ISRAEL JOURNAL OF MATHEMATICS TBD (XXXX), 1–32
DOI: 10.1007/s000000000000000000000000

LOCAL TO GLOBAL PRINCIPLESFOR HOMOMORPHISMS OF ABELIAN SCHEMES

BY

Wojciech Gajda
Faculty of Mathematics and Computer Sciences, Adam Mickiewicz University
Uniwersytetu Poznańskiego 4, 61-614 Poznań, Polande-mail: gajda@amu.edu.pl

AND

Sebastian Petersen
Institut für Mathematik, Universität Kassel
Fachbereich 10, Wilhelmshöher Allee 73, 34-119 Kassel, Germany
e-mail: petersen@mathematik.uni-kassel.de
Dedicated to Moshe Jarden with admiration on the occasion of his 80th birthday

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

Received February 6, 2022 and in revised form February 28, 2023

1

2

W. GAJDA AND S. PETERSEN

Isr. J. Math.

1. 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). 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 subvarietiesof 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.
(a) The following are equivalent:(i) There exists a k(S)-isogeny (resp. surjective homomorphism,
resp. non-zero homomorphism, resp. homomorphism with κ-dimensional kernel) A → B.
(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 genericfibers of the base changed abelian schemes AT → T, BT → T,
respectively (cf. Section 3).
(b) The following are equivalent:(i) There exists a k(S)-isogeny (resp. surjective homomorphism,
resp. non-zero homomorphism, resp. homomorphism with κ-dimensional kernel) Ak(S) → Bk(S) .
(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,k(T ) → BT,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) andsome consequences of the Hilbert irreducibility theorem (cf. Lemma 4.1), which

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

3

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 bea dense open subscheme of S. Let m ∈ {0, 1, . . . , dim(S)}. Assume that k isinfinite 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 thefiber AT is not a simple k(T )-variety.
(b) The following are equivalent:(i) Ak(S) is not a simple k(S)-variety.
(ii) For every m-dimensional smooth connected subscheme T of U thefiber AT,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′ /kof 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 theabelian 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 thisintroduction.
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. It constitutes a

4

W. GAJDA AND S. PETERSEN

Isr. J. Math.

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 Fité’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 kfinite 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 bea dense open subscheme of S. Assume that
Endk(S) (A) = Endk(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 conjectureof 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 casewhen 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 overglobal 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)
Ak(S) → Bk(S) .
(b) For every closed point s ∈ S there exists a surjective k(s)-homomorphism (resp. k(s)-isogeny)
As,k(s) → Bs,k(s) .

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

5Structure of the paper. In Sections 2 and 3 we gathered material which isneeded 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 ofk finite, m=0 and work under extra assumptions, either the minuscule weights
conjecture or trivial endomorphisms for generic fibers.
Acknowledgments.
The authors were supported by a research grantUMO-2018/31/B/ST1/01474 of the National Centre of Sciences of Poland.
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 themanuscript and for valualble comments.

2. 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 denoteby 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 isviewed 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 ] 6= ∅}.
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 K(S) instead of R(S). If T
is a finite free Zℓ -module, V = T ⊗Zℓ Qℓ and Γ is a subgroup GLT (Zℓ ), then wedenote by ΓZar the Zariski closure of Γ inside the algebraic group GLV /Qℓ sothat ΓZar is an algebraic group of Qℓ . If G is an algebraic group over Qℓ , thenwe denote by G◦ the connected component of the identity element of G.

6

W. GAJDA AND S. PETERSEN

Isr. J. Math.

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 isan E-isomorphism AE → BE . In subsequent sections we are mainly interested
in the case E = K. We denote by TwistE/K (A) the set of all isomorphismclasses of E/K-twists of A and define Twist(A) := TwistK/K (A). There arenatural operations ρA : Gal(K) → AutK (AE ) and ρB : Gal(K) → AutK (BE )and an operation
Gal(K) × HomE (AE , BE ) → HomE (AE , BE ),σ

f := ρB (σ) ◦ f ◦ ρA (σ)−1 .

Now assume that B/K is an E/K-twist of A and choose an E-isomorphism
f : AE → BE . Then
ξ : Gal(E/K) → AutE (AE ),ξ(σ) = f −1 ◦ σ f
is a 1-cocycle (i.e., ξ(ση) = ξ(σ) ◦ σ ξ(η)) whose cohomology class does notdepend on the choice of f . Let
ρ̃B : Gal(E/K) → AutK (AE ),ρ̃B (σ) = f −1 ◦ ρB (σ) ◦ f
be the operation on A derived from ρB via transport of structure via f . Thenan easy calculation shows that(1)ρ̃B (σ) = ξ(σ)ρA (σ)∀σ ∈ Gal(E/K)

is simply the action ρA twisted by the 1-cocycle ξ. It is well-known that thissets up a bijective map
ΞE/K : TwistE/K (A) → H 1 (Gal(E/K), AutE (AE )).
There is a natural map
j : H 1 (Gal(E/K), {±IdA }) → H 1 (Gal(E/K), AutE (AE )).
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 ΞE/K . We call B a quadratic
isogeny twist of A, if there exists a quadratic characterχ ∈ Hom(Gal(K/K), {±IdA })and a K-isogeny
B → Aχ .
Sometimes we tacitly identify {±IdA } with µ2 (Q) = {±1}.

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

7

Remark 2.1: Let A be an abelian variety over a finite field K. Then there isa unique non-trivial quadratic character Gal(K) → {±IdA } and accordingly aunique 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/Kan abelian variety. For a rational prime ℓ 6= p we denote by
ρA,ℓ : Gal(K) → GLVℓ (A) (Qℓ )the corresponding ℓ-adic Galois representation, where Tℓ (A) is the ℓ-adic Tate
module of A and Vℓ (A) = Tℓ (A)⊗Zℓ Qℓ . Then the Zariski closure ρA,ℓ (Gal(K))Zarof ρA,ℓ (Gal(K)) in GLVℓ (A) and its identity component (ρA,ℓ (Gal(K))Zar )◦ are
algebraic groups over Qℓ .
If K is finite, then we define the L-series of A by
L(A/K, T ) := det(IdVℓ (A) − ρA,ℓ (Fr)T )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 6= 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 Vℓ (f ) : Vℓ (A) → Vℓ (B) be the homomorphism of Qℓ -vector
spaces induced by f . Then:(a) dimQℓ (im(Vℓ (f ))) = 2 dim(im(f )).
(b) dimQℓ (ker(Vℓ (f ))) = 2 dim(ker(f )).
(c) f : A → B is an isogeny if and only if the map Vℓ (f ) is bijective. In
particular, we have ρA,ℓ ∼= ρB,ℓ provided B is K-isogenous to A.
Proof. We note that Tℓ (A) = Hom(Qℓ /Zℓ , A(K)). For the purpose of that
proof we put Tℓ (M ) := Hom(Qℓ /Zℓ , M ) for an arbitrariy abelian group M .
Let C = ker(f ) and I = im(f ). Then C ◦ and I are abelian varieties. From theexact sequence
0 → C ◦ (K) → C(K) → C(K)/C ◦ (K) → 0
we derive an exact sequence
0 → Tℓ (C ◦ (K)) → Tℓ (C(K)) → Tℓ (C(K)/C ◦ (K))

