Big monodromy theorem for abelian varieties
over finitely generated fields
Sara Arias-de-Reyna
University of Luxembourg
6 rue Richard Coudenhove-Kalergi
L-1359 Luxembourgsara.ariasdereyna@uni.lu

Wojciech Gajda∗
Department of Mathematics,
Adam Mickiewicz University,
61614 Poznań, Poland

Sebastian Petersen
Universität Kassel,
Fachbereich Mathematik,
34132 Kassel, Germany

gajda@amu.edu.plbasti.petersen@googlemail.com

June 19, 2012

Abstract
An abelian variety over a field K is said to have big monodromy, if
the image of the Galois representation on `-torsion points, for almost allprimes `, contains the full symplectic group. We prove that all abelianvarieties over a finitely generated field K with the endomorphism ring Z
and semistable reduction of toric dimension one at a place of the base fieldK have big monodromy. We make no assumption on the transcendence
degree or on the characteristic of K. This generalizes a recent result of
Chris Hall.

Introduction
It has been known in number theory, since times immemorial that Galois representation attached to the action of the absolute Galois group on torsion points∗ the corresponding author

2000 MSC: 11E30, 11G10, 14K15.
Key words and phrases: Abelian variety, Galois representation.

1

of an abelian group scheme carries a lot of basic arithmetic and geometric information. The first aim which one encounters naturally, while studying such
representations is to determine their images in terms of linear algebraic groups.
There exists a vast variety of results in the literature concerning computations ofGalois representations for abelian varieties defined over number fields and theirapplications to some classical questions such as Hodge, Tate and Mumford-Tateconjectures, see for example [30], [?], [?], [?] or [?]. In this paper we are interested in computing images of Galois representations attached to abelian varieties
defined over finitely generated fields in arbitrary characteristic, i.e., to families
of abelian varieties.
Let K be a field and denote by GK its absolute Galois group. Let A/K be an
abelian variety and ` 6= char(K) a prime number. We denote by ρA[`] : GK −→Aut(A[`]) the Galois representation attached to the action of GK on the `-torsion
points of A. We define MK (A[`]) := ρA[`] (GK ) and call this group the mod-`monodromy group of A/K. We fix a polarization and denote by e` :A[`] × A[`] →µ` the corresponding Weil pairing. Then MK (A[`]) is a subgroup of the group of
symplectic similitudes GSp(A[`], e` ) of the Weil pairing. We will say that A/K
has big monodromy if there exists a constant `0 such that MK (A[`]) contains
the symplectic group Sp(A[`], e` ), for every prime number ` ≥ `0 . Note thatthe property of having big monodromy does not depend on the choice of thepolarization.
Certainly, the most prominent result on computing monodromy groups is theclassical theorem of Serre (cf. [27], [28]): If A is an abelian variety over a finitely
generated field K of characteristic zero with End(A) = Z and dim(A) = 2, 6 orodd, then A/K has big monodromy. In this paper we consider monodromies forabelian varieties over finitely generated fields which have been recently investigated by Chris Hall [14], [15]. To simplify notation, we will say that an abelianvariety A over a finitely generated field K is of Hall type, if End(A) = Z and K
has a discrete valuation at which A has semistable reduction of toric dimension
one.
In the special case, when K = F (t) is a rational function field over anotherfinitely generated field, it has been shown by Hall that certain hyperelliptic
Jacobians have big monodromy; namely the Jacobians JC of hyperelliptic curves
C/K with affine equation C : Y 2 = (X − t)f (X), where f ∈ F [X] is a monicsquarefree polynomial of even degree ≥ 4 (cf. [14, Theorem 5.1]). Furthermore,
Hall has proved recently [15] the following theorem which in our notation reads:
If K is a global field, then every abelian variety A/K of Hall type has big
monodromy. We strengthen these results in our main theorem as follows.
Main Theorem. [cf. Thm. 3.6] If K is a finitely generated field (of arbitrary
characteristic) and A/K is an abelian variety of Hall type, then A/K has big
monodromy.
Our proof of the main Theorem follows Hall’s proof of [15] to some extent,e.g., we have borrowed a group theory result from [15] (cf. Theorem 3.4). In
addition to that we had to apply a substantial quantity of new methods toachieve the extension to all finitely generated fields, such as for instance finite2

