Finiteness properties of torsion fields of abelianvarieties∗
Wojciech Gajda and Sebastian Petersen
March 24, 2024

Abstract
Let A be an abelian variety defined over a field K. We study finite
generation properties of the profinite group Gal(Ktor (A)/K) and of
certain closed normal subgroups thereof, where Ktor (A) is the torsion
field of A over K. In fact, we establish more general finite generation properties for monodromy groups attached to smooth projective
varieties via étale cohomology. We apply this in order to give an independent proof and generalizations of a recent result of Checcoli and
Dill about small exponent subfields of Ktor (A)/K in the number fieldcase. We also give an application of our finite generation results in
the realm of permanence principles for varieties with the weak Hilbert
property.

1

Introduction

For an abelian variety A over a field K we denote by K(Ator ) the fieldobtained from K by adjoining the coordinates of all torsion points in A(K).
We define Ktor (A):=K(Ator ) ∩ Ksep where
Q Ksep is the separable closure of†
K. For a number field k, we define k = ℓ kℓ , where ℓ runs over all prime
numbers and kℓ is the compositum of those finite abelian Galois extensionsof k that are unramified outside ℓ and of degree prime to ℓ. For example, Q†
is the compositum of the fields Q(exp( 2πiℓ )) for ℓ ∈ L, by Kronecker-Weber∗

Key words: Abelian variety, torsion field, Galois representation
AMS subject classification: 14K15, 11G10, 12E30, 12E25

1theorem. The aim of this manuscript is to establish the following theorem
about finite generation properties of Galois groups of such torsion fields andto give two applications thereof.
Theorem 1.1. (cf. Theorem 4.2 and Remark 4.1) Let κ be a field, K/κ
a finitely generated field extension and A/K an abelian variety. Then thefollowing hold true.
(a) The profinite group Gal(κKtor (A)/κK) is topologically finitely generated.
(b) If the absolute Galois group Gal(κ) is topologically finitely generated(e.g., when κ is a finite or algebraically closed field), then Gal(Ktor (A)/K)
is topologically finitely generated.
(c) If κ is a local field, then Gal(Ktor (A)/K) is topologically finitely generated.
(d) If κ is a number field, then there exists a finite Galois extension k/κ
such that Gal(k † Ktor (A)/k † K) is topologically finitely generated.
Remark 1.2. If in the situation of Theorem 1.1 κ is a number field, thenthe profinite group Gal(Ktor (A)/K) is certainly not topologically finitely generated because Ktor (A) contains all roots of unity and thus Gal(Ktor (A)/K)
has an open normal subgroup of Ẑ× as a quotient.
Remark 1.3. We will in fact establish more general finite generation properties for monodromy groups attached to smooth projective varieties via étale
cohomology. We refer the reader to Section 3 for the results and do not go
into the technical details within the introduction.
For every field extension Ω/K define Ee (Ω/K) to be the set of all intermetiate fields F of Ω/K such that F/K is Galois and Gal(F/K) is a groupof exponent ≤ e. Our first application of Theorem 1.1 adresses a questionof Habegger mentioned in Section 4 of the recent preprint [6] of Checcoli
and Dill. There Checcoli and Dill estabished the following Theorem (cf. [6,Theorem 1]):Let K be a number field and A/K an abelian variety and let e ∈ N. There exists a finite extension M/K such that F ⊂ Mab , for all F ∈ Ee (K(Ator )/K).
Based on Theorem 1.1 we give an independent proof and generalization of[6, Theorem 1] and results from [6, Section 4] as follows.
2