8

W. GAJDA AND S. PETERSEN

Isr. J. Math.

because the functor Hom(Qℓ /Zℓ , −) is left exact. But Tℓ (C(K)/C ◦ (K)) = 0
because the group C(K)/C ◦ (K) is finite. Thus Tℓ (C ◦ (K)) = Tℓ (C(K)). Fromthe exact sequence
0 → C(K) → A(K) → I(K) → 0(2)

we obtain an exact sequence 0 → Tℓ (C(K)) → Tℓ (A(K)) → Tℓ (I(K)), again
because the functor Hom(Qℓ /Zℓ , −) is left exact. Using Tℓ (C ◦ (K)) = Tℓ (C(K))
and tensoring with Qℓ we see that we have an exact sequence
0 → Vℓ (C ◦ ) → Vℓ (A) → Vℓ (I).

(3)

Let c = dim(C), a = dim(A) and i = dim(I). Then a = i + c by (2).
Also dimQℓ (Vℓ (C)) = 2c, dimQℓ (Vℓ (A)) = 2a and dimQℓ (Vℓ (I)) = 2i (cf. [15,Remark 8.4]). The image of the right hand map of the exact sequence (3)has dimension 2a − 2c = 2i, too, hence Vℓ (A) → Vℓ (I) is surjective. By theleft exactness of Hom(Qℓ /Zℓ , −) we also see that Vℓ (I) → Vℓ (B) is injective.
Thus Vℓ (I) = im(Vℓ (f )) and Vℓ (C ◦ ) = ker(Vℓ (f )). It follows that
dimQℓ (ker(Vℓ (f )))= dimQℓ (Vℓ (C ◦ ))=2 dim(C ◦ )= 2 dim(C)=2 dim(ker(f )),dimQℓ (im(Vℓ (f )))= dimQℓ (Vℓ (I))=2 dim(I)=2 dim(im(f )),as desired. This finishes the proof of (a) and (b). For (c) put b := dim(B) andnote that:f is an isogeny ⇔ f is surjective with a finite kernel⇔ dim(ker(f ))=0 and dim(im(f ))=b
⇔ dimQℓ (ker(Vℓ (f )))=0 and dimQℓ (im(Vℓ (f )))=2b⇔ Vℓ (f ) is bijective.
There is the following celebrated result due to Faltings, Tate and Zarhin.
Theorem 2.3 (cf. [7], [22], [23]): Let K be a finitely generated field of characteristic p ≥ 0 and A/K and B/K be abelian varieties.
(a) ρA,ℓ is semisimple for every rational prime ℓ 6= p.
(b) The natural homomorphism
HomK (A, B) ⊗ Zℓ → HomGal(K) (Tℓ (A), Tℓ (B))
is bijective.

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

9Corollary 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 : A → B.
(b) There exists an isomorphism ρA,ℓ ∼=ρB,ℓ of Qℓ -representations of Gal(K).
If K is finite, then conditions (a) and (b) are also equivalent to the following
condition:(c) L(A/K, T ) = L(B/K, T ).
Proof. The implication (a)⇒(b) is Lemma 2.2. The implication (b)⇒(a) is
a consequence of Theorem 2.3, cf. [7, Korollar 2]. If K is finite, then theequivalence (b)⇔(c) has been established by Tate [22, Theorem 1].
Remark 2.5: Let K be a finitely generated field of characteristic p ≥ 0 and A/Kan abelian variety. Let χ : Gal(K) → {±IdA } be a quadratic character. Then:(a) ρAχ ,ℓ ∼= χ ⊗ ρA,ℓ .
(b) If K is finite and χ is the (unique) non-trivial quadratic characterof Gal(K), then L(Aχ /K, T ) = L(A/K, −T ).
Proof. (a) is immediate from equation (1). Part (b) is an immediate consequence thereof.
Lemma 2.6: Let K be a finitely generated field of characteristic p ≥ 0. Let A, B
be abelian varieties over K and ℓ 6= 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 : Gal(K) → {±IdA } such that
ρB,ℓ ∼= χ ⊗ ρA,ℓ .
If K is finite, then the equivalent conditions (a) and (b) are also equivalent to(c) L(B/K, T ) = L(A/K, T ) or L(B/K, T ) = L(A/K, −T ).
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 } anda K-isogeny B → Aχ . From Lemma 2.2 and Remark 2.5 we conclude that
ρB,ℓ ∼= χ ⊗ ρA,ℓ .
= ρAχ ,ℓ ∼

10

W. GAJDA AND S. PETERSEN

Isr. J. Math.

For the proof of the implication (b)⇒(a) assume that there exists a quadratic
character χ : Gal(K) → {±IdA } such that ρB,ℓ ∼= χ ⊗ ρA,ℓ . From Remark 2.5∼
we conclude that ρB,ℓ = ρAχ ,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 ρB,ℓ ∼= ρA,ℓand hence L(A/K, T ) = L(B/K, T ). If χ is non-trivial, then ρB,ℓ ∼= ρAχ ,ℓ , hence
L(B/K, T ) = L(Aχ , T ) = L(A/K, −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 isisogenuous to B by Corollary 2.4. If L(B/K, T ) = L(A/K, −T ) and χ thenontrivial 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 2.7: Let A and B be abelian varieties over a field K and ℓ a prime
different from the characteristic of K. Note that
ρA×B,ℓ (σ) = ρA,ℓ (σ) × ρB,ℓ (σ)

for all σ ∈ Gal(K).

