Local to global principles for homomorphisms of abelian schemes

Let $A$ and $B$ be abelian varieties defined over the function field $k(S)$ of a smooth algebraic variety $S/k.$ We establish criteria, in terms of restriction maps to subvarieties of $S,$ for existence of various important classes of $k(S)$-homomorphisms from $A$ to $B,$ e.g., for existence of $k(S)$-isogenies. Our main tools consist of Hilbertianity methods, Tate conjecture as proven by Tate, Zarhin and Faltings, and of the minuscule weights conjecture of Zarhin in the case, when the base field is finite.


Introduction
Let S be a smooth variety over a finitely generated field k of arbitrary characteristic.Let A and B be S-abelian schemes with generic fibers A and B (respectively) defined over the function field k(S). 1 In this paper we consider existence of certain classes of k(S)-homomorphisms from A to B, e.g., k(S)-isogenies, and provide local criteria in terms of restriction maps to subvarieties of S. Furthermore we study existence of abelian subvarieties of A in a similar way.Our first main result is the following local to global principle.
Theorem A (Thm. 4.5).Let S be a smooth variety over a finitely generated field k.Let A , B be abelian schemes over S with generic fibers A and B, respectively.Let U be a dense open subscheme of S. Let m ∈ {0, 1, • • • , dim(S)}.Assume that k is infinite or that m ≥ 1.Let κ ∈ N.
(ii) For every m-dimensional smooth connected subscheme T of U there exists a k(T )-isogeny (resp.surjective homomorphism, resp.non-zero homomorphism, resp.homomorphism with κ-dimensional kernel) AT → BT , where AT , BT denote the generic fibres of the base changed abelian schemes AT → T, BT → T, respectively (cf.Section 3).
(ii) For every m-dimensional smooth connected subscheme T of U there exists a k(T )-isogeny (resp.surjective homomorphism, resp.non-zero homomorphism, resp.homomorphism with κdimensional kernel) A T,k(T ) → B T,k(T ) .
Ingredients of the proof of Theorem A include standard methods based on the Tate conjecture (proven by Tate, Zarhin and Faltings cf.Theorem 2.3) and some consequences of the Hilbert irreducibility theorem (cf.Lemma 4.1), which were inspired by Drinfeld's "conventional formulation of Hilbertianity" in [6, Section A. 1.] and by Section 2 of a recent paper of Cadoret and Tamagawa [4].As a formal consequence we obtain: Corollary B (Cor. 4.6).Let S be a smooth variety over a finitely generated field k.Let A be an abelian scheme over S with generic fiber A. Let U be a dense open subscheme of S. Let m ∈ {0, 1, • • • , dim(S)}.Assume that k is infinite or that m ≥ 1.
(a) The following are equivalent: (i) A is not a simple k(S)-variety.
(ii) For every m-dimensional smooth connected subscheme T of U the fibre AT is not a simple k(T )-variety.
(b) The following are equivalent: (i) A k(S) is not a simple k(S)-variety.
(ii) For every m-dimensional smooth connected subscheme T of U the fibre A T,k(T ) is not a simple k(T )-variety Our second main result is the following local to global principle for quadratic isogeny twists of abelian varieties.We call an abelian variety B/k a quadratic isogeny twist of an abelian variety A/k, if there exists a quadratic twist A ′ /k of A and a K-isogeny B → A ′ (cf.Section 2).
Theorem C (Thm.4.7).Let S be a smooth variety over a finitely generated field k.Let A , B be abelian schemes over S with generic fibers A and B respectively.Let U be a dense open subscheme of S. Let m ∈ {0, 1, • • • , dim(S)}.Assume that k is infinite or that m ≥ 1.The following are equivalent: (a) A is a quadratic isogeny twist of B (b) For every m-dimensional smooth connected subscheme T of U the abelian variety AT is a quadratic isogeny twist of BT .
The implication (a)⇒(b) holds true also in the case where k is finite and m = 0.
We remark, that results in Section 4 of the paper are a bit more general than Theorem A, Corollary B and Theorem C in that they also cover the situation where S is an arithmetic scheme, but we do not go into the details within this introduction.
It is clear that in the above statements the case when k is a finite field and m=0 can not be covered by the Hilbertianity methods.(b) For every closed point s ∈ S there exists a surjective k(s)-homomor- Structure of the paper.In Sections 2 and 3 we gathered material which is needed in the sequel including basic facts on: twists of abelian varieties, Galois representations and abelian schemes.Section 4 is a central part of the paper.It contains proofs of main results by Hilbertianity methods in the case when k is an infinite field or m ≥ 1.In the final section we discuss the remaining case of k finite, m=0 and work under extra assumptions, either the minuscule weights conjecture or trivial endomorphisms for generic fibres.
S.P. thanks the Mathematics Department at Adam Mickiewicz University in Poznań for hospitality during research visits.We thank Jȩdrzej Garnek, Marc Hindry and Bartosz Naskrȩcki for useful discussions on the topic of this paper.Finally we want to thank the anonymous referee for a careful reading of the manuscript and for valualble comments.

