Whenkis Hilbertian, the answer to this represention-theoretic reformulation of the original question is always positive (and, actually, much more holds)

FACTOR

ANNA CADORET AND AKIO TAMAGAWA

Abstract. We discuss (variants of) the following question raised by R¨ossler and Szamuely. Letk be a field of characteristicp ≥0, Sa geometrically connected variety over kwith generic pointη andA→S an abelian scheme. Assume all the closed fibers ofA→Shave a non-trivial commonk-isogeny factor. Does this imply that the geometric generic fiberAηhas a non-trivialk-isotrivialk(η)-isogeny factor? This question can be reformulated in representation-theoretic terms and, under this form, makes sense for instance for the ´etale cohomology group Vℓ := H^u(Xη,Qℓ(v)) (whereℓ is a prime̸=p) of a smooth proper schemeX →S. Whenkis Hilbertian, the answer to this represention-theoretic reformulation of the original question is always positive (and, actually, much more holds). The difficult case is whenkis finite. In this setting, we show that, after possibly replacingS by a connected ´etale cover, the weight 0 part of the representation of the geometric monodromy onVℓis endowed with a canonical semisimple representation of the absolute Galois groupπ1(k) ofk, which we call theghost attached to Vℓ. Forℓ varying, the ghosts form aQ-rational compatible system of semisimpleℓ-adic representations ofπ1(k).

Ghosts control completely the problem, namely the representation-theoretic reformulation of the original question is positive if and only if the ghost is trivial. In general, ghosts are non-trivial but for abelian schemes whose geometric generic fiber has no non-trivialk-isotrivialk(η)-isogeny factor the triviality of the ghost is predicted by Zarhin’s microweights conjecture.

2010Mathematics Subject Classification. Primary: 14K15, 14K02; Secondary: 14F20.

1. introduction

1.1. Notation. Given a field K, write K for its algebraic closure (which we assume to be fixed) and π1(K) for its absolute Galois group. Let k be a field of characteristic p ≥ 0 and let S be a smooth, separated and geometrically connected scheme of finite type over k. Set S :=S×kk. Let η denote the generic point ofS, and |S|the set of closed points of S. Fort∈S with residue fieldk(t), regarded as a morphism t= Spec(k(t))→S, let t denote the geometric pointt= Spec(k(t))→ S; by functoriality of

´

etale fundamental group, these induce a morphism of profinite groupsπ1(t, t)→π1(S, t)'π1(S, η) (well defined up to conjugacy¹), which is injective if t∈ |S|. Let`be a prime 6=p.

For an algebraic groupGover a field, letG^◦⊂Gdenote its neutral component andG↠π₀(G) :=G/G^◦ its group of connected components.

1.2. Abelian schemes. LetA→S be an abelian scheme. The starting point of this note is the following question originally addressed by R¨ossler and Szamuely in [RSza19, Rem. 4.2.2] fork a finite field.

1.2.1.Question. Assume the A_s, s∈ |S| have a non-trivial common k-isogeny factor. Does this imply thatA_η has a non-trivial k-isotrivial k(η)-isogeny factor?

More precisely, one can ask the following.

1.2.2.Question. Assume the A_s, s ∈ |S| have a common k-isogeny factor A. Does this imply that A×_kk(η) is a k(η)-isogeny factor of Aη?

One can upgrade² Question1.2.2as follows. LetB₁, . . . , B_r→S be abelian schemes.

1.2.3.Question. Assume that for every s ∈ |S| there exists 1 ≤ is ≤ r such that Bis,s is a k-isogeny factor of A_s. Does this imply that there exists1≤i≤r such that B_i,η is a k(η)-isogeny factor of A_η?

1The choice of geometric points does not play any part in this paper; we implicitly assume that ´etale paths are fixed between them and we will usually omit them from the notation for ´etale fundamental group.

2More precisely, up to replacing k by a finite field extension, Acan be assumed to descend to Aoverk, and Question 1.2.3withi= 1 andB1=A×kS→S implies Question1.2.2.

1

1.3. Representation-theoretic formulation. For a profinite group ∆, let Mod_Qℓ(∆) denote the cat- egory of finitely generated Qℓ-modules equipped with a continuous Qℓ-linear action of ∆. For V ∈ Mod_Qℓ(∆), let ∆_V ⊂ Aut_Qℓ(V) denote the image of the corresponding morphism ∆ → Aut_Qℓ(V).