Thus
ρA×B,ℓ (Gal(K)) ⊂ GLVℓ (A) (Qℓ ) × GLVℓ (B) (Qℓ ) ⊂ GLVℓ (A)×Vℓ (B) (Qℓ ).
Let pA and pB be the projections of the product GLVℓ (A) (Qℓ ) × GLVℓ (B) (Qℓ ).
Then
ρA,ℓ (σ) = pA (ρA×B,ℓ (σ))

and ρB,ℓ (σ) = pB (ρA×B,ℓ (σ))

for all σ ∈ Gal(K). Thus the actions of Gal(K) on Vℓ (A) and Vℓ (B) and theinduced action on HomQℓ (Vℓ (A), Vℓ (B)) factor through G = ρA×B,ℓ (Gal(K)).
Let G = ρA×B,ℓ (Gal(K))Zar . If K is finitely generated, then the natural map(4) HomK (A, B) ⊗Z Qℓ → HomQℓ (Vℓ (A), Vℓ (B))G = HomQℓ (Vℓ (A), Vℓ (B))Gis bijective by the Tate conjecture (cf. Theorem 2.3).
The following lemma and its proof are inspired by [20, Prop 2.10].

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

11

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 (ρA×B,ℓ )−1 (G◦ ).
Then
(5)

HomK ◦ (AK ◦ , BK ◦ ) = HomK (AK , BK ).

If K is finitely generated, then the above specialization map (4) induces abijective map(6)◦

HomK (AK , BK ) ⊗Z Qℓ → HomQℓ (Vℓ (A), Vℓ (B))G .

Proof. For (5) it is enough to prove that
HomK ◦ (AK ◦ , BK ◦ ) = HomK ′ (AK ′ , BK ′ )for every finite Galois extension K ′ /K with K ◦ ⊂ K ′ . The profinite group
Gal(K ′ ) is of finite index in Gal(K) and thus
ρA×B,ℓ (Gal(K ′ ))Zar
is a closed subgroup scheme of finite index in G◦ . Thus finitely many cosets ofρA×B,ℓ (Gal(K ′ ))Zar cover G◦ . As G◦ is connected, it follows that
ρA×B,ℓ (Gal(K ′ ))Zar = G◦ .
Let H = HomQℓ (Vℓ (A), Vℓ (B)) and consider the diagram
HomK ◦ (AK ◦ , BK ◦ )

HG

◦

Id

HomK ′ (AK ′ , BK ′ )HG◦

H Gal(K )Id◦′

H Gal(K )The horizontal arrows are injective (cf. Theorem [15, Prop. 12.2]). Thus, forevery f ∈ HomK ′ (AK ′ , BK ′ ), the map Vℓ (f ) ∈ H is invariant under Gal(K ◦ )and, by the injectivity of the horizontal maps of the diagram, f itself is invariant
under Gal(K ◦ ) which implies f ∈ HomK ◦ (AK ◦ , BK ◦ ). This finishes up the proofof (5). If K is finitely generated, then (5) together with the Tate conjecture(cf. Theorem 2.3) implies that (6) is bijective.

12

W. GAJDA AND S. PETERSEN

Isr. J. Math.

3. 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 pointut : Spec(E) → T −→ S
of S. We fix throughout this section a rational prime ℓ ∈ L(T ). This is possibleby the following lemma. We note that automatically ℓ ∈ L(S).
Lemma 3.1: For every scheme X, the set L(X) is non-empty.
Proof. If there exists p ∈ L ∪ {0} such that char(k(x)) = p for all x ∈ X,
then L(X) = L \ {p} is not empty. Otherwise, there exist x1 , x2 ∈ X suchthat char(k(x1 )) 6= char(k(x2 )) and char(k(x1 )) > 0, hence puttingℓ := char(k(x1 ))we get x2 ∈ X[ℓ−1 ] and thus ℓ ∈ L(X), so that L(X) is non-empty also in thatcase.
Let A /S be an abelian scheme with generic fiber A/F . Let AT := A ×S T
and let AT := A ×S Spec(E) be the generic fiber of AT → T . We then havecartesian squares
AT

AT

Spec(E)

T

u

A

A

S

Spec(F )Let n ∈ N. Then the restriction of A [ℓn ] to S[ℓ−1 ] is a finite étale S[ℓ−1 ]scheme of rank ℓ2 dim(A)n . In particular, the action of Gal(F ) on A[ℓn ](F ) factors
through π1 (S[ℓ−1 ]). Likewise ρA,ℓ factors through π1 (S[ℓ−1 ]), i.e., we can view
it as a homomorphism
ρA,ℓ : π1 (S[ℓ−1 ]) → GLVℓ (A) (Qℓ ).
There exists a finite connected étale cover S ′ → S[ℓ−1 ] such that A [ℓn ] × S ′splits up in a coproduct of ℓ2n dim(A) copies of S ′ , i.e., it is a constant groupVol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

13

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 [ℓn ] ×S S ′ is a constant group scheme, thenatural maps
A[ℓn ](F ) ← A [ℓn ](S ′ ) → AT [ℓn ](E)derived from these points are bijective. We thus get a specialization isomorphism
sA,T,ℓn : A[ℓn ](F ) → AT [ℓn ](E)and accordingly a specialization isomorphism
sA,T,ℓ∞ : Tℓ (A) → Tℓ (AT ).
These are equivariant for the action of π1 (S[ℓ−1 ]). In particular we havedim(A) = dim(AT ).
Definition 3.2: Let G be a group, S0 an open subscheme of S and ρ : π1 (S0 ) → G
a homomorphism. Let T0 := u−1 (S0 ). We define u∗ ρ to be the composite
morphism1u∗
π1 (S0 ) → G.
π1 (T0 ) −→If u is clear from the context we put ρT := u∗ ρ. We say that T (or u) isρ-generic if T0 is not empty andρT (π1 (T0 )) = ρ(π1 (S0 )).
We often apply this notation in the case where S0 = S[ℓ−1 ] and ρ = ρA,ℓ .
Note that in that case u−1 (S0 ) = T [ℓ−1 ] is automatically non-empty by our
choice of ℓ.
Remark 3.3: The specialization isomorphism sA,T,ℓ∞ : Tℓ (A) → Tℓ (AT ) gives
an isomorphism∼ ρA ,ℓ .
u∗ ρA,ℓ =Tof representations of Gal(E).
For the rest of this section let B be an abelian scheme over S with genericfiber B/F . Define BT := B ×S T and let BT := B ×T Spec(E) be the genericfiber of BT → T .
1 Note: The fundamental group of the empty scheme is the trivial group.

14

W. GAJDA AND S. PETERSEN

Isr. J. Math.