Preliminaries Notation
For a field K we denote by K a separable closure of K.If E/K is a Galois extension, we denote by Gal(E/K) its Galois group and define Gal(K) := Gal(K/K).A K-variety is a separated algebraic K-scheme which is reduced and irreducible.A K-curve is a K-variety of dimension 1.For a scheme S and s ∈ S we denote by k(s) the residue field of s.
Let n ∈ Z.Then, as usual, we denote by S[n −1 ] the open subscheme of S with underlying set {s ∈ S : n ∈ k(s) × } (where n is viewed as an element of k(s) via the ring homomorphism Z → k(s)).We let L be the set of all rational primes and define L(S) := {ℓ ∈ L : S[ℓ −1 ] = ∅}.If S is reduced and irreducible we denote, following EGA, by R(S) the function field of S. If S is a K-variety, we sometimes write then we denote by Γ Zar the Zariski closure of Γ inside the algebraic group GLV /Q ℓ so that Γ Zar is an algebraic group of Q ℓ .If G is an algebraic group over Q ℓ , then we denote by G • the connected component of the identity element of G.

Twists of abelian varieties.
Let K be a field and A and B abelian varieties over K. Let E/K be a Galois extension.We call B an E/K-twist of A if there is an E-isomorphism AE → BE.In subsequent sections we are mainly interested in the case E = K.We denote by Twist E/K (A) the set of all isomorphism classes of E/K-twists of A and define Twist(A) := Twist K/K (A).There are natural operations ρA : Gal(K) → AutK (AE) and ρB : Gal(K) → AutK (BE) and an operation Now assume that B/K is an E/K-twist of A and choose an E-isomorphism f : AE → BE.Then whose cohomology class does not depend on the choice of f .Let be the operation on A derived from ρB via transport of structure via f .Then an easy calculation shows that is simply the action ρA twisted by the 1-cocycle ξ.It is well-known that this sets up a bijective map There is a natural map For a quadratic character χ ∈ Hom(Gal(E/K), {±IdA}) we denote by Aχ an E/K-twist of A corresponding to j(χ) under Ξ E/K .We call B a quadratic isogeny twist of A, if there exists a quadratic character χ ∈ Hom(Gal(K/K), {±IdA}) and a K-isogeny B → Aχ.Sometimes we tacitly identify {±IdA} with µ2(Q) = {±1}.
Remark 2.1.Let A be an abelian variety over a finite field K. Then there is a unique non-trivial quadratic character Gal(K) → {±IdA} and accordingly a unique non-trivial quadratic twist of A. This is clear since Gal(K) ∼ = Ẑ.