Corollary 1.4. Let κ be a field. Let K/κ be a finitely generated field extension, A/K an abelian variety and e ∈ N. If Gal(κ) is topologically finitely
generated (e.g., when k is finite or algebraically closed) or if κ is a local
field, then there exists a finite separable extension M/K such that F ⊂ M,
for all F ∈ Ee (Ktor (A)/K).
Corollary 1.5. Let K/Q be a finitely generated field extension, A/K an
abelian variety and e ∈ N. Then there exists a number field k and a finite
extension M/K such that F ⊂ k † M ⊂ Mab for all F ∈ Ee (Ktor (A)/K).
Our second application of Theorem 1.1 concerns permanence principles forvarieties that satisfy the weak Hilbert property. We shall give a new proof
and generalize [2, Theorem 1.7] considerably. Relations with a conjecture of
Zannier [26, Section 2] will be explained in Remark 5.3. We refer the reader
to Section 5 for the details.
The strategy of proof for Theorem 1.1(a) is as follows.
One can constructQ
an adelic Galois representation ρ : Gal(K) → ℓ∈L GL2g (Zℓ ) such that
GA (L) := ρ(Gal(L)) is isomorphic to Gal(LKtor (A)/L) for every extension
field L/K. From [5] one gets that some open subgroup H of GA (κK) satisfies
the following technical condition (+): For almost all ℓ ∈ L the profinitegroup prℓ (H) (projection on ℓ-th factor of the product) is generated by itsℓ-Sylow
subgroups. In Section 2 we prove that every closed subgroup ofQGL
2g (Zℓ ) satisfying condition (+) is topologically finitely generated.
ℓ∈L
This then accounts for the proof of parts (a) and (b) of Theorem 1.1, andthe proof of Theorem 1.1(d) is similar, relying on [21] instead of [5]. The
proof of Theorem 1.1(d) in case |κ| < ∞ is then straightforward. The proofof Theorem 1.1(c) (case where κ is a local field) relies on Theorem 1.1(a) andthe potential semistability from [4]. The corollaries follow from Theorem 1.1
by applying a seminal group theoretical result of Zelmanov [27] and Wilson[25]: Every periodic compact (Hausdorff) group is locally finite.
Acknowledgement
W.G. thanks Marc Hindry, Pawel Mleczko and participants of SFARA for
enlightening discussions on related topics. S.P. thanks Lior Bary-Soroker,
Arno Fehm and Moshe Jarden for interesting discussions and help withsome special questions. We thank Arno Fehm in particular for bringing
the work [27] of Zelmanov to our attention. Some of the topics of this3

paper started in the research of both authors supported by the grant UMO2018/31/B/ST1/01474 of the National Centre of Sciences of Poland.
Notation
Let L be the set of all rational prime numbers. For a field K let K be
an algebraic closure of K, Ksep (resp. Kper ) the separable (resp. perfect)
closure of K inside K, and Gal(K) = Gal(Ksep /K) the absolute Galois
group of K. We denote by Kab the maximal abelian extension of K in
Ksep . A K-variety is a separated algebraic K-scheme which is geometricallyreduced and geometrically irreducible.
For a profinite group G and ℓ ∈ L we let S (ℓ) (G) be the normal subgroup
topologically generated by the ℓ-Sylow subgroups of G; if ℓ is clear from thecontext we simply write G+ instead of S (ℓ) (G), following [23]. We define
exp(G) := inf{n ∈ N : g n = 1 for all g ∈ G} ∈ N ∪ {∞} to be the exponentof G. We let FSQ(G) (resp. JH(G)) be the class of all finite simple quotientsof G (resp. of all Jordan-Hölder factors of G). We let Lieℓ be the class of(1)
all finite simple groups of Lie type in characteristic ℓ. We let GLn (Zℓ ) be
the kernel of the natural surjection GLn (Zℓ ) → GLn (Fℓ ).