Lemma 3.4: The canonical map
HomS (A , B) → HomF (A, B), f 7→ fF
is bijective.
Proof. Let U be the category of all dense affine open subschemes of S. Then
Spec(F ) = lim U←−
U∈U(projective limit of schemes, cf. [11, 8.2]). It follows from [11, 8.8.2] that thenatural map(7)

lim HomU (AU , BU ) → HomF (A, B),−→

U∈U

induced by the natural maps HomU (AU , BU ) → HomF (A, B), f 7→ fF , isbijective. Moreover, by [8, Prop. 2.7], for every U ∈ U the natural map
HomS (A , B) → HomU (AU , BU ), f 7→ fU
is bijective, and therefore the natural map(8)

HomS (A , B) → lim HomU (AU , BU )−→U∈U

is bijective. The assertion is immediate from the bijectivity of (7) and (8).
Lemma 3.5: The canonical map(9)

HomS (A , B) → HomT (AT , BT )is injective.
Proof. Choose a point t ∈ T . We even prove that the composite homomorphism(10)

HomS (A , B) → HomT (AT , BT ) → Homk(t) (At , Bt )is injective. Let f be in the kernel of (10). Thenft : A ×S Spec(k(t)) → B ×S Spec(k(t))
is the zero homomorphism. Hence f must be the zero homomorphism by therigidity lemma [15, 20.1].

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

15

Note that, by Lemmata 3.4 and 3.5 above, there is a canonical injective
specialization map(11)

rA ,B,T,S : HomF (A, B)∼=HomS (A , B)−→ HomT (AT , BT ) ∼= HomE (AT , BT ).

Let ρℓ := ρA×B,ℓ ,G := ρℓ (π1 (S[ℓ−1 ]))Zar
and let S ′ be the finite étale cover of S[ℓ−1 ] corresponding to the subgroup◦−1
ρ−1]). Let F ′ be the function field of S ′ . Then, by Lemma 2.8,ℓ (G ) of π1 (S[ℓHomF ′ (AF ′ , BF ′ ) = HomF (AF , BF ).
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
(12)

r A ,B,T,S :HomF (AF , BF ) = HomF ′ (AF ′ , BF ′ )rA ,B,T ′ ,S ′−→HomE ′ (AT,E ′ , BT,E ′ ) → HomE (AT,E , BT,E )which tacitly depends on the choice of T ′ and on the choice of an embedding
E ′ → E. This map
rA ,B,T,S : HomF (AF , BF ) → HomE (AT,E , BT,E )is injective.
Lemma 3.6: Let f : A → B be a homomorphism. Let fT : AT → BT be thehomomorphism fT := rA ,B,T,S (f ) obtained from f by specialization.
(a) dim(im(f )) = dim(im(fT )),(b) dim(ker(f )) = dim(ker(fT )),(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
Vℓ (A)Vℓ (AT )Vℓ (f )Vℓ (fT )Vℓ (B)Vℓ (BT )16

W. GAJDA AND S. PETERSEN

Isr. J. Math.

whose vertical arrows are bijective. Together with Lemma 2.2 we get:11dim(ker(Vℓ (f ))) = dim(ker(Vℓ (fT ))) = dim(ker(fT )),2211
dim(im(f )) = dim(im(Vℓ (f ))) = dim(im(Vℓ (fT ))) = dim(im(fT )).
22

dim(ker(f )) =This proves (a) and (b), and (c) is immediate from that.
Remark 3.7: For f ∈ HomF (AF , BF ) 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 .
Lemma 3.8: If both fields E and F are finitely generated and u : T → S is
ρA×B,ℓ -generic, then the canonical maps
rA ,B,T,S ⊗ Zℓ : HomF (A, B) ⊗ Zℓ → HomE (AT , BT ) ⊗ Zℓandr A ,B,T,S ⊗ Zℓ : HomF (AF , BF ) ⊗ Zℓ → HomE (AT,E , BT,E ) ⊗ Zℓ
are bijective. In particular, under these assumptions coker(rA ,B,T,S ) andcoker(r A ,B,T,S ) are finite groups of order prime to ℓ.
Proof. We identify, Tℓ (A) with Tℓ (AT ) and Tℓ (B) with Tℓ (BT ) along the naturaland equivariant specialization isomorphisms. Let ρℓ := ρA×B,ℓ ,G := ρℓ (π1 (S[ℓ−1 ]))Zar

and GT := (ρℓ )T (π1 (T [ℓ−1 ]))Zar .

Let
rℓ = rA ,B,T,S ⊗ Zℓ
and rℓ = r A ,B,T,S ⊗ Zℓ . We then have a commutative diagram
HomF (A, B) ⊗ Zℓ

HomZℓ (Vℓ (A), Vℓ (B))Grℓ

HomE (AT , BT ) ⊗ Zℓ

HomZℓ (Vℓ (A), Vℓ (B))GT ,where the vertical maps are bijective (cf. Tate conjecture, Remark 2.7). The
lower horizonal map is bijective because G = GT by our assumption that T isρℓ -generic. Hence the upper horizontal map rℓ is bijective, too.

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

17

Next consider the commutative diagramrℓ

HomF (AF , BF ) ⊗ Zℓ

HomZℓ (Vℓ (A), Vℓ (B))G◦HomE (AT,E , BT,E ) ⊗ Zℓ◦

HomZℓ (Vℓ (A), Vℓ (B))GT

where the vertical maps are bijective (cf. Tate conjecture, Lemma 2.8). The
lower horizonal map is bijective because
G◦ = G◦T .
Hence, rℓ is bijective. The Z-module HomE (AT,E , BT,E ) is free and finitely
generated. Hence, the statement about the cokernels of rA ,B,T,S and r A ,B,T,S
follows as well.

4. 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 isassumed to be smooth. Note that S and Z are reduced and irreducible. Let dbe 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 connectedsubschemes 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 thatcase S is simply a smooth k-variety.
(2) The case where Z is an open subscheme of Spec(R) and R is the ringof integers in a number field; in that case S is sometimes called anarithmetic scheme.
The following lemma is inspired by Drinfeld’s “conventional formulation ofHilbertianity” in [6, Section A.1.].

18

W. GAJDA AND S. PETERSEN

Isr. J. Math.