Galois representations.
Let K be a field of characteristic p ≥ 0 and A/K an abelian variety.For a rational prime ℓ = p we denote by the corresponding ℓ-adic Galois representation, where T ℓ (A) is the ℓ-adic Tate module of A and V ℓ (A) = T ℓ (A) ⊗ Z ℓ Q ℓ .Then the Zariski closure ρ A,ℓ (Gal(K)) Zar of ρ A,ℓ (Gal(K)) in GL V ℓ (A) and its identity component (ρ A,ℓ (Gal(K)) Zar ) • are algebraic groups over Q ℓ .If K is finite, then we define the L-series of A by where Fr ∈ Gal(K) is the Frobenius element.This characteristic polynomial L(A/K, T ) has integer coefficients and does not depend on the rational prime ℓ = p by the Weil conjectures.We have the following elementary but useful fact.
Lemma 2.2.Let K be a field of characteristic p ≥ 0 and ℓ a prime different from p. Let A/K and B/K be abelian varieties.Let f : A → B be a homomorphism and let V ℓ (f ) : V ℓ (A) → V ℓ (B) be the homomorphism of Q ℓ -vector spaces induced by f .Then: In particular, we have ρ A,ℓ ∼ = ρ B,ℓ provided B is K-isogenous to A.
Proof.We note that T ℓ (A) = Hom(Q ℓ /Z ℓ , A(K)).For the purpose of that proof we put T ℓ (M ) := Hom(Q ℓ /Z ℓ , M ) for an arbitrariy abelian group M .Let C = ker(f ) Let I = im(f ).Then C • and I are abelian varieties.
From the exact sequence we derive an exact sequence From the exact sequence we obtain an exact sequence 0 and tensoring with Q ℓ we see that we have an exact sequence [15,Remark 8.4]).The image of the right hand map of the exact sequence (3) has dimension 2a as desired.This finishes the proof of a) and b).For c) put b := dim(B) and note that: There is the following celebrated result due to Faltings, Tate and Zarhin.
(b) The natural homomorphism is bijective.
Corollary 2.4.Let K be a finitely generated field of characteristic p ≥ 0 and A/K and B/K be abelian varieties.The following statements are equivalent.
(a) There exists a K-isogeny f : If K is finite, then conditions (a) and (b) are also equivalent to the following condition Remark 2.5.Let K be a finitely generated field of characteristic p ≥ 0 and A/K an abelian variety.Let χ : Gal(K) → {±IdA} be a quadratic character.Then: Lemma 2.6.Let K be a finitely generated field of characteristic p ≥ 0.
Let A, B be abelian varieties over K and ℓ = p a rational prime.Then the following statements are equivalent.
(a) B is a quadratic isogeny twist of A.
(b) There exists a quadratic character χ : If K is finite, then the equivalent conditions (a) and (b) are also equivalent to Proof.We prove the implication (a)⇒(b): Assume B is a quadratic isogeny twist of A. Then there exists a quadratic character χ : Gal(K) → {±IdA} and a K-isogeny B → Aχ.From Lemma 2.2 and Remark 2.5 we conclude that For the proof of the implication (b)⇒(a) assume that there exists a quadratic character χ : From Remark 2.5 we conclude that ρ B,ℓ ∼ = ρ Aχ,ℓ and then Corollary 2.4 implies that there exists an isogeny B → Aχ.It follows that B is a quadratic isogeny twist of A.
From now on assume that K is finite.Then Gal(K) = Ẑ and thus there exist exactly two quadratic characters thereof.We prove the implication (b)⇒(c): If χ is the trivial character then ρ B,ℓ ∼ = ρ A,ℓ and hence Remark 2.7.Let A and B be abelian varieties over a field K and ℓ a prime different from the characteristic of K.
Let pA and pB be the projections of the product Thus the actions of Gal(K) on V ℓ (A) and V ℓ (B) and the induced action on is bijective by the Tate conjecture (cf.Theorem 2.3).
The following lemma and its proof are inspired by [20, Prop 2.10].
Lemma 2.8.Let A and B be abelian varieties over a field K and keep the notation from Remark 2.7.Let K • be the fixed field of the group If K is finitely generated, then the above specialization map (4) induces a bijective map Proof.For (5) it is enough to prove that for every finite Galois extension ) and consider the diagram The horizontal arrows are injective (cf.Theorem [15, Prop.12.2]).Thus, for every ) and, by the injectivity of the horizontal maps of the diagram, This finishes up the proof of ( 5).If K is finitely generated, then (5) together with the Tate conjecture (cf.Theorem 2.3) implies that ( 6) is bijective.

Abelian schemes
Throughout this section let S and T be noetherian, normal and connected schemes.Note that then the local rings of S are domains and thus [10, 6.1.10](together with the connectedness of S) implies that S is in fact irreducible.Furthermore S is reduced.Similarly T is reduced and irreducible.
Let F (resp.E) be the function field of S (resp.T ).Let u : T → S be a morphism.Furthermore consider the point of S. We fix throughout this section a rational prime ℓ ∈ L(T ).This is possible by the following lemma.We note that automatically ℓ ∈ L(S).
Lemma 3.1.For every scheme X, the set L(X) is non-empty.
Let A /S be an abelian scheme with generic fibre A/F .Let AT := A ×S T and let AT := A ×S Spec(E) be the generic fibre of AT → T .We then have cartesian squares There exists a finite connected étale cover splits up in a coproduct of ℓ 2n dim(A) copies of S ′ , i.e. , it is a constant group scheme.There is a point t ′ : Spec(E) → S ′ over t and a point ξ ′ : F → S ′ lying over the generic point of S. As A [ℓ n ] ×S S ′ is a constant group scheme, the natural maps derived from these points are bijective.We thus get a specialization isomorphism s A,T,ℓ n : and accordingly a specialization isomorphism These are equivariant for the action of π1(S[ℓ −1 ]).In particular we have dim(A) = dim(AT ).
We often apply this notation in the case where S0 = S[ℓ −1 ] and ρ = ρ A,ℓ .Note that in that case u −1 (S0) = T [ℓ −1 ] is automatically non-empty by our choice of ℓ.
of representations of Gal(E).
For the rest of this section let B be an abelian scheme over is bijective.The assertion is immediate from the bijectivity of ( 7) and (8).
Proof.Choose a point t ∈ T .We even prove that the composite homomorphism is injective.Let f be in the kernel of (10).Then is the zero homomorphism.Hence f must be the zero homomorphism by the rigidity lemma [15, 20.1].
Note that, by the above Lemmata 3.4 and 3.5, there is a canonical injective specialization map Let T ′ be an irreducible component of T ×S S ′ and E ′ the function field of T ′ .Then E ′ /E is a finite separable extension and we can consider the composite map r A ,B,T,S :Hom → which tacitly depends on the choice of T ′ and on the choice of an embedding E ′ → E. This map (c) f has finite kernel (resp. is surjective, resp. is an isogeny) if and only if fT has finite kernel (resp. is surjective, resp. is an isogeny).
Proof.With the above specialization isomorphisms we construct a diagram whose vertical arrows are bijective.Together with Lemma 2.2 we get: This proves (a) and (b), and (c) is immediate from that. .
The lower horizonal map is bijective because G = G T by our assumption that T is ρ ℓ -generic.Hence the upper horizontal map r ℓ is bijective, too.