For a continuous homomorphism of profinite groups φ : Γ → ∆, write φ^∗ or −|Γ : Mod_Qℓ(∆) → Mod_Qℓ(Γ) for the obvious restriction functor. This applies especially to the morphism of profinite groups f : π1(Y) → π1(S) induced by a morphism of k-schemes f : Y → S, where Y is a smooth, separated, geometrically connected scheme of finite type over k. In particular, the notation s^∗ or

−|π1(s) : Mod_Qℓ(π1(S)) → Mod_Qℓ(π1(s)) will refer to the morphism s : π1(s) → π1(S) induced by s : Spec(k(s)) → S while the notation a^∗_S or −|π1(S) : Mod_Qℓ(π1(k)) → Mod_Qℓ(π1(S)) (resp. a^∗_s or −|π1(s) : Mod_Qℓ(π1(k)) → Mod_Qℓ(π1(s))) will refer to the morphism aS : π1(S) → π1(k) (resp.

as :π1(s)→π1(k)) induced by the structural morphismS →Spec(k) (resp. Spec(k(s))→Spec(k)). For V ∈Mod_Qℓ(∆), writeV^ssfor its semisimplification in Mod_Qℓ(∆).

Let aMod_Qℓ(∆) denote the category of almost ∆-modules, defined as follows. The objects of aMod_Qℓ(∆) are the elements in

lim_−→

U⊂∆

Mod_Qℓ(U),

where the direct limit is over all open subgroups of ∆. Given V₁, V₂ ∈aMod_Qℓ(∆), HomaMod_Q

ℓ(∆)(V1, V2) = lim_−→

U⊂∆

HomMod_Q

ℓ(U)(V1|U, V2|U),

where we choose an open subgroup U0 ⊂ ∆ such that V1, V2 come from and are regarded as objects of Mod_Qℓ(U₀) and the direct limit is taken over all open subgroups U ⊂ U₀. (Clearly the definition is independent of the choice ofU0.) Note that aMod_Qℓ(∆) is abelian and that one can regard Mod_Qℓ(∆) as a subcategory of aMod_Qℓ(∆)via the tautological faithful functor Mod_Qℓ(∆)→aMod_Qℓ(∆). We will say that a property of an objectM(resp. of a morphismM →N) in Mod_Qℓ(∆) is persistent if, for every open subgroupU ⊂∆, it is preserved by the restriction functor−|U : Mod_Qℓ(∆)→ Mod_Qℓ(U). For a persis- tent property P, we will say that an objectM (resp. a morphismM →N) in aMod_Qℓ(∆) has almost P if there exists an open subgroupU ⊂∆ such that M ∈Mod_Qℓ(U) (resp. M, N ∈Mod_Qℓ(U) and M →N is a morphism in Mod_Qℓ(U)) and M (resp. M → N) has P in Mod_Qℓ(U). We will say that an object M (resp. a morphism M → N) in Mod_Qℓ(∆) has almost P if its image via Mod_Qℓ(∆) → aMod_Qℓ(∆) has almost P. For instance, the property of being semisimple is persistent³ so that one can define the semisimplification M^ss of M ∈ aMod_Qℓ(∆) as the image in aMod_Qℓ(∆) of the U-semisimplification of M, where U ⊂ ∆ is any open subgroup such that M ∈ Mod_Qℓ(U). For a continuous homomorphism of profinite groups φ: Γ → ∆, write φ^∗ or −|Γ : aMod_Qℓ(∆) → aMod_Qℓ(Γ) for the obvious restriction functor.

With this terminology, consider the following questions.

LetV ∈Mod_Qℓ(π₁(S)).

1.3.1.Question. Assume there exists a non-zero W ∈Mod_Qℓ(π1(k))such that for everys∈ |S|,a^∗_sW(=

s^∗a^∗_SW) is almost a submodule ofs^∗V. Does this imply that there exists a non-zero W^′ ∈aMod_Qℓ(π₁(k)) such thata^∗_SW^′ is almost a submodule of V?

1.3.2.Question. Assume there exists W ∈Mod_Qℓ(π1(k))such that for every s∈ |S|, a^∗_sW(=s^∗a^∗_SW) is almost a submodule of s^∗V. Does this imply that a^∗_SW is almost a submodule ofV?

LetV, W₁, . . . , W_r ∈Mod_Qℓ(π₁(S)).

1.3.3.Question. Assume that for every s ∈ |S| there exists 1 ≤ i_s ≤ r such that s^∗W_is is almost a submodule ofs^∗V. Does this imply that there exists 1≤i≤r such thatWi is almost a submodule of V?

3Since it is equivalent to the connected component of the Zariski-closure of the image of ∆ to be reductive - here, we use thatQℓis a field of characteristic 0. For a more elementary argument, seee.g. (the proof of) [BH06, 2.7 (1)].