generation properties of fundamental groups of schemes and Galois theory ofcertain division fields of abelian varieties, which are gathered in Section 2 andSection 3 of the paper. Furthermore, at a technical point in the case char(K) =0, we perform a tricky reduction argument (described in detail in Section 3) at aplace of K whose residue field is a number field. The paper carries an appendix
with a self contained proof of the group theoretical Theorem 3.4 due to Hall,
which can be of independent value for the reader.
Theorem A plays an important role in our paper [?], where we make progresson the conjecture of Geyer and Jarden (cf. [9]) on torsion of abelian varieties
over large algebraic extensions of finitely generated fields.
As a further application, we combine our monodromy computation with recent
results of Ellenberg, Hall and Kowalski in order to obtain the following result
on endomorphism rings and simplicity of fibres in certain families of abelianvarieties. If K is a finitely generated transcendental extension of another field
F and A/K is an abelian variety, then we call A weakly isotrivial with respecte
to F , if there is an abelian variety B/Fe and an K-isogeny
BKe → AKe .
Corollary. [cf. Cor. 4.3] Let F be a finitely generated field and K = F (t)the function field of P1 /F . Let A/K be an abelian variety. Let U ⊂ P1 be an
open subscheme such that A extends to an abelian scheme A/U . For u ∈ U (F )denote by Au /F the corresponding special fiber of A. Assume that A is not
weakly isotrivial with respect to F and that either of the conditions i) or ii)
listed below is satisfied.
i) A is of Hall type.
ii) char(K) = 0, End(A) = Z and dim(A) = 2, 6 or odd.
Then the sets:
X1 := {u ∈ U (F ) |End(Au ) 6= Z}

andX2 := {u ∈ U (F ) |Au /F is not geometrically simple}

are finite.
Note that Ellenberg, Elsholtz, Hall and Kowalski proved the statement of theCorollary in the special case when A is the Jacobian variety of the hyperelliptic
curve given by the affine equation Y 2 = (X − t)f (X), with f ∈ F [X] squarefree
and monic of even degree ≥ 4 (cf. [5, Theorem 8]). It is the case, where themonodromy of A is known by [14, Theorem 5.1]. We obtain part (i) of theCorollary as a consequence of the main theorem, our Proposition 4.2 below andalso Propositions 4 and 7 of [5]. In order to prove (ii) we use Serre’s Theorem[27], [28] instead of the main Theorem.
We warmly thank Gerhard Frey, Dieter Geyer, Cornelius Greither and Moshe
Jarden for conversations and useful comments on the topic of this paper. The
mathematical content of the present work has been much influenced by seminal
results of J.-P. Serre contained in [27], [28], [29], [30] and by the inspiring paper[15] of C.Hall. We acknowledge this with pleasure.