Next consider the commutative diagram Hom
where the vertical maps are bijective (cf.Tate conjecture, Lemma 2.8).
The lower horizonal map is bijective because Hence, r ℓ is bijective.The Z-module Hom E (A T,E , B T,E ) is free and finitely generated.Hence, the statement about the cokernels of r A ,B,T,S and r A ,B,T,S follows as well.

Hilbertianity
Throughout this section let Z be a regular noetherian connected scheme.Let S be a connected scheme and f : S → Z a morphism of finite type which is assumed to be smooth.Note that S and Z are reduced and irreducible.Let d be the relative dimension of S/Z.Let K be the function field of Z. Assume that K is finitely generated.We denote by Smm(S/Z) the set of all connected subschemes T of S such that the restriction f |T : T → Z is smooth of relative dimension m.Note that every T ∈ Smm(S/Z) is regular and connected, hence reduced and irrecucible.The aim of this section is to consider specializations to subvarieties in Smm(S/Z).The results of this section are in our opinion most interesting in the following cases: 1.The case where Z = Spec(k) for a finitely generated field k; in that case S is simply a smooth k-variety.

The case where
Z is an open subscheme of Spec(R) and R is the ring of integers in a number field; in that case S is sometimes called an arithmetic scheme.
where U ′ = U ×Z Z ′ , X ′ = X ×Z Z ′ and g ′ and p ′ are the restrictions of g and p respectively.As K is Hilbertian there is a point a ∈ A d (K) such that (gK • pK) −1 (a) = Spec(F ) and g −1 K (a) = Spec(E) where E/K and F/E are finite separable field extensions.For a suitable choice of Z ′ the closed immersion a : Spec(K) → A d,K extends to a closed subscheme Y of Z ′ × A d (cf.[11, 8.8.2 and 8.10.5]).Put T := g −1 (Y ) and XT := p −1 (T ).After replacing Z ′ by one of its dense open subschemes we can assume that the maps p|XT : XT → T and g : T → Y are finite and étale, because the corresponding maps on the generic fibres are finite and étale.The closed subscheme T := g −1 (Y ) of U ′ is connected because it is finite and étale over Z ′ and its generic fibre T × Z ′ Spec(K) = g −1 K (a) = Spec(E) is connected.Likewise the closed subscheme XT of X ′ is connected, because it is finite and étale over Z ′ and its generic fibre XT × Z ′ Spec(K) = Spec(F ) is connected.Proof.We consider the homomorphism ρ ′ : π1(S) By the Frattini property, a subscheme T ∈ Smm(S/Z) is ρ-generic if and only if it is ρ ′ -generic.In the proof of the corollary we can hence assume that G is finite.Case A: Assume that K is Hilbertian and m = 0. Let X be the finite étale cover of S corresponding to the kernel of ρ.By Lemma 4.1 there exists T ∈ Smm(S/Z) such that T ⊂ U and such that T ×S X is connected.This implies that T is ρ-generic.Case B: Assume that m ∈ {1, 2, • • • , d} (and K arbitrary).As f : S → Z is smooth we can, after replacing U by a smaller open set, assume that there exists an étale Z-morphism U → Z × A d .Composing with an appropriate projection we get a smooth morphism Note that Sm0(S/Z×Am) ⊂ Smm(S/Z).The function field K(x1, • • • , xm) of Z × Am is Hilbertian (even if K is not).We can thus apply Case A with Z replaced by Z × Am to finish up the proof in Case B. [5,Thm. 8.33], [19, §10.6]).Hence, the above Corollary 4.2 can be applied to ℓ-adic representations of π1(S[ℓ −1 ]), e.g., to ρ A×B,ℓ .From now on until the end of this section, let A and B be abelian schemes over S. For T ∈ Smm(S/Z) we denote (as in the previous section) by AT /R(T ) (resp.by BT /R(T )) the generic fibre of AT → T (resp. of BT → T ).Finally, for ℓ ∈ L(S) we define ρ ℓ := ρ A×B,ℓ .
The following lemma is at the core of the rest of the arguments of this section.
Proof.Let f : AT → BT be an R(T )-homomorphism such that we have dim(ker(f ))∈∆1 and dim(im(f ))∈∆2, and assume that T is ρ ℓ -generic.By Lemma 3.8 the specialization maps are injective with finite cokernels C = coker(r) and C = coker(r).Let s = |C|.Then s • f lies in the image of r.Thus there exists a R(S)homomorphism F : A → B such that r(F ) = s • f .Using Lemma 3.6 once more, we see that The proof of the resp case can be carried out in a completely analoguous way using r instead of r and taking Remark 3.7 into account.
We are ready for our first local-global statement.(a) The following are equivalent: (i) A is not a simple R(S)-variety.
(ii) For every T ∈ Smm(S/Z) with T ⊂ U the fibre AT is not a simple R(T )-variety.
(b) The following are equivalent: is not a simple R(S)-variety.