1.3.4.Remark. LetY be a smooth, separated, geometrically connected scheme of finite type over kand f :Y →Sa dominantk-morphism. Then Questions1.3.1,1.3.2have (resp. Question1.3.3has) a positive answer if it has a positive answer forf^∗V ∈Mod_Qℓ(π₁(Y)) (resp. f^∗V, f^∗W₁, . . . , f^∗W_r ∈Mod_Qℓ(π₁(Y))).

When f is surjective, the converse is also true.

1.3.5. For an abelian variety A over a fieldK of characteristic p≥0 and a prime`6=p, let T_ℓ(A_K) := lim_←−

n

A[`ⁿ]_K'H¹(A_K,Zℓ)^∨

denote the `-adic Tate module of A_K and setVℓ(A_K) :=Tℓ(A_K)⊗Qℓ. Whenkis finitely generated, the Tate conjectures for abelian varieties, which we recall below, ensure that Question1.2.1(resp. Question 1.2.2, resp. Question 1.2.3) is equivalent to Question 1.3.1(resp. Question 1.3.2, resp. Question 1.3.3) withV :=V_ℓ(Aη) (resp. V :=V_ℓ(Aη), resp. V :=V_ℓ(Aη) andWi:=V_ℓ(Bi,η),i= 1, . . . , r).

Fact. (Tate conjectures for abelian varieties) Let K be a finitely generated field of characteristic p ≥0 and `6=p a prime. Then,

- (Semisimplicity) For every abelian varietyA over K, Vℓ(A_K) is a semisimple π1(K)-module.

- (Fullness) For every pair of abelian varieties A, B over K, the natural morphism of Zℓ-modules Hom_K(A, B)⊗_ZZℓ →Hom_Zℓ(T_ℓ(A_K), T_ℓ(B_K))^π1(K)

is an isomorphism.

For the proof see [Ta66] (forKfinite), [Z75], [Z76], [Mo77] (forp >0), [F83], [FW92, IV] (forK a number field) and [FW92, VI] (for p= 0).

The difficulty of the above questions depends on the arithmetic complexity ofk. If kis an infinite finitely generated field then Question1.3.3always has a positive answer (Proposition 1.4.1and Subsection2.1).

If k contains an infinite finitely generated field (equivalently, k is not an algebraic extension of a finite field) then Question1.2.3always has a positive answer (Corollary 1.4.2and Subsection2.2) but Question 1.3.3 and even Questions 1.3.1, 1.3.2 may fail (as the Tate conjectures are not available in general). If k is finite, Question1.3.1has (hencea fortiori Questions 1.3.2,1.3.3 have) a negative answer in general (Subsections1.5.1,1.5.2) but we expect Questions1.2.1,1.2.2always have a positive answer. The failure of a positive answer to Questions 1.3.1, 1.3.2 when k is finite is measured by what we call the ghost attached to V. The main achievement of this paper is the introduction and study of ghosts (Theorem 1.5.3.3 and Section5).

1.4. Fields containing an infinite finitely generated subfield. The proofs of Proposition1.4.1and Corollary1.4.2below are given in Section 2.

1.4.1. Proposition. Assume k is Hilbertian. Then Question 1.3.3 always has a positive answer.

Proposition 1.4.1 is a consequence of the Hilbertian property and a Frattini argument of Serre. By standard descent arguments and the Tate conjectures for abelian varieties, one deduces from Proposition 1.4.1the following, which answers positively Questions 1.2.1,1.2.2 except fork⊂Fp.

1.4.2.Corollary. Assume kcontains an infinite finitely generated field. Then Question 1.2.3 always has a positive answer.

1.5. Finite fields. If k is finite, Questions 1.3.1, 1.3.2, 1.3.3 cannot have a positive answer without additional assumptions.

1.5.1. For instance, one has to work with semisimplifications⁴, as shown by the following counterexample to Question 1.3.3. Let S = Gm,k, χ : π₁(S) ↠ π₁(k) → Z^×_ℓ the `-adic cyclotomic character and ψ:π1(S)→Zℓ(1) a 1-cocyle lifting a class in

H¹(π₁(S),Zℓ(1))'lim

←−Gm(S)/Gm(S)^ℓn

4By Fact1.3.5, if Ais an abelian variety over a finitely generated fieldk the action of π1(k) onVℓ(A_k) is semisimple.

More generally, the Grothendieck-Serre conjecture predicts that the action ofπ1(k) on H^u(X_k,Qℓ(v)) should be semisimple for every smooth, proper schemeX overkbut this conjecture is still widely open.

whose restriction in H¹(π1(S),Zℓ(1)) is non-zero (for instance, take ψ lifting the generic Kummer class that is the image of T ∈ Gm(S) ' k[T, T⁻¹]^× in H¹(π₁(S),Zℓ(1))). Let V denote the π₁(S)-module

defined by (

χ ψ 0 1

) and W₁ theπ₁(S)-module defined by (

χ 0 0 1

) .