Lemma 4.1: If K is Hilbertian, then for every dense open subscheme U of S andevery finite étale morphism p : X → U there exists a connected subscheme Tof 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 × Ad
(cf. [12, Exposé II, §1]). For any dense open subscheme Z ′ of Z we can consider
the following commutative diagram of schemes
XK

X′

pK

p′

UK

U′

gK

g′

Ad,K

Spec(K)Z ′ × Ad

Z ′,where U ′ = U ×Z Z ′ , X ′ = X ×Z Z ′ and g ′ and p′ are the restrictions of g andp respectively. As K is Hilbertian there is a point a ∈ Ad (K) such that(gK ◦ pK )−1 (a) = Spec(F )−1and gK(a) = Spec(E)where E/K and F/E are finite separable field extensions. For a suitable choiceof Z ′ the closed immersion a : Spec(K) → Ad,K extends to a closed subscheme Yof Z ′ × Ad (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 thatthe maps p|XT : XT → T and g : T → Y are finite and étale, because thecorresponding maps on the generic fibers−1(gK ◦ pK )−1 (a) = Spec(F ) → gK(a) = Spec(E) → Spec(K)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 fiber−1(a) = Spec(E)T ×Z ′ Spec(K) = gK

is connected. Likewise the closed subscheme XT of X ′ is connected, because it
is finite and étale over Z ′ and its generic fiber XT ×Z ′ Spec(K) = Spec(F ) isconnected.

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

19

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 existsa T ∈ Smm (S/Z) such that T ⊂ U and such that T is ρ-generic. If K is
Hilbertian, then the same holds true for m = 0.
Proof. We consider the homomorphismρρ′ : π1 (S) −→ G → G/Φ(G).
The image of ρ′ is finite because Φ(G) is open in G. 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 coverof 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 issmooth we can, after replacing U by a smaller open set, assume that there exists
an étale Z-morphism U → Z × Ad . Composing with an appropriate projection
we get a smooth morphism
S → Z × Ad → Z × Am .
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.
Remark 4.3: If G is a compact subgroup of GLn (Qℓ ), then Φ(G) is open in G
(cf. [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 ρA×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 fiber of AT → T (resp. of BT → T ). Finally,
for ℓ ∈ L(S) we defineρℓ := ρA×B,ℓ .
The following lemma is at the core of the rest of the arguments of this section.

20

W. GAJDA AND S. PETERSEN

Isr. J. Math.

Lemma 4.4: Let m ∈ {0, 1, . . . , d} and T ∈ Smm (S/Z). Let ∆1 , ∆2 be subsets of N. If there exists an R(T )-homomorphism f : AT → BT (resp. R(T )homomorphism f : AT,R(T ) → BT,R(T ) ) such thatdim(ker(f )) ∈ ∆1anddim(im(f )) ∈ ∆2

and if T is ρℓ -generic, then there exists an R(S)-homomorphism F : A → B(resp. R(S)-homomorphism F : AR(S) → BR(S) ) such thatdim(ker(F )) ∈ ∆1anddim(im(F )) ∈ ∆2 .

Proof. Let f : AT → BT be an R(T )-homomorphism such that we havedim(ker(f ))∈∆1 and dim(im(f ))∈∆2 , and assume that T is ρℓ -generic. By
Lemma 3.8 the specialization maps
r := rA ,B,T,S : HomR(S) (A, B) → HomR(T ) (AT , BT ),r := r A ,B,T,S : HomR(S) (AR(S) , BR(S) ) → HomR(T ) (AT,R(T ) , BT,R(T ) ),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 an R(S)-homomorphism
F : A → B such that r(F ) = s ◦ f . Using Lemma 3.6 once more, we see thatdim(ker(F )) = dim(ker(s ◦ f )) = dim(ker(f )) ∈ ∆1 ,dim(im(F )) = dim(im(s ◦ f )) = dim(im(f )) ∈ ∆2 .
The proof of the respective 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.
Theorem 4.5: Let U be a dense open subscheme of S. Let m ∈ {0, 1, . . . , d}.
Assume that K is Hilbertian or that m ≥ 1. Let ∆1 and ∆2 be subsets of N.
The following are equivalent:(a) There exists an R(S)-homomorphism F : A → B (resp. R(S)-homomorphism F : AR(S) → BR(S) ) such that dim(ker(F )) ∈ ∆1 and one hasdim(im(F )) ∈ ∆2 .
(b) For every T ∈ Smm (S/Z) with T ⊂ U there exists a R(T )-homomorphism f : AT → BT (resp. R(T )-homomorphism f : AT,R(T ) → BT,R(T ) )such that dim(ker(f )) ∈ ∆1 and one has dim(im(f )) ∈ ∆2 .

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

21

Proof. Assume (a) holds true and let T ∈ Smm (S/Z). There exists ℓ ∈ L(T )by Lemma 3.1. The existence of f as in (b) is then immediate from themere existence of the specialization maps rA ,B,T,S and r A ,B,T,S (cf. definitions of maps (11) and (12)) plus the fact that they “respect dimension data”(cf. Lemma 3.6 and Remark 3.7). This proves (a)⇒(b).
We now prove the implication (b)⇒(a). Assume (b) is true. Choose ℓ ∈ L(U )(cf. Lemma 3.1). By Corollary 4.2 (applied with S[ℓ−1 ] instead of S andwith U [ℓ−1 ] instead of U )) there exists T ∈ Smm (S[ℓ−1 ]/Z) such that
T ⊂ U [ℓ−1 ]and such that T is ρℓ -generic. By (b) there exists an R(T )-homomorphism
f : AT → BT (resp. R(T )-homomorphism f : AT,R(T ) → BT,R(T ) ) such thatdim(ker(f )) ∈ ∆1 and dim(im(f )) ∈ ∆2 . Now (a) follows by Lemma 4.4.
Proof of Theorem A. Apply Theorem 4.5 with
∆1 ={0} and ∆2 ={dim(B)}(resp. with ∆1 =N and ∆2 ={dim(B)}, resp. with ∆1 ={0, 1, · · ·, dim(A)−1} and∆2 =N, resp. with ∆1 ={κ} and ∆2 =N).
Corollary 4.6: Let U be a dense open subscheme of S. Let m ∈ {0, 1, · · ·, d}.
Assume that K is Hilbertian or that m ≥ 1.
(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 fiber AT is not a simple
R(T )-variety.
(b) The following are equivalent:(i) AR(S) is not a simple R(S)-variety.
(ii) For every T ∈ Smm (S/Z) with T ⊂ U the fiber AT,R(T ) is not a
simple R(T )-variety.
Proof. Note that A is non-simple if and only if there existsκ ∈ {1, 2, . . . , dim(A)−1}
and a homomorphism A → A with κ-dimensional kernel. Thus the corollary is
a formal consequence of Theorem 4.5.

22

W. GAJDA AND S. PETERSEN

Isr. J. Math.