Specialization to a finite field
In Theorem 4.5, Corollary 4.6 and Theorem 4.7 the case when k is finite and m=0 has been excluded.This case is more difficult and cannot be handled with Hilbertianity alone.In this section we generalize parts of results 4.5, 4.6, 4.7 (under additional assumptions) to this "critical" case by using recent results of Khare-Larsen [13] and Fité [9].
Definition 5.1.Let A be an abelian variety over a finitely generated field K and let ℓ = char(K) be a rational prime.Let G ℓ be the connected component of the Zariski closure of ρ A,ℓ (Gal(K)).We say that A satisfies condition M W C(A) if for all rational primes ℓ = char(K) the action of G ℓ,Q ℓ on each irreducible factor of the representation V ℓ (A) ⊗ Q ℓ Q ℓ is minuscule in the sense of Bourbaki [1] cf. [13, page 1].
Remark 5.2.Let A be an abelian variety over a finitely generated field K satisfying M W C(A). Let B be an abelian variety.
(a) For every finite extension K ′ /K the abelian variety A K ′ satisfies M W C(A K ′ ) because the connected component of the Zariski closure of ρ A,ℓ (Gal(K ′ )) agrees with the connected component of the Zariski closure of ρ A,ℓ (Gal(K)) as Gal(K ′ ) is of finite index in Gal(K).The following is a global function field analogue of a result of Khare and Larsen [13,Thm. 1].We include the proof for the reader's convenience.Proof.Let K = R(S).Fix one rational prime ℓ = char(K).After replacing S by one of its connected finite étale covers, we can assume that the Zariski closures From HomG(VA, VB) ∼ = HomK (A, B) ⊗ Q ℓ and the fact that HomK (A, B) is Z-free it follows that HomK(A, B) is non-zero, as desired.
We can now treat higher dimensional S by combining Proposition 5.4 with the results of the previous section.Proof.The implication (a)⇒(b) is immediate from Lemma 3.6.We prove the implication (b)⇒ (a) in case "surjective".After replacing S by one of its connected finite étale covers we can assume that End By the Poincaré reducibility theorem (cf.[15, Prop.12.1] and the passage below) A (resp.B) is k(S)-isogenous to i∈I A n i i (resp.j∈J B m i j ) where I and J are finite sets, the Ai (i ∈ I) are mutually not k(S)-isogenous simple abelian varieties over k(S) and the Bj (j ∈ J)) are mutually not k(S)-isogenous simple abelian varieties over k(S).We consider the commutative diagram where the sij : Hom 0 k(S) (Ai, Aj) → Hom 0 k(S) (A i,k(S) , A j,k(S) ) are the canonical maps.The horizontal maps in the diagram are bijective.The left hand vertical map is bijective by (13).Thus the sij are bijective, too.For i = j we have Ai ≃ Aj, hence Hom 0 k(S) (A i,k(S) , A j,k(S) ) = Hom 0 k(S) (Ai, Aj) = 0, and thus the A i,k(S) are mutually non-isogenous over k(S).As Ai is simple it follows that End 0 k(S) (A i,k(S) ) = End 0 k(S) (Ai) is a division algebra over Q and thus the A i,k(S) are simple.Likewise the B j,k(S) are simple and mutually non-isogenous over k(S).Note that M W C(Ai) and M W C(Bj) holds true (cf.Remark 5.2).After replacing S by one of its dense open subschemes each Ai (resp.Bj ) extends to an abelian scheme Ai (resp.Bj) over S. It is then clear that for every closed point s ∈ S the fibre A . In what follows we can thus asssume, without loss of generality, that A = i∈I A n i i and that B = j∈J B m j j .From (b) we know that for every closed point s ∈ S there exists a surjective k(s)-homomorphism A s,k(s) → B s,k(s) and thus for every j ∈ J a surjective homomorphism A s,k(s) → B j,s,k(s) .Our assumption End k(S) (A) = End k(S) (B) = Z implies that ρ A,ℓ and ρ B,ℓ are absolutely irreducible.Now, because of the claim, Theorem 5.7 implies that there is a finite Galois extension L/K and a character χ : Gal(L/K) → Q × ℓ of finite order such that ρ A,ℓ = χ ⊗ ρ B,ℓ .But now, by Theorem 2.3, we have an isomorphism of Gal(L/K)-modules HomL(AL, BL) Thus Gal(L/K) acts on HomL(AL, BL) ∼ = Z via χ and this implies that χ is a quadratic character.Lemma 2.6 (implication (b)⇒ (a) applied over K) now implies that B is a quadratic isogeny twist of A.
Again one can eliminate the assumption dim(S) = 1 by the Hilbertianity approach of the previous section.Remark 5.10.It is easy to show that if A and B in Corollary 5.9 are elliptic curves with nontrivial endomorphisms, then the claim still holds.It is so, since then j-invariants j(A) and j(B) belong to k, hence the curves are isotrivial cf.[21, V.3.1 and Exerc.V.5.8], so a twist between fibres at u ∈ U extends to a twist between the curves.It is an interesting question, if Corollary 5.9 holds true for nonsimple abelian varieties, e.g., for products of mutually nonisogenous elliptic curves over k(S).
k(S) (B) = Z.The following are equivalent: (a) A is a quadratic isogeny twist of B. (b) For every closed point u of U the abelian variety Au is a quadratic isogeny twist of Bu.If abelian varieties A and B meet the so-called minuscule weights conjecture of Zarhin (cf.condition MWC, Definition 5.1), then we apply a global function field analogue (cf.Proposition 5.4) of a result of Khare and Larsen [13, Thm.1] and prove the following result.It completes part (b) of Theorem A in case when k is finite and m=0.We discuss the current status of Zarhin's conjecture in Remark 5.3.In particular, it holds true for ordinary abelian varieties over global fields of positive characteristics.Theorem E (Thm.5.6).Let S be a smooth variety over a finite field k.Let A , B be abelian schemes over S with generic fibers A and B respectively.Assume that A satisfies M W C(A) and B satisfies M W C(B).The following are equivalent:(a) There exists a surjective k(S)-homomorphism (resp.k(S)-isogeny) A k(S) → B k(S) .
T ) by Corollary 2.4 and Remark 2.5.We prove the implication (c)⇒(a): If L(B/K, T ) = L(A/K, T ), then A is isogenuous to B by Corollary 2.4.If L(B/K, T ) = L(A/K, −T ) and χ the nontrivial quadratic character, then L(B/K, T ) = L(Aχ/K, T ) and Corollary 2.4 implies that B is isogenous to Aχ, as desired.