2

Concepts from group theory

We recall some information about subgroups of GLn (Fℓ ). Of central importance is the following theorem of Larsen and Pink. We do not state it in its
most general form.
Theorem 2.1. (Larsen and Pink, cf. [19, Theorem 0.2]) Let n ∈ N. Thereexists a constant J ′ (n), depending only on n, such that for every ℓ ∈ L andevery subgroup Γof GLn (Fℓ ) there are normal subgroups Γ ▷ Γ1 ▷ Γ2 ▷ Γ3 ofΓ such that(a) |Γ/Γ1 | ≤ J ′ (n),(b) Γ1 /Γ2 = L1 × · · · × Ls is a finite product of groups Lj ∈ Lieℓ ,(c) Γ2 /Γ3 is abelian of order prime to ℓ and(d) Γ3 is an ℓ-group.
4Remark 2.2. The proof of [19, Theorem 0.2] in [19, p. 1155–1156] actually
gives more information. Let F := Fℓ and G/F the algebraic subgroup ofGLn,F introduced in that proof. Let Z be the center of the reductive groupG◦red = G◦ /Radu (G◦ ) and S = G◦red /Z.
(a) The number s in Theorem 2.1(b) satisfies s ≤ dim(S) ≤ dim(GLn )=n2 .
(b) The group Γ2 /Γ3 in Theorem 2.1(c) is contained in the torus Z(F).
Furthermore, as F is algebraically closed, Z ∼= Ghm for some h ∈ Nwith h ≤ dim(GLn ) = n2 .
Definition 2.3. For a profinite group Γ we define d(Γ) to be the minimal
number d such that Γ can be topologically generated by d elements, i.e., suchthat Γ contains a dense subgroup that can be generated by d elements.
A lattice in Qnℓ is a free Zℓ -submodule Λ of Qnℓ such that the canonicalmap Λ ⊗Zℓ Qℓ → Qnℓ is an isomorphism. We identify the group of Zℓ automorphims GLΛ (Zℓ ) of Λ with the subgroup {f ∈ GLn (Qℓ ) : f (Λ) = Λ}of GLn (Qℓ ) in the sequel.
Remark 2.4. Let Γ be a compact subgroup of GLn (Qℓ ). Then there exists
a lattice Λ in Qn such that Γ ⊂ GLΛ (Zℓ ). In particular Γ is isomorphic toa closed subgroup of GLn (Zℓ ).
Lemma 2.5. For every n ∈ N there exists a bound b(n), depending only
on n, such that for every ℓ ∈ L, every compact subgroup Γ of GLn (Qℓ ) is
topologically finitely generated with d(Γ) ≤ b(n).
Proof. By Remark 2.4 we can assume that Γ ⊂ GLn (Zℓ ). The profinite(1)
group P := Γ ∩ GLn (Zℓ ) is topologically finitely generated with d(P ) ≤ n2
(cf. [15, Prop. 8.1, Ex. 6.3], [22]). The group Γ := Γ/P is isomorphic to a
subgroup of GLn (Fℓ ), hence Theorem 2.1 applies to it. Let Γ1 , Γ2 and Γ3 beas in that theorem. Clearly d(Γ/Γ1 ) ≤ J ′ (n). The groups Lj from Theorem
2.1(b) can be generated by two elements (cf. [24], [16, §1]), and s ≤ n2 by
Remark 2.2(a). Hence d(Γ1 /Γ2 ) ≤ 2n2 . From Remark 2.2(b) Γ2 /Γ3 is a
subgroup of (F× )h . The Pontryagin dual of F× is pro-cyclic and Γ2 /Γ3 is aquotient of the Pontryagin dual of (F× )h , hence d(Γ2 /Γ3 ) ≤ h ≤ n2 . Finallyd(Γ3 ) ≤ 14 n2 by [20, Theorem B].
Corollary 2.6. Let ℓ ∈ L and let Γ be a compact subgroup of GLn (Qℓ ). Ifℓ > J ′ (n) and Γ = Γ+ , then FSQ(Γ) ⊂ {Z/ℓ} ∪ Lieℓ .
5