3

1

Notation and background material

In this section we fix notation and gather some background material on Galois
representations that is important for the rest of this paper.
Let X be a scheme. For x ∈ X we denote by k(x) the residue field at x. If X
is integral, then R(X) stands for the function field of X, that is, for the residue
field at the generic point of X. If X happens to be a scheme of finite type overa base field F , then we often write F (x) instead of k(x) and F (X) instead ofR(X).
e the separable (resp. algebraic)
If K is a field, then we denote by Ksep (resp. K)
closure of K and by GK its absolute Galois group. A finitely generated field
is by definition a field which is finitely generated over its prime field. For an
abelian variety A/K we let EndK (A) be the ring of all K-endomorphisms of A.
We denote by End(A) := EndKe (AKe ) the absolute endomorphism ring.
If Γ is an object in an abelian category and n ∈ Z, then nΓ : Γ → Γ is the morphism “multiplication by n” and Γ[n] is the kernel of nΓ . Recall that there is anequivalence of categories between the category of finite étale group schemes over
K and the category of finite (discrete) GK -modules, where we attach Γ(Ksep ) toa finite étale group scheme Γ/K. For such a finite étale group scheme Γ/K wesometimes write just Γ instead of Γ(Ksep ), at least in situations where we aresure that this does not cause any confusion. For example, if A/K is an abelianvariety and n an integer coprime to char(K), thenS we often write A[n] rather
than A(Ksep )[n]. Furthermore we put A[n∞ ] := i∈N A[ni ].
If M is a GK -module (for example M = µn or M = A[n] where A/K is anabelian variety), then we shall denote the corresponding representation of the
Galois group GK byρM : GK → Aut(M )
ker(ρ

)

and define MK (M ) := ρM (GK ). We define K(M ) := Ksep M to be the fixed
field in Ksep of the kernel of ρM . Then K(M )/K is a Galois extension andG(K(M )/K) ∼= MK (M ).
If R is a commutative ring with 1 (usually R = F` or R = Z` ) and M is a
finitely generated free R-module equipped with a non-degenerate alternating
bilinear pairing e : M × M → R0 into a free R0 -module of rank 1 (which is amultiplicatively written R-module in our setting below), then we denote bySp(M, e) = {f ∈ AutR (M ) |∀x, y ∈ M : e(f (x), f (y)) = e(x, y)}

the corresponding symplectic group and by
GSp(M, e) = {f ∈ AutR (M ) |∃ε ∈ R× : ∀x, y ∈ M : e(f (x), f (y)) = εe(x, y)}

the corresponding group of symplectic similitudes.
Let n be an integer coprime to char(K) and ` be a prime different from char(K).
Let A/K be an abelian variety. We denote by A∨ the dual abelian variety and4let en : A[n] × A∨ [n] → µn and e`∞ : T` A × T` A∨ → Z` (1) be the corresponding
Weil pairings. If λ : A → A∨ is a polarization, then we deduce Weil pairings eλn :A[n]×A[n] → µn and eλ`∞ : T` A×T` A → Z` (1) in the obvious way. If ` does notdivide deg(λ) and if n is coprime to deg(λ), then eλn and eλ`∞ are non-degenerate,
alternating, GK -equivariant pairings. Hence we have representations
ρA[n] : GK → GSp(A[n], eλn ),ρT` A : GK → GSp(T` A, eλ`∞ )with images MK (A[n]) ⊂ GSp(A[n], eλn ) and MK (T` A) ⊂ GSp(T` A, eλ`∞ ). We
shall say that an abelian variety (A, λ) over a field K has big monodromy, if there
is a constant `0 > max(char(K), deg(λ)) such that MK (A[`]) ⊃ Sp(A[`], eλ` ) for
every prime number ` ≥ `0 .
Now let S be a noetherian regular 1-dimensional connected scheme with function
field K = R(S) and A/K an abelian variety. Denote by A → S the Néron model(cf. [2]) of A. For s ∈ S let As := A ×S Spec(k(s)) be the corresponding fiber.
Recall that we say that A has good reduction at s provided As is an abelianvariety. In general, we denote by A◦s the connected component of As . If T isa maximal torus in A◦s , then dim(T ) does not depend on the choice of T [13,IX.2.1] and we call dim(T ) the toric dimension of the reduction As of A at s.
Finally recall that one says that A has semi-stable reduction at s, if A◦s is anextension of an abelian variety by a torus.
We shall also need the following connections between the reduction type of A andproperties of the Galois representations attached to A. Let s be a closed point of
S. The valuation v attached to s admits an extension to the separable closure
Ksep ; we choose such an extension v and denote by D(v) the corresponding
decomposition group. This is the absolute Galois group of the quotient fieldhhof the valuation ring OS,s of v. Hence) of the henselization OS,s
Ks = Q(OS,sthe results mentioned in [13, I.0.3] for the henselian case carry over to givethe following description of D(v): If I(v) is the kernel of the canonical map
D(v) → Gk(s) defined by v, then D(v)/I(v) ∼= Gk(s) . Let p be the characteristicof the residue field k(s) (p is zero or a prime number). I(v) has a maximal pro-p
subgroup P (v) (P (v) = 0 if p = 0) andY
I(v)/P (v) ∼Z` (1).
µ (k(s)sep ) ∼
= lim
=
←− n
n∈pZ
/