Theorem 4.7: 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 aquadratic isogeny twist of BT .
The implication (a)⇒(b) holds true also in the case where Z = Spec(k) with afinite 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 . Chooseℓ ∈ L(T ) (cf. Lemma 3.1). It follows by Lemma 2.6, that there exists aχ ∈ Hom(Gal(R(S)), {±IdB }) such that
ρA,ℓ ∼= χ ⊗ ρB,ℓ .
For every σ ∈ ker(Gal(R(S)) → π1 (U [ℓ−1 ])) we haveρA,ℓ (σ) = IdTℓ (A)and ρB,ℓ (σ) = IdTℓ (B) ,hence χ(σ) = +IdB , i.e., χ factors through π1 (U [ℓ−1 ]). Now we haveρAT ,ℓ ∼= χT ⊗ ρBT ,ℓ= u∗ ρA,ℓ = u∗ (χ ⊗ ρB,ℓ ) = u∗ χ ⊗ u∗ ρB,ℓ ∼where u:T →S is the embedding and χT :π1 (T [ℓ−1 ])→{±IdBT } is a quadratic
character. By Lemma 2.6 AT is a quadratic isogeny twist of BT .
We now prove the other implication (b)⇒(a): Assume that for every
T ∈ Smm (S/Z) with T ⊂ U the fiber AT is a quadratic isogeny twist of BT .
Let ℓ ∈ L(U ) (cf. Lemma 3.1). By Corollary 4.2 there exists T ∈ Smm (S/Z)
with T ⊂ U [ℓ−1 ] such that T is ρA×B,ℓ -generic. Let
K = ker(ρA×B,ℓ : π1 (S[ℓ−1 ]) → GLTℓ (A×B) (Zℓ ))and
KT = ker(ρA×B,ℓ ◦ u∗ ).
Then u∗ : π1 (T ) → π1 (S[ℓ−1 ]) induces an isomorphism
π1 (T )/KT → π1 (S[ℓ−1 ])/K .
By Lemma 2.6 there exists χT ∈ Hom(π1 (T ), {±IdBT }) such that
ρAT ,ℓ ∼= χT ⊗ ρBT ,ℓ .

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

23

It follows thatχT (σ) = +IdBT ,for every σ ∈ KT , because σ ∈ ker(ρAT ,ℓ ) ∩ ker(ρBT ,ℓ ). Hence χT factors
through π1 (T )/KT and thus induces a quadratic character χ of π1 (S[ℓ−1 ])/K
and thus of π1 (S[ℓ−1 ]), such that
ρA,ℓ ∼=χ ⊗ ρB,ℓ .
Thus (a) holds true by Lemma 2.6.

5. 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 handledwith 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 resultsof Khare–Larsen [13] and Fité [9].
Definition 5.1: Let A be an abelian variety over a finitely generated field K and
let ℓ 6= char(K) be a rational prime. Let Gℓ be the connected component of the
Zariski closure of ρA,ℓ (Gal(K)). We say that A satisfies condition M W C(A)if for all rational primes ℓ 6= char(K) the action of Gℓ,Qℓ on each irreducible
factor of the representation Vℓ (A)⊗Qℓ Qℓ is minuscule in the sense of Bourbaki [1]cf. [13, p. 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 AK ′ satisfies
M W C(AK ′ ) because the connected component of the Zariski closureof ρA,ℓ (Gal(K ′ )) agrees with the connected component of the Zariski
closure of ρA,ℓ (Gal(K)) as Gal(K ′ ) is of finite index in Gal(K).
(b) If f : A → B is a surjective K-homomorphism, then B satisfies
M W C(B), because Vℓ (f ) : Vℓ (A) → Vℓ (B) is a surjective homomorphism of representations.
(c) If B is an abelian subvariety of A, then there exists surjective homomorphism A → B by [15, Prop. 12.1] and thus B satisfies M W C(B)by (b).

24

W. GAJDA AND S. PETERSEN

Isr. J. Math.

Remark 5.3: If K is a number field, then M W C(A) holds true for every abelianvariety 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] proved his conjecture 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]. On the other hand, the analogue of the conjecture of Zarhin inpositive characteristic seems to fail in general, as indicated by a result of Bültel(cf. [2, Thm. 1.2], [2, Comments and examples, p. 637]) who constructed anabelian variety of dimension 56 over a finitely generated field of positive characteristic with a simple factor of exceptional type G2 (there are no minusculeweights in this case !) in its ℓ-adic monodromy group.2
The following is a global function field analogue of a result of Khare andLarsen [13, Thm. 1]. We include the proof for the reader’s convenience.
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)-homomorphismAs,k(s) → Bs,k(s) has Dirichlet density 1, then there exists a non-zero k(S)homomorphism Ak(S) → Bk(S) .
Proof. Let K = R(S). Fix one rational prime ℓ 6= char(K). After replacing S
by one of its connected finite étale covers, we can assume that the Zariski
closures GA of ρA,ℓ (Gal(K)), GB of ρB,ℓ (Gal(K)) and G of ρA×B,ℓ (Gal(K)) areconnected. The action of GQℓ on VA := Vℓ (A)⊗Qℓ Qℓ and on VB := Vℓ (B)⊗Qℓ Qℓis minuscule. By [14, Thm. 1.2] and [18] the set of all closed points s ∈ S suchthat the Frobenius element F rs ∈ G(Qℓ ) generates a Zariski dense subgrouphF rs i of a maximal torus of G has positive density. Thus we can choose a closed
2 For more examples of abelian varieties of this type the reader can consult [3] whichappeared in the arXiv after our paper had been submitted for publication.

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

25

point s ∈ S such that hF rs i is a Zariski dense subgroup of a maximal torus T
of G and such that there exists a non-zero homomorphism f : As,k(s) → Bs,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 rsm , and F rsm
also generates a Zariski dense subgroup of T. It follows that the map
Vℓ (f ) ⊗ Qℓ : VA → VB
induced by f is fixed by the torus T. Moreover, Vℓ (f )⊗ Qℓ is a non-zero elementof HomT (VA , VB ). Thus
dim(HomT (VA , VB )) > 0.
Now G is a reductive group because ρA×B,ℓ is semisimple. By our assumption
A satisfies M W C(A) and B satisfies M W C(B). Thus we can apply [13, Prop. 2]
to conclude thatdim(HomG (VA , VB )) > 0.
From HomG (VA , VB ) ∼= HomK (A, B) ⊗ Qℓ 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 withthe results of the previous section.
Theorem 5.5: Let S be a smooth variety over a finite field k. Let A , B beabelian 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 Ak(S) → Bk(S) .
(b) For every closed point s ∈ S there exists a non-zero k(s)-homomorphismAs,k(s) → Bs,k(s) .
Proof. After replacing S by one of its connected finite étale covers we canassume that
Homk(S) (A, B) = Homk(S) (Ak(S) , Bk(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 6= char(K)be a rational prime. By Corollary 4.2 there exists a T ∈ Sm1 (S/k) of S suchthat T is ρA×B,ℓ -generic. Then T is automatically ρA,ℓ -generic and ρB,ℓ -generic.
In particular,
ρA,ℓ (Gal(K))Zar = ρAT ,ℓ (Gal(k(T )))Zar .