Remark 3 . 7 .Lemma 3 . 8 .
For f ∈ Hom F (A F , B F ) and fT := r A ,B,T,S (f ) one can compare dimension data of f and fT in a completely analogous way.This is a consequence of Lemma 3.6 and the construction of r A ,B,T,S .If both fields E and F are finitely generated and u : T → S is ρ A×B,ℓ -generic, then the canonical mapsr A ,B,T,S ⊗ Z ℓ : HomF (A, B) ⊗ Z ℓ → HomE(AT , BT ) ⊗ Z ℓ and r A ,B,T,S ⊗ Z ℓ : Hom F (A F , B F ) ⊗ Z ℓ → Hom E (A T,E , B T,E ) ⊗ Z ℓare bijective.In particular, under these assumptions coker(r A ,B,T,S ) and coker(r A ,B,T,S ) are finite groups of order prime to ℓ. Proof.We identify, T ℓ (A) with T ℓ (AT ) and T ℓ (B) with T ℓ (BT ) along the natural and equivariant specialization isomorphisms.Let ρ ℓ := ρ A×B,ℓ , G := ρ ℓ (π1(S[ℓ −1 ])) Zar and G T := (ρ ℓ )T (π1(T [ℓ −1 ])) Zar .Let r ℓ = r A ,B,T,S ⊗ Z ℓ and r ℓ = r A ,B,T,S ⊗ Z ℓ .We then have a commutative diagram HomF (A, B) ⊗ Z ℓ r ℓ / / HomE(AT , BT ) ⊗ Z ℓ HomZ

Lemma 4 . 1 .
If K is Hilbertian, then for every dense open subscheme U of S and every finite étale morphism p : X → U there exists a connected subscheme T of U with the following properties.(a) T is a subscheme of (f |U ) −1 (Z ′ ) for some dense open subscheme Z ′ of Z.(b) f |T : T → Z ′ is finite and étale.(c) p −1 (T ) = X ×S T is connected.Proof.As f is smooth, after replacing U by a smaller dense open set and after replacing X accordingly, there exists an étale Z-morphism g : U → Z × A d (cf.[12, Exposé II, §1]).For any dense open subscheme Z ′ of Z we can consider the following commutative diagram of schemes

Corollary 4 . 2 .
Let G be a profinite group and assume that the Frattini subgroup Φ(G) of G is open in G. Let ρ : π1(S) → G be a group homomorphism.For every dense open subscheme U of S and every m ∈ {1, • • • , d} there exists a T ∈ Smm(S/Z) such that T ⊂ U and such that T is ρgeneric.If K is Hilbertian, then the same holds true for m = 0.