`6=p prime

Hence the maximal pro-`-quotient I` (v) of I(v) is isomorphic to Z` (1), if ` 6= p
is a prime.
Proposition 1.1. Let ` 6= p be a prime number. Assume that A has semi-stablereduction at s.
a) The image ρA[`] (P (v)) = {Id} and ρA[`] (I(v)) is a cyclic `-group.
b) Let g be a generator of ρA[`] (I(v)). Then (g − Id)2 = 0.
c) Assume that ` does not divide the order of the component group of As .
The toric dimension of A at s is equal to 2 dim(A) − dimF` (Eig(g, 1)) if
Eig(g, 1) = ker(g − Id) is the eigenspace of g at 1.
5

Proof. Part a) and b) are immediate consequences of [13, IX.3.5.2.].
Assume from now on that ` does not divide the order of the component groupof As . This assumption implies A◦s [`] ∼= As [`].
As we assumed A to be semi-stable at s, there is an exact sequence
0 → T → A◦s → B → 0
where T is a torus and B is an abelian variety and dim(T )+dim(B) = dim(As ) =dim(A). Now dimF` (T [`]) = dim(T ) and dimF` (B[`]) = 2 dim(B) = 2 dim(A) −2 dim(T ). Taking into account that we have an exact sequence
0 → T [`] → A◦s [`] → B[`] → 0(note that T (ek) ∼k × )dim(T ) is divisible by `), we find the relation dimF` (As [`]) == (e◦dimF` (As [`]) = 2 dim(A) − dim(T ). This implies c), because As [`] = A[`]I(v)([31, p. 495]) and obviously A[`]I(v) = Eig(g, 1) .
In general, if V is a finite dimensional vector space over F` , and g ∈ EndF` (V ),then one defines drop(g) = dim(V ) − dim(Eig(g, 1)). One calls g a transvection,
if it is unipotent of drop 1. We shall say that an abelian variety A over a field
K is of Hall type, provided End(A) = Z and there is a discrete valuation v on
K such that A has semistable reduction of toric dimension 1 at v (i.e. at the
maximal ideal of the discrete valuation ring of v). We have thus proved thefollowing
Proposition 1.2. If A is an abelian variety of Hall type over a finitely generated
field K, then there is a constant `0 such that MK (A[`]) contains a transvection
for every prime number ` ≥ `0 .

2

Finiteness properties of division fields

If A is an abelian variety over a field K (of arbitrary characteristic) and p =char(K), then we denote by A6=p the group of points in A(Ksep ) of order prime
to p. ThenY[
K(A6=p ) =K(A[`∞ ]) =K(A[n]).
n∈pZ/`6=p prime

If p = 0, then K(A6=p ) = K(Ator ). In this section we prove among other things:
If K is finitely generated of positive characteristic, then G(K(A6=p )/K) is a
finitely generated profinite group.
In this section, a function field of n variables over a field F will be a finitely
generated field extension E/F of transcendence degree n. As usual we call sucha function field E/F of n variables separable if it has a separating transcendency
base. The following Lemma is an easy consequence of [7, Proposition 3.1].
Lemma 2.1. (cf. [7]) Let F be a field and K/F a function field of one variable.
Assume that K/F is separable. Let p = char(F ). Let A/K be an abelian variety.
Let F 0 be the algebraic closure of F in K(A6=p ). Then G(K(A6=p )/F 0 K) is a
finitely generated profinite group.
6