By our choice of ψ, V|_π1(S) is not almost trivial in Mod_Qℓ(π₁(S)) hence W₁ and V are not almost iso- morphic in Mod_Qℓ(π₁(S)), while, as kis finite, s^∗W₁ and s^∗V are isomorphic in Mod_Qℓ(π₁(s)) for every s∈ |S|(indeed, π₁(s) is procyclic generated by the Frobeniusϕ_s:x7→x^|k(s)|and χ(ϕ_s)6= 1).

A counterexample to Questions 1.3.1,1.3.2 is obtained similarly, by replacing Gm with a non-isotrivial elliptic scheme E → S with positive Mordell-Weil rank and χ with the 2-dimensional representation arising fromE →S.

Hence the right questions should rather be

1.5.1.1.Question. Assume there exists a non-zero semisimple W ∈ Mod_Qℓ(π1(k)) such that for every s∈ |S|, a^∗_sW(=s^∗a^∗_SW) is almost a submodule of (s^∗V)^ss. Does this imply that there exists a non-zero semisimpleW^′ ∈aMod_Qℓ(π1(k))such that a^∗_SW^′ is almost a submodule of V^ss?

1.5.1.2.Question. Assume there exists a semisimple W ∈ Mod_Qℓ(π₁(k)) such that for every s ∈ |S|, a^∗_sW(= s^∗a^∗_SW) is almost a submodule of (s^∗V)^ss. Does this imply that a^∗_SW is almost a submodule of V^ss?

1.5.1.3.Question. Assume that for every s∈S there exists 1 ≤i_s ≤r such that (s^∗W_is)^ss is almost a submodule of (s^∗V)^ss. Does this imply that there exists 1 ≤i≤r such that W_i^ss is almost a submodule of V^ss?

1.5.2. But Question1.5.1.1also fails, even for motivic representations (that is subquotients of H^u(Xη,Qℓ(v)) forX →S a smooth proper morphism andu≥0, v integers). For instance, consider an abelian scheme A→S of relative dimension g with large geometric monodromy that is such that the Zariski-closure of the image of π1(S) acting on V_ℓ(Aη) is Sp_2g,Qℓ. Set E := V_ℓ(Aη)^∨⊗V_ℓ(Aη) ⊂H^2g(Aη ×Aη,Qℓ(g)) and V := E/QℓId; then, for each connected ´etale cover S^′ → S, E^π1(S′) =QℓId and V^π1(S′) = 0 (semisim- plicity), whileV^π1(s)6= 0 for anys∈ |S|(as either the Frobenius endomorphism ofA_s gives a non-trivial element ofV^π1(s), orE^π1(s) =E and V^π1(s)=V).

1.5.3. Actually, there is a dichotomy depending on whether the dimension of W (resp. of the W_i, i = 1, . . . , r) in Questions 1.5.1.1, 1.5.1.2 (resp. Question 1.5.1.3) is equal to or strictly smaller than the dimension of V. In the former case, Question 1.5.1.3 always has (hence Questions 1.5.1.1, 1.5.1.2 always have) a positive answer (Proposition1.5.3.1). In the latter case, the failure to a positive answer to Questions1.5.1.1,1.5.1.2 is measured by the ghost ofV (Theorem1.5.3.3).

In this paper, the densityδS(Σ) (resp. upper densityδ^u_S(Σ)) of a subset Σ⊂ |S|always refers to Dirichlet density (resp. Dirichlet upper density) - see Section 3.

1.5.3.1.Proposition. Letkbe a finite field. Fix a prime`6=p. LetV, W₁, ..., W_r∈Mod_Qℓ(π₁(S)); write G for the Zariski-closure of the image of π1(S) acting on V ⊕W1 ⊕ · · · ⊕Wr. Assume that the set of all s∈ |S| for which there exists 1 ≤is ≤r such that (s^∗Wis)^ss and (s^∗V)^ss are almost isomorphic has upper density >1−_|π0¹_(G)|.

- (1.5.3.1.1) Then there exists 1≤i≤r such that W_i^ss and V^ss are almost isomorphic.

- (1.5.3.1.2) Assume Wi|_π1(S) ∈ Mod_Qℓ(π1(S)) is trivial for i = 1, . . . , r and V|_π1(S) is semisimple in Mod_Qℓ(π₁(S)). Then V|_π1(S) is almost trivial.