( 5 . 4 . 7 .
ii) For every T ∈ Smm(S/Z) with T ⊂ U the fibre A T,R(T ) is not a simple R(T )-variety Proof.Note that A is non-simple if and only if there exists κ ∈ {1, 2, • • • , dim(A)−1} and a homomorphism A → A with κ-dimensional kernel.Thus the corollary is a formal consequence of Theorem 4.Theorem Let U be a dense open subscheme of S. Let m∈{0, 1, • • • , d}.Assume that K is Hilbertian or that m ≥ 1.The following are equivalent: (a) A is a quadratic isogeny twist of B (b) For every T ∈ Smm(S/Z) with T ⊂ U the abelian variety AT is a quadratic isogeny twist of BT .The implication (a)⇒(b) holds true also in the case where Z = Spec(k) with a finite field k and m = 0. Proof.We prove the implication (a)⇒(b): Assume that A is a quadratic isogeny twist of B and let T ∈ Smm(S/Z).Assume that T ⊂ U .Choose ℓ ∈ L(T ) (cf.Lemma 3.1).It follows by Lemma 2.6, that there exists a χ ∈ Hom(Gal(R(S)), {±IdB})

Proposition 5 . 4 .
Let S be a smooth curve over a finite field k.Let A , B be abelian schemes over S with generic fibers A and B respectively.Assume that A satisfies M W C(A) and B satisfies M W C(B).If the set of all closed points s ∈ S such that there exists a non-zero k(s)-homomorphism A s,k(s) → B s,k(s) has Dirichlet density 1, then there exists a non-zero k(S)-homomorphism A k(S) → B k(S) .

Theorem 5 . 5 .Theorem 5 . 6 .
Let S be a smooth variety over a finite field k.Let A , B be abelian schemes over S with generic fibers A and B respectively.Assume that A satisfies M W C(A) and B satisfies M W C(B).The following are equivalent: (a) There exists a non-zero k(S)-homomorphism A k(S) → B k(S) .(b) For every closed point s ∈ S there exists a non-zero k(s)-homomorphism A s,k(s) → B s,k(s) .Proof.After replacing S by one of its connected finite étale covers we can assume that Hom k(S) (A, B) = Hom k(S) (A k(S) , B k(S) ).The implication (a)⇒(b) is immediate from Lemma 3.6 and Remark 3.7.We prove the other implication (b)⇒(a): Let K = R(S) and let ℓ = char(K) be a rational prime.By Corollary 4.2 there exists a T ∈ Sm1(S/k) of S such that T is ρ A×B,ℓ -generic.Then T is automatically ρ A,ℓ -generic and ρ B,ℓ -generic.In particular, ρ A,ℓ (Gal(K)) Zar = ρ A T ,ℓ (Gal(k(T ))) Zar .In particular, AT /k(T ) satisfies M W C(AT ).Similarly BT /k(T ) satisfies M W C(BT ).By (b), for every closed point t ∈ T there exists a non-zero homomorphism A k(t) → B k(t) .By Proposition 5.4 there exists a non-zero homomorphism A T,k(T ) → B T,k(T ) .Now (a) follows by Lemma 4.4.We can upgrade Theorem 5.5 a bit, to treat not only non-zero homomorphisms but also other important classes of homomorphisms, much in the spirit of Corollary B. Let S be a smooth variety over a finite field k.Let A , B be abelian schemes over S with generic fibers A and B respectively.Assume that A satisfies M W C(A) and B satisfies M W C(B).The following are equivalent: (a) There exists a surjective k(S)-homomorphism (resp.k(S)-isogeny) A k(S) → B k(S) .(b)For every closed point s ∈ S there exists a surjective k(s)-homomorphism (resp.k(s)-isogeny) A s,k(s) → B s,k(s) .