Proof. By Remark 2.4 we can assume that Γ ⊂ GLn (Zℓ ). The group P :=(1)
Γ ∩ GLn (Zℓ ) is pro-ℓ and Γ = Γ/P is isomorphic to a subgroup of GLn (Fℓ )+satisfying Γ = Γ . Hence the assertion is immediate from Theorem 2.1.
Theorem 2.7. (cf. [1], [18], [23, Théorème 5]) If 5 ≤ ℓ1 < ℓ2 , thenLieℓ1 ∩ Lieℓ2 = ∅.
Definition
2.8. Let n ∈ N and L ⊂ L. Let G be a compact subgroup of
Q
GL
(Q
n
ℓ ) and prℓ the projection on the ℓ-th factor.
ℓ∈L
(a) Q
We call G independent (resp. group theoretically independent) if G =ℓ∈L prℓ (G) (resp. if FSQ(prℓ1 (G))∩FSQ(prℓ2 (G)) = ∅ for all ℓ1 ̸= ℓ2
in L).
(b) We say that G satisfies condition (+) if prℓ (G) = prℓ (G)+ for all but
finitely many ℓ in L.
(c) We say that G satisfies condition (+) potentially if G has an open
subgroup H such that H satisfies condition (+)
Remark 2.9. If G is group theoretically independent, then G is independent(cf. [23, Lemme 2]).
Lemma
2.10. Every group theoretically independent compact subgroup GQof ℓ∈L GLn (Qℓ ) is topologically finitely generated.
Proof. Let b = b(n) be the constant from Lemma 2.5. For every ℓ ∈ L there(ℓ)(ℓ)
exists a system (g1 , · · · , gb ) of topological generators of prℓ (G) by LemmaQ(ℓ)
2.5. Consider the gj = (gj )ℓ∈L ∈ ℓ∈L prℓ (G) and the closure H of thesubgroup ⟨g1 , · · · , gb ⟩ generated by the elements gj . Then prℓ (G) = prℓ (H)for all ℓ ∈ L. From this and our assumption on G we see that the groups G
and H are both group theoretically independent. It follows that G and H
are independent (cf. Remark 2.9), and thus G = H, as desired.
Q
Lemma 2.11. Let G beacompact
subgroupofℓ∈L GLn (Qℓ ), L0 a finite
Q
Q
subset of L and pr : ℓ∈L GLn (Qℓ ) → ℓ∈L\L0 GLn (Qℓ ) the projection.
If pr(G) is topologically finitely generated, then G is topologically finitely
generated.
Proof. ker(pr) ∩ G is isomorphic to a compact subgroup of the finite productQℓ∈L0 GLn (Qℓ ). It follows from Lemma 2.5 that ker(pr) ∩ G is topologically6finitely generated. The assertion is now immediate from the exact sequence
1 → ker(pr) ∩ G → G → pr(G) → 1.
Q
Proposition 2.12. If a compact subgroup G of ℓ∈L GLn (Qℓ ) satisfies condition (+), then it is topologically finitely generated.
Proof. Let Gℓ = prℓ (G). There exists a finite subset L0 of L such thatGℓ = G+ℓ for all ℓ ∈ L \ L0 . We can furthermore assume that L0 contains
all rational primes ≤ J ′ (n) and the primes 2 and 3. For all ℓ ∈ L \ L0 wehave FSQ(Gℓ ) = Lieℓ ∪ {Z/ℓ} by Corollary 2.6. By Theorem 2.7 we see that
FSQ(Gℓ1 )∩FSQ(Gℓ2 ) = ∅ for all ℓ1 , ℓ2 Q
∈ L\L0 with ℓ1 ̸= Q
ℓ2 . Hence the imagepr(G) of G under the projection pr : ℓ∈L GLn (Qℓ ) → ℓ∈L\L0 GLn (Qℓ ) isgroup theoretically independent. By Lemma 2.10 the profinite group pr(G)is topologically finitely generated, and this suffices by Lemma 2.11
For further use, we finally discuss in which circumstances propery (+) descends to normal subgroups.
Definition 2.13. For a profinite group G define LLie (G) to be the set of allℓ ∈ L such that JH(G) ∩ Lieℓ ̸= ∅.
We note that LLie (G) is finite for example when |G| < ∞ or when G is
pro-solvable.
Q
Lemma 2.14. Let G be a compact subgroup of ℓ∈L GLn (Qℓ ) and N aclosed normal subgroup of G.
(a) If G satisfies condition (+) and LLie (G/N ) is finite, then N satisfies
condition (+).
(b) If G satisfies condition (+) potentially, then there exists an open normal subgroup H of G such that H satisfies condition (+).
(c) If G satisfies condition (+) potentially and LLie (G/N ) is finite, thenN satisfies condition (+) potentially and is topologically finitely generated.
Proof. Assume througout that LLie (G/N ) is finite. Let Gℓ = prℓ (G) andNℓ = prℓ (N ). Assume G satisfies condition (+). Then, for all but finitely
many ℓ, we have Gℓ = G+ℓ and JH(Gℓ /Nℓ ) ∩ Lieℓ = ∅, so that [21, Lemma7

1.6] implies Nℓ = Nℓ+ , whence N satisfies condition (+). This proves (a).
Now assume that G satisfies condition (+) potentially. Then there exists
an openH0 of G such that H0 satisfies condition (+). If we let
T subgroup−1
H = g∈G g H0 g, then H is an open normal subgroup of G and satisfies(+) by (a). Thus (b) holds true. Furthermore H/N ∩H is a normal subgroupof G/N . It thus follows that LLie (H/N ∩ H) is finite, hence (a) implies thatthat N ∩ H satisfies condition (+). As N ∩ H is open in N , it follows that
N satisfies condition (+) potentially. Lemma 2.12 implies that N ∩ H is
topologically finitely generated. As N ∩ H is open in N , it follows that N
is topologically finitely generated. This finishes up the proof of (c).