Proposition1.5.3.1is a consequence of (an`-adic version of) the Cebotarev density theorem and the fact that semisimple modules are determined by their characteristic polynomials; its proof is given in Section 4. For an enhancement of Proposition1.5.3.1, see Remark4.4.

1.5.3.2.Corollary. Let k be a finite field. Let A, B₁, . . . , B_r → S be abelian schemes; write G for the Zariski-closure of the image of π1(S) acting on Vℓ(Aη)⊕Vℓ(B1,η)⊕ · · · ⊕Vℓ(Br,η). Assume that the set of all s ∈ |S| for which there exists 1 ≤ is ≤ r such that Bis,s is k-isogenous to As has upper density

>1−_|π0¹_(G)|. Then there exists 1≤i≤r such that B_i,η isk(η)-isogenous toA_η.

Corollary1.5.3.2 follows from Proposition1.5.3.1 applied toV =V_ℓ(A_η), W_i =V_ℓ(B_i,η),i= 1, . . . , r and Fact 1.3.5(forp >0).

Since elliptic curves are automatically simple, Corollary1.5.3.2 answers positively Question 1.2.3 for el- liptic curves.

1.5.3.3. As shown by the counterexample in Subsection1.5.2, Questions 1.5.1.1,1.5.1.2 may have a neg- ative answer, even for motivic representations, as soon as dim(W) < dim(V). However, when V is part of a Q-rational compatible system Vℓ, ` 6= p of (pointwise pure) lisse Qℓ-sheaves on S - typically V<sub>ℓ</sub> = H<sup>u</sup>(X<sub>η</sub>,Qℓ(v)), ` 6= p for smooth proper morphisms X → S and integers u ≥0, v (but see Sub- section 5.7), we show that the failure to a positive answer to Questions 1.5.1.1, 1.5.1.2 is measured by a ‘hidden motive’, which we call the ghost of V. This is the content of the theorem below, which is the main result of this paper; its proof is given in Section 5.

Theorem. Let k be a finite field. Fix a Q-rational compatible system V_ℓ, ` 6= p in Mod<sub>Qℓ</sub>(π<sub>1</sub>(S)), pointwise pure of weight w∈Z. Let G<sub>ℓ</sub> ⊂GL<sub>Vℓ</sub> denote the Zariski-closure of the image of π1(S) acting on Vℓ. Fix a maximal torus Tℓ ⊂ Gℓ. After possibly replacing k with a finite field extension which is independent of `, V_ℓ^Tℓ is canonically equipped with a structure of π1(k)-module, which is semisimple, independent of T_ℓ up to isomorphism and has the following properties.

- (1.5.3.3.1)For every s∈ |S|, a^∗_s(V_ℓ^Tℓ) is a submodule of (s^∗V_ℓ)^ss;

- (1.5.3.3.2) Let W be a semisimple π1(k)-module in Mod_Qℓ(π1(k)). Assume the set of all s∈ |S| such that a^∗_sW is almost a submodule of (s^∗V_ℓ)^ss has upper density 1. Then W is almost a submodule of V_ℓ^Tℓ;

- (1.5.3.3.3) The characteristic polynomial of the Frobenius F ∈ π1(k) acting on V_ℓ^Tℓ is in Q[T] and independent of `.

We will callV_ℓ^Tℓ equipped with this structure of almost π1(k)-module, theghost attached to Vℓ and de- note it by Ghost(V_ℓ). (1.5.3.3.1) and (1.5.3.3.2) say thatGhost(V_ℓ) is the largest common almost factor of the (s^∗Vℓ)^ss,s∈ |S|.

In terms of ghosts, Questions1.5.1.1,1.5.1.2 forV_ℓ can be reformulated as follows.

1.5.3.4.Question. Does Ghost(V_ℓ)(=V_ℓ^Tℓ)6= 0 imply V^G

◦ ℓ

ℓ 6= 0?

1.5.3.5.Question. Does Ghost(V_ℓ)(= V_ℓ^Tℓ) = V^G

◦ ℓ

ℓ hold? Equivalently, does G^◦_ℓ act on V_ℓ with no non- trivial zero weights?

We feel the introduction of ghosts and their use to reformulate Questions 1.5.1.1, 1.5.1.2 is the most striking contribution of this paper.

