal subgroup in n1(V) that lies in the kernel of every ye for all e G P; in particular, it has finite index. The preimage A of r is an open normal subgroup of finite index in Cal(K) and the corresponding subfield of A-invariants E = is a finite Galois extension of K that contains Ke for all e G P. This implies that for all £ G P the compositum EK(Xe) is abelian over E, because K(Xe) is abelian over Ke. But EK(Xe) = E(XEe) where

XE = X xkE

is an abelian variety over E. Applying Theorem 2.11 to XE and E (instead of X and K), we conclude that XE is an abelian variety of CM-type and isogenous over E = K to an abelian variety that is defined over a finite field. The same is true for X, since XE = X xKE. □

6. Concluding remarks

Theorem 1.2 and Corollaries 2.3 (iii) and 2.7 imply readily that the following results remain true for all prime characteristics p, including p = 2.

• Assertions (Sections 1.3 and 4.4), Corollaries 1-6, Theorem 4.1, and Remark 1 of [20] remain true over any field E that is finitely generated over a finite field of arbitrary characteristic, including 2. [20, Corollary 7] remains true for any field F of arbitrary prime characteristic, including 2.

• [15, Theorem 1.1 (ii)].

• [28, Theorems 1.1, 1.4, 1.6, 1.9, 2.1].

7. Corrigendum to [27]

P.317, Remark 1.3, second sentence: one has to assume additionally that the kernel of the morphism is W. P.317, line -12: read Hom(V*,X*) instead of Hom(V,X).

P. 326, Proof of Theorem 3.4, second line: read 8g-dimensional instead of 4g-dimensional.

Acknowledgements

The final version of this paper was prepared during a stay at Max-Planck-Institut fur Mathematik (Bonn) in September 2013: the author is grateful to the MPI for the hospitality and support.

The author is grateful to Alexey Parshin, Chad Schoen and Doug Ulmer for their interest in this paper. Special thanks go to the referees, whose comments helped to improve the exposition.

This work was partially supported by a grant from the Simons Foundation (#246625 to Yuri Zarkhin).

References

[1] Bosch S., Lütkebohmert W., Raynaud M., Neron Models, Ergeb. Math. Grenzgeb., 21, Springer, Berlin, 1990

[2] Curtis Ch.W., Reiner I., Representation Theory of Finite Groups and Associative Algebras, WHey Classics Library, John Wiley & Sons, New York, 1962

[3] Faltings G., Complements to Mordell, In: Rational Points, Bonn, 1983/84, Aspects Math., E6, Vieweg, Braunschweig, 1984, 203-227

[4] Grothendieck A. (Ed.), Revêtements Étales et Groupe Fondamental (SGA 1), Lecture Notes in Math., 224, Springer, Berlin, 1971

[5] Grothendieck A., Modèles de Néron et monodromie, Éxpose IX dans SGA7, I, Lecture Notes in Math., 288, Springer, Berlin, 1972, 313-523

[6] Lang S., Algebra, Addison-Wesley, Reading, 1965

[7] Lenstra H.W.Jr., Oort F., Zarhin Yu.G., Abelian subvarieties, J. Algebra, 1996, 180(2), 513-516

[8] Moret-Bailly L., Pinceaux de Variétés Abéliennes, Astérisque, 129, Paris, 1985

[9] Mumford D., Abelian Varieties, 2nd ed., Oxford University Press, London, 1974

[10] Oort F., The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field, J. Pure Appl. Algebra, 1973, 3, 399-408

[11] Serre J.-P., Sur les groupes des congruence des variétés abéliennes, Izv. Akad. Nauk SSSR Ser. Mat., 1964, 28, 3-20

[12] Serre J.-P., Corps Locaux, 2nd ed., Publications de l'Université de Nancago, VIII, Hermann, Paris, 1968

[13] Serre J.-P., Représentations Linéares des Groupes Finis, 3rd ed., Hermann, Paris, 1978

[14] Serre J.-P., Abelian ?-Adic Representations and Elliptic Curves, 2nd ed., Advanced Book Classics, Addison-Wesley, Redwood City, 1989

[15] Skorobogatov A.N., Zarhin Yu.G., A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces, J. Algebraic Geometry, 2008, 17(3), 481-502

[16] Tate J., Endomorphisms of abelian varieties over finite fields, Invent. Math., 1966, 2, 134-144

[17] Zarhin Ju.G., Endomorphisms of Abelian varieties over fields of finite characteristic, Math. USSR-Izv., 1975, 9, 255-260

[18] Zarkhin Yu.G., Abelian varieties in characteristic p, Math. Notes, 1976, 19(3), 240-244

[19] Zarkhin Yu.G., Endomorphisms of Abelian varieties and points of finite order in characteristic p, Math. Notes, 1977, 21(6), 415-419

[20] Zarkhin Yu.G., Torsion of Abelian varieties in fininite characteristic, Math. Notes, 1977, 22(1), 493-498

[21] Zarkhin Yu.G., Homomorphisms of Abelian varieties and points of finite order over fields of finite characteristic, In: Problems in Group Theory and Homological Algebra, Yaroslav. Gos. Univ., Yaroslavl, 1981, 146-147 (in Russian)

[22] Zarhin Yu.G., A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction, Invent. Math., 1985, 79(2), 309-321

[23] Zarhin Yu.G., Endomorphisms and torsion of abelian varieties, Duke Math. J., 1987, 54(1), 131-145

[24] Zarhin Yu.G., Hyperelliptic Jacobians without complex multiplication in positive characteristic, Math. Res. Lett., 2001, 8(4), 429-435

[25] Zarkhin Yu.G., Endomorphism rings of certain Jacobians in finite characteristic, Sb. Math., 2002, 193(8), 1139-1149

[26] Zarhin Yu.G., Non-supersingular hyperelliptic Jacobians, Bull. Soc. Math. France, 2004, 132(4), 617-634

[27] Zarhin Yu.G., Homomorphisms of abelian varieties over finite fields, In: Higher-Dimensional Geometry over Finite Fields, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 16, IOS Press, Amsterdam, 2008, 315-343

[28] Zarhin Yu.G., Homomorphisms of abelian varieties over geometric fields of finite characteristic, J. Inst. Math. Jussieu, 2013, 12(2), 225-236

[29] Zarkhin Yu.G., Parshin A.N., Finiteness problems in Diophantine geometry, Amer. Math. Soc. Transl., 1989, 143, 35-102

[30] Zariski O., Samuel P., Commutative Algebra, I, Van Nostrand, Princeton, 1958