3

Representations attached to cohomology

Throughout this section κ is a field of characteristic p ≥ 0, K/κ a finitely
generated extension and L′ = L \ {p}. Let X/K be a smooth projective
variety. Let i ∈ N, j ∈ Z. For every ℓ ∈ L′ consider the ℓ-adic étale
cohomology group Vℓ = H i (XK , Qℓ (j)).Q Consider the representaion ρℓ :Gal(K) → GLVℓ (Qℓ ). Let ρ : Gal(K) → ℓ∈L′ GLVℓ (Qℓ ) be the homorphism
induced by the ρℓ . For every field extension E/K there is a restriction map
rE/K : Gal(E) → Gal(K) and we define G(E) = ρ(rE/K (Gal(E))).
Lemma 3.1. Let ℓ ∈ L′ and let E/K be a separable algebraic field extension.
′
Let εℓ : Gal(K) → Q×ℓ be the cyclotomic character and let ρℓ = ρℓ ⊗ εℓ :Gal(K) → GLVℓ (Qℓ ). If E contains the ℓ-th roots of unity and ρℓ (Gal(E)) =ρℓ (Gal(E))+ , then ρ′ℓ (Gal(E)) = ρ′ℓ (Gal(E))+ .
Proof. By Lemma 2.4 there exists a lattice T in Vℓ such that ρℓ (Gal(K)) ⊂′GLT (Zℓ ). From εℓ (Gal(K)) ⊂ Z×ℓ we conclude that ρℓ (Gal(K)) ⊂ GLT (Zℓ ).
Let p : GLT (Zℓ ) → GLT (Fℓ ) be the projection and consider the residual
representations ρℓ = p ◦ ρℓ and ρ′ℓ = p ◦ ρ′ℓ . From ρℓ (Gal(E)) = ρℓ (Gal(E))+
it follows that ρℓ (Gal(E)) = ρℓ (Gal(E))+ . This implies ρ′ℓ (Gal(E)) =ρ′ℓ (Gal(E))+ because εℓ (Gal(E)) ⊂ 1 + ℓZℓ by our assumption on E. From
this it follows that ρ′ℓ (Gal(E)) = ρ′ℓ (Gal(E))+ because ker(p) is pro-ℓ.
There exists n ∈ N such that dimQℓ (Vℓ ) = n for all ℓ ∈ L′ by the Weil conjectures (cf. [8, Thm. 1.6], [13, Rem. 1.4]). We can thus choose isomorphisms
GLVℓ (Qℓ ) ∼= GLn (Qℓ ) and apply results from Section 2.
Proposition 3.2. (cf. [5, Theorem 7.5], [21, Theorem 3.1])
8