Theorem 5 . 7 .
(Rajan,[17, Thm.2]) Suppose that the Zariski closure H1 of ρ1(Gal(K)) is connected and that the upper Dirichlet density of SM (ρ1, ρ2) is positive.Then the following holds true.(a)There is a finite Galois extension L/K such that ρ1| Gal(L) ∼ =ρ2|Gal(L)and the connected component of the Zariski closure of ρ2(Gal(K)) is conjugate to H1.(b)If ρ1 is absolutely irreducible, then there is a Dirichlet characterχ : Gal(L/K) → Q × ℓ of finite order such that ρ2 ∼ = χ ⊗ Q ℓ ρ1.We have the following global function field analogue of [9, Cor.2.7].Proposition 5.8.Let S be a smooth curve over a finite field k.Let A , B be abelian schemes over S with generic fibers A and B, respectively.Let ℓ = char(k) be a rational prime and let K = k(S).Let U be a dense open subscheme of S. Assume that End k(S) (A) = End k(S) (B) = Z and that the Zariski closures of ρ A,ℓ (Gal(K)) and of ρ B,ℓ (Gal(K)) are connected.If the set of all closed points u ∈ U such that Au is a quadratic isogeny twist of Bu has Dirichlet density 1, then A is a quadratic isogeny twist of B. Proof.Let Γ be the set of all u ∈ U such that Au is a quadratic isogeny twist of Bu.Assume that Γ has Dirichlet density 1.We claim that SM (ρ ℓ,A , ρ ℓ,B ) has positive upper Dirichlet density.Consider the two sets Γ± := {u ∈ Γ : T r(ρ A,ℓ (F ru)) = ±T r(ρ B,ℓ (F ru))}.For every u ∈ Γ we have T r(ρ A,ℓ (F ru)) ∈ {±T r(ρ B,ℓ (F ru))} by Lemma 2.6 (implication (a)⇒(c) applied over k(u)).Thus Γ = Γ+ ∪ Γ−.Moreover Γ+ ⊂ SM (ρ ℓ,A , ρ ℓ,B ).If the inclusion were wrong, then it would follow that Γ+ has upper Dirichlet density zero and that Γ− has Dirichlet density 1.But then the Chebotarev density theorem would imply that T r(ρ A,ℓ (g)) = −T r(ρ B,ℓ (g)) for all g ∈ Gal(K), which is obviously false for g = Id.Thus the inclusion Γ+ ⊂ SM (ρ ℓ,A , ρ ℓ,B ) holds true.

Corollary 5 . 9 .
Let S be a smooth variety over a finite field k.Let A , B be abelian schemes over S with generic fibers A and B respectively.Let U be a dense open subscheme of S. Assume that End k(S) (A) = End k(S) (B) = Z.The following are equivalent: (a) A is a quadratic isogeny twist of B. (b) For every closed point u of U the abelian variety Au is a quadratic isogeny twist of Bu.Proof.The implication (a)⇒(b) is known from Theorem 4.7.We prove the other implication.Assume that for every closed point u of U the abelian variety Au is a quadratic isogeny twist of Bu.Then, for every smooth connected curve T on U the abelian variety AT is a quadratic isogeny twist of BT by Proposition 5.8.Thus Theorem 4.7 implies (a).
It constitutes a separate question which we address under two sets of additional assumptions.If A and B do not have nontrivial endomorphisms geometrically, then we establish a global function field analogue (cf.Proposition 5.8 below) of Fite's result [9, Cor.2.7] following its proof quite closely.By combining Proposition 5.8 with the Hilbertianity approach we augment Theorem C by the case k finite and m=0.
Theorem D (Cor.5.9).Let S be a smooth variety over a finite field k.Let A , B be abelian schemes over S with generic fibers A and B respectively.Let U be a dense open subscheme of S. Assume that End k(S) (A) = End [4,,rk 5.3.If K is a number field, then M W C(A) holds true for every abelian variety A/K due to a result of Pink[16, Cor.5.11]which proves Zarhin's minuscule weights conjecture [24, Conjecture 0.4] in the number field case.The same holds true if K is a finitely generated field of characteristic zero; using Hilbertianity one can easily reduce to the number field case.According to [24, Conjecture 0.4] every abelian variety over a global field of any characteristic should satisfy M W C(A). Zarhin [24, 4.2.1]provedhisconjecture for K a global field of positive characteristic and A/K an ordinary abelian variety.In a recent preprint[4]Cadoret and Tamagawa formulated an anologue of Zarhin's conjecture over finitely generated fields and checked that M W C(A) holds true for another case of A over K finitely generated of positive characteristic, see[4, Cor.6.3.2.3].
[18]scule.By[14, Thm.1.2]and[18]theset of all closed points s ∈ S such that the Frobenius element F rs ∈ G(Q ℓ ) generates a Zariski dense subgroup F rs of a maximal torus of G has positive density.Thus we can choose a closed point s ∈ S such that F rs is a Zariski dense subgroup of a maximal torus T of G and such that there exists a non-zero homomorphism f : A s,k(s) → B s,k(s) .Furthermore f is defined over a finite extension k ′ of k(s).Now |k ′ | = |k(s)| m , for a natural number m and thus f is fixed under the action of F r m s , and F r m s also generates a Zariski dense subgroup of T. It follows that the mapV ℓ (f ) ⊗ Q ℓ : VA → VB induced by f is fixed by the torus T. Moreover V ℓ (f ) ⊗ Q ℓ is a non-zero element of Hom T (VA, VB).Thus dim(Hom T (VA, VB)) > 0.Now G is a reductive group because ρ A×B,ℓ is semisimple.By our assumption A satisfies M W C(A) and B satisfies M W C(B).