1.5.3.6. When V = V_ℓ(A_η) for an abelian scheme A → S, the ghost corresponds to the largest possible common k-isogeny factor of the fibersAs,s∈S (see Section6). From Zarhin’s microweights conjecture ([Z85, Conj. 0.4]), the ghost should correspond to the largest (weakly) k-isotrivial isogeny factor ofA_η (which would answer positively R¨ossler and Szamuely’s original question in the case of a finite fieldkas well). The known cases of the microweights conjecture give a partial positive answer to Questions 1.2.1,

1.2.2(Corollary 6.2.3). We can also show that, ifGacts on the Tate module of its simple isogeny factors with a 1-dimensional center (this occurs for types I, II, III in Albert’s classification - see Corollary6.3.3) then the only possible commonk-isogeny factors of the A_s are supersingular (Corollary 6.3.3).

Acknowledgements: The first author is partly funded by the ANR project ECOVA, ANR-15-CE40- 0002-01 and the CNRS-JSPS project ASPIC. This paper was completed while she was a member of the CMLS - Ecole Polytechnique and a visiting researcher at RIMS; she thanks both institutes for pro- viding remarkable working conditions. One of her stays at RIMS was supported by the International Research Unit of Advanced Future Studies at Kyoto University, which she is grateful to. The second author was partly supported by JSPS KAKENHI Grant Numbers JP22340006, JP15H03609. The au- thors thank Tam´as Szamuely for suggesting expository improvements and Emiliano Ambrosi for several accurate comments. They are very grateful to the referee for a thorough reading, insightful comments and questions.

2. Base fields containing an infinite finitely generated field

2.1. Proof of Proposition 1.4.1. Write M :=⊕1≤i≤rW_i^∨⊗V and fori= 1, . . . , r, let Σ_i ⊂W_i^∨⊗V denote the open subset corresponding to injective morphismsWi ,→V. We are to show that there exists an open subgroup Π⊂π₁(S) such that (t1≤i≤rΣ_i)^Π6=∅. By assumption, it is enough to show that there existss∈ |S|such thatπ1(s)M is open in π1(S)M (Recall the notation of Subsection1.3). The existence of such an s is explained in [S81, §1]; it follows from the Hilbertian property of k (in [S81, §1], k is a number field but the argument only requiresk to be Hilbertian) and the fact thatπ₁(S)_M is a compact

`-adic Lie group, which ensures that its Frattini subgroup is open (see [S89,§10.6]).

Remark. The proof of Proposition 1.4.1already suggests that the assumption ‘for every s∈ |S|, there exists (...)’ can be weakened. For instance, when p = 0, k is finitely generated, S is a curve and Lie(π₁(S)_M) is perfect, it is enough to assume [CT13, Thm. 1.1] that there exist an integer d≥1, and infinitely many s∈S with [k(s) :k]≤d such that there exists 1≤is ≤r such that Wis|π1(s) is almost a submodule ofV|π1(s). The assumption that Lie(π1(S)_M) is perfect is satisfied when theWi,i= 1, ..., r and V are motivic [D71, Cor. 4.2.9 (a)], [CT12b, Thm. 5.7]. See also [MP12], especially Prop. 1.15 and Rem. 1.17 forp-adic variants.

2.2. Proof of Corollary 1.4.2. If k is itself infinite finitely generated, Corollary1.4.2 directly follows from Proposition1.4.1 and Fact 1.3.5. Otherwise, as S is of finite type overk and A,B₁, . . . , B_r are of finite type over S, there exists an infinite finitely generated field k^# ⊂ k such that A, B₁, . . . , B_r → S are defined overk^#. So, in view of Proposition1.4.1, to prove Corollary 1.4.2it is enough to prove the following. Letkbe an infinite finitely generated field and let Ω be an algebraically closed field containing k. Assume that for everys ∈ |S|, there exists 1 ≤ is ≤r such thatBis is an isogeny factor of As over Ω. Then Bis is an isogeny factor of As over k(s). This in turn follows from the case of infinite finitely generated base fields observing that for every pair of abelian varietiesA,B over a fieldF

- The scheme of homomorphismsSch/F →Ab,T →Hom_AbSch/T(BT,AT) is representable by a commu- tative ´etale group scheme overF. In particular, for every field extension K of F, every K-morphism fromB toA is automatically defined over a finite (separable) field extension ofF.

- By faithfully flat descent, for every F-morphism φ : B → A and every field extension K of F, the following are equivalent:

(1) φ:B→Ainduces an isogeny onto its image;

(2) φ_K :B_K →A_K induces an isogeny onto its image.

3. Density and upper density

Unless otherwise stated, the following notation will be used in the remaining part of the paper.

Let k be a finite field and let F ∈π1(k) denote the geometric Frobenius. For every integer m ≥ 1, let k⊂k_m be the unique field extension of degreem= [k_m:k] (in a given algebraic closurek ofk).