(a) The profinite group G(κK) satisfies condition (+) potentially.
(b) If κ is a number field, then there exists a finite Galois extension k/κ
such that G(k † K) satisfies condition (+) potentially.
Proof. Part (a) in case j = 0 is immediate from [5, Theorem 7.5]. As κ
contains all roots of unity H i (XK , Qℓ ) and H i (XK , Qℓ (j)) are isomorphic
as Gal(κK)-modules. Thus part (a) follows in general.
From now on assume that κ is a number field. Part (b) in case j = 0 isestablished in [21, Theorem 3.1]. By Lemma 3.1 part (b) follows in general.
Theorem 3.3. The profinite group G(κK) is topologically finitely generated. If Gal(κ) is topologically finitely generated (e.g., when κ is finite or
algebraically closed), then G(K) is topologically finitely generated.
Proof. By Proposition 3.2(a) and Lemma 2.14(c) the profinite group G(κK)
is topologically finitely generated. There exists an epimorphism
Gal(κsep K/K) → G(K)/G(κK)
and the profinite group Gal(κsep K/K) is isomorphic to an open subgroup ofGal(κ). Hence, if Gal(κ) is topologically finitely generated, then G(K)/G(κK)
is topologically finitely generated and it follows that G(K) is finitely generated.
Lemma 3.4. If κ is a local field and K = κ, then G(K) is topologicallyfinitely generated.
Proof. If K ∈ {R, C} is an archimedian local field, then Gal(K) is finite
and thus the assertion is trivially satisfied by Theorem 3.3. So assume that
K is a non-archimedian local field. Let q be the residue
Q characteristic of
the local field K. Let L = L′ \ {q}, ρ∗ : Gal(K) → ℓ∈L GLVℓ (Qℓ ) the
homomorphism induced by the ρℓ for ℓ ∈ L and G∗ (K) = ρ∗ (Gal(K)).
By Lemma 2.11 it is enough to show that G∗ (K) is topologically finitely
generated. Let I ⊂ Gal(K) be the inertia group and P the maximal normal
pro-q subgroup of I. By the semistable reduction theorem [4, Prop. 6.3.2]
there exists an open subgroup J of I such that for every ℓ ∈ L the actionof J on H i (XK , Qℓ ) is unipotent. Furthermore H i (XK , Qℓ ) is isomorphic to
H i (XK , Qℓ (j)) as a J-module because εℓ (J) = {1}. It follows that the action9

of J on H i (XK , Qℓ (j)) is unipotent. Thus ρℓ (J ∩ P ) = 0 for all ℓ ∈ L. Henceρ∗ (P ) is finite. As Gal(K)/I and I/P are topologically finitely generated,
it follows that G∗ (K) is topologically finitely generated, as desired.
Theorem 3.5. If κ is a local field, then G(K) is topologically finitely generated.
Proof. After replacing κ by a finite extension (and replacing the rest accordingly) we can assume that K/κ is separable (cf. [11, 4.6.7]) and primary.
Then there exists a geometrically connected smooth κ-scheme S with function field K. By the usual spreading-out principles, after replacing S bya non-empty open subscheme, we can assume that X extends to a smoothprojective S-scheme X such that f : X → S has geometrically connected
fibres and such that for every ℓ ∈ L′ the sheaf Ri f∗ Zℓ (j) is lisse and offormation compatible with any base change S ′ → S (cf. [13, Cor. 2.6]). In
particular ρℓ factors through π1 (S). After replacing κ by a finite separable
extension and replacing the rest accordingly we can assume that there exists a point s ∈ S(κ). Let σ be the section of π1 (S) → Gal(κ) induced by s(well-defined up to conjugation). There is a diagram
G(K)
O

ρ

1

/ π1 (Sκ )/ π1 (S) l-Gal(κ)/1σwith exact row. Now ρ(π1 (Sκ )) = G(κK) is topologically finitely generated by Theorem 3.3. By the base change compatibility the representation ρℓ ◦ σ of Gal(κ) is isomorphic to the representation of Gal(κ) on
H q (Xs,κ , Qℓ (j)) where Xs = f −1 (s). Hence ρ(σ(Gal(κ))) is topologicallyfinitely generated by Lemma 3.4. From the diagram we see that G(K) =ρ(π1 (Sκ )) · ρ(σ(Gal(κ))), hence G(K) is topologically finitely generated.
Remark 3.6. If κ is a finite extension of Qp , then Gal(κ) is known tobe finitely generated (cf. [14, Theorem 3.4]). Hence Theorem 3.5 in thatcase follows directly from Theorem 3.3. If κ is a finite extension of Fp ((t)),
however, then Gal(κ) is not finitely generated, and thus the arguments fromthe proof of Theorem 3.5 are needed essentially in that case.
Theorem 3.7. If κ is a number field, then there exists a finite Galois extension k/κ with the following property: If Ω/k † K is a Galois extension and
LLie (Gal(Ω/k † K)) is finite, then G(Ω) is topologically finitely generated.
10

Proof. Immediate from Proposition 3.2(b) and Lemma 2.14(c).