26

W. GAJDA AND S. PETERSEN

Isr. J. Math.

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 Ak(t) → Bk(t) . By Proposition 5.4 there exists a non-zero homomorphism AT,k(T ) → BT,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 spiritof Corollary B.
Theorem 5.6: Let S be a smooth variety over a finite field k. Let A , B beabelian 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)
Ak(S) → Bk(S) .
(b) For every closed point s ∈ S there exists a surjective k(s)-homomorphism (resp. k(s)-isogeny) As,k(s) → Bs,k(s) .
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
Homk(S) (A, B) = Homk(S) (Ak(S) , Bk(S) ),Endk(S) (A) = Endk(S) (Ak(S) ),(13)

Endk(S) (B) = Endk(S) (Bk(S) ).
By the Poincaré reducibility theorem (cf. [15, Prop. 12.1] and the passage
Q
Q
below) A (resp. B) is k(S)-isogenous to i∈I Ani i (resp. j∈J Bjmi ) 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 diagram0
Endk(S)(Ak(S) )Q0ni ×nji,j∈I Homk(S) (Ai,k(S) , Aj,k(S) )Q

ni ×nji,j∈I sij

End0k(S) (Ak(S) )Q

0ni ×nji,j∈I Homk(S) (Ai , Aj )where thesij : Hom0k(S) (Ai , Aj ) → Hom0k(S) (Ai,k(S) , Aj,k(S) )Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

27

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 6= j we have Ai 6≃ Aj , hence
0(Ai,k(S) , Aj,k(S) ) = Hom0k(S) (Ai , Aj ) = 0,
Homk(S)and thus the Ai,k(S) are mutually non-isogenous over k(S). As Ai is simple
it follows that End0k(S) (Ai,k(S) ) = End0k(S) (Ai ) is a division algebra over Q
and thus the Ai,k(S) are simple. Likewise the Bj,k(S) are simple and mutually non-isogenous over k(S). Note that M W C(Ai ) and M W C(Bj ) hold 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 forQevery closed point s ∈ S the fiber As,k(s) is k(s)-isogenous to i∈I Ani
i,s,k(s)Qm. In what follows we can thus
and Bs,k(s) is k(s)-isogenous to j∈J B j
j,s,k(s)QQmasssume, without loss of generality, that A = i∈I Ani i and that B = j∈J Bj j .
From (b) we know that for every closed point s ∈ S there exists a surjective
k(s)-homomorphism As,k(s) → Bs,k(s) and thus for every j ∈ J a surjective
homomorphism As,k(s) → Bj,s,k(s) .
By Theorem 5.5 there exists for every j ∈ J a non-zero homomorphism
Ak(S) → Bj,k(S) . Thus, for every j ∈ J, there must be a unique ν(j) ∈ I,
such that there exists a non-zero k(S)-homomorphism hj : Aν(j),k(S) → Bj,k(S)and hj must be a k(S)-isogeny, because Aν(j),k(S) and Bj,k(S) are both simple.
We now show that nν(j) ≥ mj , for all j ∈ J. Let j ∈ J be arbitrary. LetY
Ani i .
A′j =i∈I\{ν(j)}Then for every j ∈ J we havenν(j)A = Aν(j)× A′j .

There is no non-zero k(S)-homomorphism A′j,k(S) → Bj,k(S) , because the Ai,k(S)are mutually non-isogenous over k(S) and Aν(j),k(S) ≃ Bj,k(S) . The contraposition of Theorem 5.5 implies that there exists a closed point s ∈ S such that
there is no non-zero k(s)-homomorphism A′j,s,k(s) → Bj,s,k(s) . Furthermore,by (b), there is a surjective k(s)-homomorphismnAs,k(s) = A ν(j)

ν(j),s,k(s)× A′j,s,k(s) → B

mj
j,s,k(s)28

W. GAJDA AND S. PETERSEN

Isr. J. Math.

which restricts to a surjective k(s)-homomorphismnA ν(j)

ν(j),s,k(s)× {0} → B

mj.
j,s,k(s)As dim(Aν(j),s,k(s) ) = dim(Bj,s,k(s) ), this finally shows that nν(j) ≥ mj asclaimed above.
This together with the existence of the isogeny hj implies the existence ofnm→ B j , and these in turna surjective k(S)-homomorphism fj : A ν(j)
ν(j),k(S)induce a surjective k(S)-homomorphism
Y mY
Y nν(j)
→
B j
fj :
A
(14)
j∈J

j∈J

ν(j),k(S)j∈Jj,k(S)

j,k(S)= Bk(S) .

Composing the surjective k(S)-homomorphism (14) with the projection
Y nν(j)A
Ak(S) →j∈J

ν(j),k(S)we obtain a k(S)-epimorphism Ak(S) → Bk(S) as desired.
We next prove the implication (b)⇒(a) in case “isogeny”. In that case weknow by (b) that for every closed point s ∈ S there exists a k(s)-isogeny
gs : As,k(s) → Bs,k(s) .
The above arguments imply the existence of a surjective k(S)-homomorphismg : Ak(S) → Bk(S) . Let s ∈ S be a closed point. Then, by the existence of theisogeny gs ,dim(A) = dim(As,k(s) ) = dim(Bs,k(s) ) = dim(B),and thus g must automatically be an isogeny.
We shall now establish a global function field analogue of [9, Cor. 2.7], following proof in loc. cit. quite closely. For this we need the following theoremof Rajan [17]. Let K be a global field and denote by ΣK the set of all nonarchimedian discrete valuations of K. Let ℓ 6= char(K) be a rational prime
and let ρ1 : Gal(K) → GLn (Qℓ ) and ρ2 : Gal(K) → GLn (Qℓ ) be continuous
semisimple representations which are unramified outside a finite set S ⊂ ΣK .
DefineSM (ρ1 , ρ2 ) := {v ∈ ΣK : T r(ρ1 (F rv )) = T r(ρ2 (F rv ))}.