Let S be a smooth, separated and geometrically connected scheme of finite type over k. For s ∈ |S|, let k(s) denote the residue field at s, n_s := [k(s) :k], and F_s ∈ π₁(s) the geometric Frobenius (that is F_s=F^ns); we also denote by F_s its image inπ₁(S)via π₁(s),→π₁(S) (See footnote 1).

We refer to [P97, Appendix B] for the notion and basic properties of Dirichlet densities. For a subset Σ⊂ |S|, the series

FΣ(t) =∑

s∈Σ

|k(s)|^−t

converges absolutely and locally uniformly for Re(t)>dim(S). Write x_Σ(t) := F_Σ(t)

F_|S|(t), t∈R_>dim(S) and sΣ(t) := sup{xΣ(t^′)|dim(S)< t^′ < t},t∈R_>dim(S). Let

δ_S^u(Σ) := lim

t→dim(S), t∈R>dim(S)

s_Σ(t)

denote the (Dirichlet) upper density of Σ. By definition 0≤δ_S^u(Σ)≤1 and, if the limitδS(Σ) of xΣ(t), for t → dim(S), t ∈ R_>dim(S) exists, then δ_S(Σ) = δ_S^u(Σ) and one says that Σ has (Dirichlet) density δS(Σ).

3.1.Lemma.

- (3.1.1) If Σ = ∪

1≤i≤r

Σi thenδ_S^u(Σ)>0 ⇔ there exists 1≤i≤r such thatδ^u_S(Σi)>0.

- (3.1.2) If Σ2 has density then δ_S^u(Σ1∩Σ2) ≥ δ_S^u(Σ1) +δS(Σ2)−1. If, moreover, δS(Σ2) = 1, then δ_S^u(Σ₁∩Σ₂) =δ_S^u(Σ₁).

- (3.1.3) If δ^u(Σ)> ¹₂, then the gcd of ns, s∈Σ is 1.

- (3.1.4) If f :S^′ → S is a connected finite ´etale cover and δ_S^u(Σ) = 1 (resp. δ_S^u(Σ)>1−_deg(f¹ ₎), then δ_S^u′(f⁻¹(Σ)) = 1(resp. δ_S^u′(f⁻¹(Σ))>0).

- (3.1.5) If f :S^′ →S is a connected finite ´etale cover, then δ_S^u(Σ)≥ _deg(f)¹ δ_S^u′(f⁻¹(Σ)).

Proof. (3.1.1): The⇒ implication follows from the inequality δ^u_S(∪1≤i≤rΣ_i)≤ ∑

1≤i≤r

δ_S^u(Σ_i)

while the ⇐ implication follows from the fact that Σ^′ ⊂Σ implies δ_S^u(Σ^′)≤δ_S^u(Σ).

(3.1.2): By definition of upper density, there exists a strictly decreasing sequence φ :Z_≥1 → R_>dim(S) such thatφ(n)→dim(S) andxΣ1(φ(n))→δ^u_S(Σ1) (n→ ∞). Then

x_Σ1∩Σ2(φ(n)) =x_Σ1(φ(n))+x_Σ2(φ(n))−x_Σ1∪Σ2(φ(n))≥x_Σ1(φ(n))+x_Σ2(φ(n))−1→δ^u_S(Σ₁)+δ_S(Σ₂)−1 Henceδ_S^u(Σ₁∩Σ₂)≥δ^u_S(Σ₁) +δ_S(Σ₂)−1. This concludes the proof of the first assertion, and the second assertion follows immediately from the first.

(3.1.3): As δ^u(Σ) > ¹₂ > 0, Σ is non-empty. Take s0 ∈ Σ and let p1, . . . , pr be the prime factors of n_s0. For each i= 1, . . . , r, consider the connected finite ´etale Galois cover S_kpi → S corresponding to π₁(S)↠π₁(k) = ˆZ↠Z/p_iZ. As δ^u(Σ)> ¹₂ ≥ _p¹_i, there exists s_i ∈Σ such that the image ofF_si ∈π₁(S) in Z/p_iZ (which is n_si ∈ Z/p_iZ) is distinct from 0 ∈ Z/p_iZ, by the Cebotarev density theorem [P97, Thm. B.9]. Then the gcd of n_s0, n_s1, . . . , n_sr is 1, hence so is that ofn_s,s∈Σ.