4Application to torsion fields of abelian varieties

Let K be a field of characteristic p ≥ 0, L′ = L \ {p}, A/K an abelianvariety and g = dim(A). For all ℓ ∈ L (including the case ℓ = p) we consider
the Tate module Tℓ := lim A[ℓj ](K) of the Barsotti Tate group A[ℓ∞ ], put←−j∈NVℓ = Tℓ ⊗Zℓ Qℓ and note that Tℓ is a free Zℓ -module with2g
if ℓ ̸= p,
rkZℓ (Tℓ ) = dimFℓ (A[ℓ](K)) =≤ 2g if ℓ = p.
There is a natural action of Gal(Kper ) 1 on Vℓ and a restriction isomorphismrKper /K : Gal(Kper ) → Gal(K), so we get a representaion ρℓ : Gal(K) ∼=Gal(Kper ) → GLVℓ (Qℓ ). We consider the homomorphisms
YYρ : Gal(K) →GLVℓ (Qℓ ) and ρ∗ : Gal(K) →GLVℓ (Qℓ )
ℓ∈L′

ℓ∈L

induced by the ρℓ . For every field extension E/K there is a restriction map
rE/K : Gal(E) → Gal(K) and we define GA (E) = ρ(rE/K (Gal(E))) andG∗A (E) = ρ∗ (rE/K (Gal(E))).
Remark 4.1. Let Ktor (A) := K(Ator ) ∩ Ksep and E/K a field extension.
The homomorphism ρ induces an isomorphism GA (E) ∼= Gal(EKtor (A)/E).
Theorem 4.2. Let κ be a field, K/κ a finitely generated extension and A/Kan abelian variety.
(a) The profinite group GA (κK) is topologically finitely generated.
(b) If Gal(κ) is topologically finitely generated (e.g., when κ is algebraicallyclosed or finite), then GA (K) is topologically finitely generated.
(c) If κ is a local field, then GA (K) is topologically finitely generated.
(d) If κ is a number field, then there exists a finite Galois extensionk/κ with the following property: If Ω/k † K is a Galois extension and
LLie (Gal(Ω/k † K)) is finite, then GA (Ω) is topologically finitely generated.
1

For ℓ ̸= p the passage to Kper is not necessary because then the finite group schemes
A[ℓj ] are étale over K and A[ℓj ](K) = A[ℓj ](Ksep ).

11

Proof. For every ℓ ∈ L \ {p} there is an Gal(K)-equivariant isomorphism
Vℓ (A) ∼, Qℓ (1)), where A∨ is the dual abelian variety. Hence the= H 1 (A∨K
statements (a)-(d) with GA replaced by G∗A are a consequence of Theorems
3.3, 3.5 and 3.7. Then Lemma 2.11 implies that the statements (a)-(d) holdtrue as they stand.
Remark 4.3. In the situation of Theorem 4.2(d) it follows that GA (Ω) is
topologically finitely generated for Ω ∈ {k † K, kab K, (kK)ab }. We introduced
the additional field Ω in part (d) because we need the flexibility in Section 5.
Proof of Theorem 1.1. This is an immediate consequence of Remark 4.1 andTheorem 4.2.
Proposition 4.4. Let A/K be an abelian variety over a field K and E/K
an algebraic extension. Let e ∈ N. Assume that GA (E) is topologicallyfinitely generated. There exists a finite separable extension M/K such thatF ⊂ EM for all F ∈ Ee (Ktor /K).
(e)

Proof. Let Fmax be the compositum of all fields in Ee (Ktor /K). Then
(e)
(e)
(e)
Fmax /K is Galois and Gal(Fmax /K) is periodic. Gal(EFmax /E) is a quotient(e)of GA (E) and isomorphic to a subgroup of Gal(Fmax /K). Hence the profi(e)nite group Gal(EFmax /E) is topologically finitely generated and periodic.
(e)
By [27, Theorem 1] and [25, Corollary, p. 58] the group Gal(EFmax /E) is(e)finite. It follows that [EFmax : E] < ∞. Hence there exists a finite separa(e)ble extension M/K such that Fmax ⊂ EM , and this extension M has thedesired property.
Proof of Corollary 1.5. There exists a number field k such that GA (k † K)
is topologically finitely generated by Theorem 4.2(d). By Proposition 4.4
applied with E = k † K there exists a finite extension M/K such that F ⊂k † M for all F ∈ Ee (Ktor /K). Replacing M by kM we get that k † M ⊂Mab .
Proof of Corollary 1.4. From Theore 4.2(b) and (c) we conclude that GA (K)is topologically finitely generated in the situation under consideration. The
assertion now follows by Proposition 4.4 applied with E = K.