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

29

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 hold 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 thatρ2 ∼= χ ⊗Qℓ 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 6=char(k) be a rational prime and let K = k(S). Let U be a dense open subschemeof S. Assume that Endk(S) (A) = Endk(S) (B) = Z and that the Zariski closuresof ρA,ℓ (Gal(K)) and of ρB,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 twistof Bu . Assume that Γ has Dirichlet density 1. We claim that SM (ρℓ,A , ρℓ,B )has positive upper Dirichlet density. Consider the two sets
Γ± := {u ∈ Γ : T r(ρA,ℓ (F ru )) = ±T r(ρB,ℓ (F ru ))}.
For every u ∈ Γ we have T r(ρA,ℓ (F ru )) ∈ {±T r(ρB,ℓ(F ru ))} by Lemma 2.6(implication (a)⇒(c) applied over k(u)). Thus Γ = Γ+ ∪ Γ− . MoreoverΓ+ ⊂ SM (ρℓ,A , ρℓ,B ).
Because otherwise it would follow that Γ+ has upper Dirichlet density zero andthat Γ− has Dirichlet density 1. But then the Chebotarev density theorem
would imply that T r(ρA,ℓ (g)) = −T r(ρB,ℓ (g)) for all g ∈ Gal(K), which is
obviously false for g = Id. Thus the inclusion Γ+ ⊂ SM (ρℓ,A , ρℓ,B ) holds true.
Our assumption
Endk(S) (A) = Endk(S) (B) = Z
implies that ρA,ℓ and ρB,ℓ are absolutely irreducible.

30

W. GAJDA AND S. PETERSEN

Isr. J. Math.

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 suchthat ρA,ℓ = χ ⊗ ρB,ℓ . But now, by Theorem 2.3, we have an isomorphismof Gal(L/K)-modules
HomL (AL , BL ) ⊗ Qℓ ∼= Vℓ (A)∗ ⊗Qℓ Vℓ (B)∼= Vℓ (A)∗ ⊗Q Vℓ (A) ⊗ Qℓ (χ)ℓ∼= EndL (AL ) ⊗ Qℓ (χ) ∼= Qℓ (χ).
Thus Gal(L/K) acts on HomL (AL , BL ) ∼= Z via χ and this implies that χ isa 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.
Corollary 5.9: Let S be a smooth variety over a finite field k. Let A , B beabelian schemes over S with generic fibers A and B respectively. Let U be adense open subscheme of S. Assume that Endk(S) (A) = Endk(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 theother 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 connectedcurve T on U the abelian variety AT is a quadratic isogeny twist of BT by
Proposition 5.8. Thus Theorem 4.7 implies (a).
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, sincethen j-invariants j(A) and j(B) belong to k, hence the curves are isotrivial,
cf. [21, V.3.1 and Ex. V.5.8], so a twist between fibers 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).

Vol. TBD, XXXXHOMOMORPHISMS OF ABELIAN SCHEMES

31

References[1] N. Bourbaki, Éléments de mathématique. Groupes et algébres de Lie. Chapitre II:
Algèbres de Lie libres. Chapitre III: Groupes de Lie. Actualités Scientifiques et Industrielles, Vol. 1349, Hermann, Paris, 1972.
[2] O. Bültel, Construction of abelian varieties with given monodromy, Geometric and Functional Analysis 15 (2005), 634–696.
[3] O. Bültel, Further counterexamples to Zarhin’s conjecture about micro weights,
https://arxiv.org/abs/2207.08706.
[4] A. Cadoret and A. Tamagawa, Ghosts and families of abelian varieties with a common
isogeny factor, https://webusers.imj-prg.fr/~ anna.cadoret/Ghosts.pdf.
[5] L. Dixon, M. du Sautoy, A. Mann and D. Segal, Analytic pro-p Groups, Cambridge
Studies in Advanced Mathematics, Vol. 61, Cambridge University Press, Cambridge,
1999.
[6] V. Drinfeld, On a conjecture of Deligne, Moscow Mathematical Journal 12 (2012), 515–
542.
[7] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones
Mathematicae 73 (1983), 411–423.
[8] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 22, Springer, Berlin, 1990.
[9] F. Fité, On a local-global principle for quadratic twists of abelian varieties, Mathematische Annalen, https://doi.org/10.1007/s00208-022-02535-0.
[10] A. Grothendieck, Éléments de géométrie algébraique. I. Le langage des schémas, Institutdes Hautes Études Scientifiques. Publications Mathématiques 4 (1960).
[11] A. Grothendieck, Éléments de géométrie algébraique. IV. Étude locale des schémas etdes morphismes de schémas, Troisième partie, Institut des Hautes Études Scientifiques.
Publications Mathématiques 28 (1966).
[12] A. Grothendieck, Séminaire de Géométrie Algébrique du Bois-Marie SGA1 - Revêtementsétales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer, Berlin,1971.
[13] C. Khare and M. Larsen, Abelian varieties with isogenous reductions, Comptes Rendus
Mathématique. Académie des Sciences. Paris 358 (2020), 1085–1089.
[14] M. Larsen and R. Pink, A connectedness criterion for ℓ-adic Galois representations, Israel
Journal of Mathematics 97 (1997), 1–10.
[15] J. Milne, Abelian varieties, in Arithmetic Geometry (Storrs, Conn., 1984), Springer, New
York, 1986, pp. 103–150.
[16] R. Pink, ℓ-adic algebraic monodromy groups, cocharacters, and the Mumford–Tate conjecture, Journal für die Reine und Angewandte Mathematik 495 (1998), 187–237.
[17] C. S. Rajan, On strong multiplicity one for ℓ-adic representations, International Mathematics Research Notices 3 (1998), 161–172.
[18] J.-P. Serre, Quelques applications du théoréme de densité de Chebotarev, Institute desHautes Études Scientifiques. Publications Mathématiques 54 (1981), 323–401.
[19] J.-P. Serre, Lectures on the Mordell–Weil Theorem, Aspects of Mathematics, Vol. E51,
Viehweg & Sohn, Braunschweig, 1989.

32

W. GAJDA AND S. PETERSEN

Isr. J. Math.

[20] A. Silverberg and Y. Zarhin, Connectedness results for ℓ-adic representations associatedto abelian varieties, Compositio Mathematicae 97 (1995), 273–284.
[21] J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics,
Vol. 106, Springer, Dordrecht, 2009.
[22] J. Tate, Endomorphisms of abelian varieties over finite fields, Inventiones Mathematicae
2 (1966), 134–144.
[23] Y. Zarhin, Abelian varieties in finite characteristic, Central European Journal of Mathematics 12 (2014), 659–675.
[24] Y. Zarhin, Weights of simple Lie algebras in the cohomology of algebraic varieties, Mathematics of the U.S.S.R.-Izvestiya 24 (1985), 245–281.