(3.1.4): Let |S^′|^split ⊂ |S^′| denote the set of s^′ ∈ |S^′| such that [k(s^′) : k(f(s^′))] = 1 and set |S|^split :=

f(|S^′|^split) and |S|^nonsplit:= |S| \ |S|^split. Set Σ^′ := f⁻¹(Σ), (Σ^′)^split = Σ^′∩ |S^′|^split, Σ^split = Σ∩ |S|^split and Σ^nonsplit = Σ∩ |S|^nonsplit. Let G (resp. H) be the Galois group of the Galois closure ˜S → S of f :S^′ →S (resp. the cover ˜S →S^′ induced naturally). Let Γ denote the set of conjugacy class ofG. For

each J ∈Γ, write |S|J for the set of closed points s∈ |S|such that the image of Fs ∈π1(S) inG lies in J and set Σ_J := Σ∩ |S|J. For eachJ ∈Γ, takej ∈J and set

nJ :=|(G/H)^⟨j⟩|= |{g∈ G |g⁻¹jg ∈ H}|

|H| = |J∩ H||G|

|J||H| . Then one has F_(Σ′)^split(t) =∑

J∈Γn_JF_ΣJ(t), whileF_Σsplit(t) =∑

J∈Γ,nJ>0F_ΣJ(t).

We first prove:

- (3.1.6) lim_{t→dim(S), t∈R}

>dim(S)

F_|S′|(t) F_|S|(t) = 1.

- (3.1.7) |S|^split⊂ |S|has densityδ_S(|S|^split)≥ _deg(f¹ ₎. (3.1.6): Since F_|S′|^split(t) =∑

J∈Γn_JF_|S|J(t), one has F_|S′|(t)

F_|S|(t) = F_|S′|(t)

F_|S′|^split(t)·F_|S′|^split(t)

F_|S|(t) = F_|S′|(t) F_|S′|^split(t) ·∑

J∈Γ

nJ

F_|S|J(t) F_|S|(t) , hence, from [P97, Prop. B.8 and Thm. B.9],

lim

t→dim(S), t∈R>dim(S)

F_|S′|(t) F_|S|(t) =∑

J∈Γ

n_J|J|

|G| =∑

J∈Γ

|J∩ H|

|H| = 1.

(3.1.7): One has

F_|S|split(t)

F_|S|(t) = ∑

J∈Γ,nJ>0

F_|S|J(t) F_|S|(t) , hence, from [P97, Thm. B.9],

δS(|S|^split) = ∑

J∈Γ,n_J>0

|J|

|G| = | ∪g∈GgHg⁻¹|

|G| ≥ |H|

|G| = 1 deg(f).

To prove (3.1.4), first, suppose δ^u_S(Σ) = 1. Then there exists a strictly decreasing sequence φ: Z_≥1 → R_>dim(S) such thatφ(n)→dim(S) andx_Σ(φ(n))→δ_S^u(Σ) = 1 (n→ ∞). Then

0≤∑

J∈Γ

x_|S|J\ΣJ(φ(n)) =x_|S|\Σ(φ(n)) =x_|S|(φ(n))−x_Σ(φ(n))→1−1 = 0.

Thus, for each J ∈Γ, one must have

nlim→∞(x_|S|J(φ(n))−xΣJ(φ(n))) = lim

n→∞x_|S|J\ΣJ(φ(n)) = 0, δ_S(|S|J) = lim

n→∞x_|S|J(φ(n)) = lim

n→∞x_ΣJ(φ(n))≤δ^u_S(Σ_J)≤δ_S(|S|J), hence

nlim→∞x_ΣJ(φ(n)) =δ^u_S(Σ_J) =δ_S(|S|J).

Observe

x_|S′|^split(t) =

∑

J∈Γn_JF_|S|J(t)

F_|S′|(t) = F_|S|(t) F_|S′|(t)

∑

J∈Γ

n_Jx_|S|J(t), x_(Σ′)^split(t) =

∑

J∈ΓnJFΣJ(t)

F_|S′|(t) = F_|S|(t) F_|S′|(t)

∑

J∈Γ

nJxΣJ(t).

From these, (3.1.6) and [P97, Prop. B.8 and Thm. B.9], one has 1 =δ_S′(|S^′|^split) =∑

J∈Γ

n_Jδ_S(|S|J) = lim

n→∞

∑

J∈Γ

n_Jx_ΣJ(φ(n)) = lim

n→∞x_(Σ′)^split(φ(n))≤δ^u_S′((Σ^′)^split)≤δ_S^u′(Σ^′)≤1, henceδ^u_S′(Σ^′) = 1.

Next, supposeδ_S^u(Σ)>1− _deg(f)¹ . Then, as Σ = Σ^nonsplittΣ^split⊂ |S|^nonsplittΣ^split, one has 1− 1

deg(f) < δ_S^u(Σ)≤δ_S(|S|^nonsplit) +δ^u_S(Σ^split)≤1− 1

deg(f) +δ_S^u(Σ^split),