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5Application to the weak Hilbert property

In this section we use exactly the same notation as the paper [2] and themanuscript [3]. In particular, if X is a normal variety over a field K ofcharacteristic zero and W ⊂ X(K), then W is said to be strongly thin if
there exists a finite family (fj : Yj → X)j=1,··· ,s of finite ramified morphisms
and a proper closed subset CSof X such that each Yj is normal and connected
and such that W ⊂ C(K) ∪ sj=1 fj (Yj (K)). Furthermore X is said to havethe weak Hilbert property (WHP for short), if X(K) is not strongly thin.
We shall strenghten and give a new proof of [2, Theorem 1.7], avoiding theoriginal arguments from [2, Section 5]. The new proof is based on Theorem
1.1, [3, Corollary 4.3] and [2, Theorem 1.4], where [2, Theorem 1.4] in turn
relies on [7] and builds on techniques from [12].
Theorem 5.1. Let K be a number field and A/K a geometrically simple
abelian variety. Assume that rk(A(Qab K)) = ∞. Let B/K be an abelianvariety. Then AK(Btor ) has property WHP over K(Btor ).
Proof. There exists a number field k such that, if we put E = kK, theprofinite group GB (Eab ) is topologically finitely generated (cf. Theorem 4.2
and Remark 4.3). Let Ktor = K(Btor ). Clearly
GB (Eab ) = Gal(Ktor Eab /Eab ) ∼= Gal(Ktor /Eab ∩ Ktor ),where the first equality follows by Remark 4.1, so this group is topologicallyfinitely generated. Let F = Eab ∩ Ktor . Now F E/E is an abelian Galois
extension, hence K ′ := E ∩ F is a finite extension of K such that Gal(F/K ′ )is abelian. Furthermore, due to the existence of the Weil pairing on B,
the field Ktor contains all roots of unity. It follows that Qab K ⊂ F . Thusrk(A(F )) = ∞. The following diagram shows the fields constructed so far.
Eab

E

FE

K′

F

K

Qab K

Eab Ktor

Ktor

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By [2, Theorem 1.4] it follows that AF = (AK ′ )F has WHP. From the finite
generation of Gal(Ktor /F ) and [3, Corollary 4.3] we obtain the assertion.
Corollary 5.2. (cf. [2, Theorem 1.7]) Let A/Q be an elliptic curve. Let
B/Q be an abelian variety. Then AQ(Btor ) has property WHP over Q(Btor ).
Proof. In that case A is obviously geometrically simple. Furthermore wehave rk(A(Qab )) = ∞ by [9, Lemma 2.1, Theorem 2.2]. The assertion now
follows by Theorem 5.1.
Remark 5.3. Let us briefly explain the relation with the conjecture of Zannier in [26, Section 2]. Consider the situation of Theorem 5.1 with A = B.
Let T = A(K)tor . Let f : X → A be a cover and assume X is geometricallyirreducible. Assume that X is not isomorphic to an abelian variety. Then fis necessarily ramified and M = f (X(Ktor )) is a strongly thin set. By Theorem 5.1 the complement of M in A(Ktor ) is Zariski dense. In particular
the complement of M ∩ T in A(Ktor ) is Zariski dense. The conjecture would
predict the much stronger statement that M ∩ T is contained in a proper
closed subset, and this without assumptions on the rank. Nevertheless one
can see Theorem 5.1 as a result providing at least some evidence for the
conjecture of Zannier.

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Wojciech Gajda
Faculty of Mathematics and Computer Science
Adam Mickiewicz University
Uniwersytetu Poznańskiego 461614 Poznań, Poland
E-mail adress: gajda@amu.edu.pl

Sebastian Petersen
Universität Kassel
Fachbereich 10
Wilhelmshöher Allee 71–73
34121 Kassel, Germany
E-mail address: petersen@mathematik.uni-kassel.de

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