Lemma 2.2. Let (K, v) be a discrete valued field, A/K an abelian variety withgood reduction at v, n an integer coprime to the residue characteristic of v,
L = K(A[n]) and w an extension of v to L. Denote the residue field of v
(resp. w) by k(v) (resp. k(w)). Let Av /k(v) be the reduction of A at v. Thenk(w) = k(v)(Av [n]).
Proof. Let R be the valuation ring of v and S = Spec(R). Let A → S be an
abelian scheme with generic fibre A. Then Av = A ×S Spec(k(v)), A[n] is a
finite étale group scheme over S, and if T is the normalization of S in L, thenthe restriction map r : A[n](L) ∼= A[n](T ) → Av [n](k(w)) is bijective (cf. [31]).
The assertion follows easily from that.

Definition 2.3. We shall say in the sequel that a field K has property F, if
G(K 0 (A6=p )/K 0 ) is a finitely generated profinite group for every finite separable
extension K 0 /K and every abelian variety A/K 0 .
Proposition 2.4. Let F be a field that has property F. Let p = char(F ). Let K
be a function field over F . Assume that K/F is separable. Then K has property
F.
Proof. By a routine induction on trdeg(K/F ) it is enough to prove the Proposition in the special case where K/F is a function field in one variable. We
may thus assume trdeg(K/F ) = 1 and we have to show that G(K 0 (A6=p )/K 0 ) is
finitely generated for every finite separable extension K 0 /K and every abelianvariety A/K 0 . But if K 0 /K is a finite separable extension, then K 0 /F is a separable function field of one variable again. Hence it is enough to prove that
G(K(A6=p )/K) is finitely generated for every abelian variety A/K.
Let A/K be an abelian variety. Let F0 be the algebraic closure of F in K. Then
K/F0 is a regular extension. Let C/F0 be a smooth curve with function field K
and such that A has good reduction at all points of C. There is a finite Galois
extension F1 /F0 such that C(F1 ) 6= ∅. If we put K1 := F1 K, then K1 /F1 isregular. Furthermore there is an exact sequence
1 → G(K1 (A6=p )/K1 ) → G(K(A6=p )/K) → G(K1 /K)
and G(K1 /K) is finite. If we prove that G(K1 (A6=p )/K1 ) is finitely generated,then it follows that G(K(A6=p )/K) is finitely generated as well. Hence we may
assume that K1 = K, i.e. that K/F is regular and that C(F ) 6= ∅.
Choose a point c ∈ C(F ) and denote by Ac /F the (good) reduction of A at c.
As in Lemma 2.1 denote by F 0 the algebraic closure of F in K(A6=p ).
Claim. F 0 ⊂ F (Ac,6=p ).
Let x ∈ F 0 . Then x is algebraic over F and x ∈ K(A[n]) for some n whichis coprime to p. If Fn denotes the algebraic closure of F in K(A[n]), thenx ∈ Fn . Let w be the extension to K(A[n]) of the valuation attached to c. Thenk(w) = F (Ac [n]) by Lemma 2.2. Obviously Fn ⊂ k(w). Hence x ∈ F (Ac [n]) ⊂F (Ac,6=p ). This finishes the proof of the Claim.

7

The profinite group G(F (Ac,6=p )/F ) is finitely generated, because F has property F by assumption. Hence its quotient G(F 0 /F ) is finitely generated aswell. Note that G(F 0 K/K) = G(F 0 /F ). On the other hand G(K(A6=p )/F 0 K)
is finitely generated by Lemma 2.1. From the exact sequence
1 → G(K(A6=p )/F 0 K) → G(K(A6=p )/K) → G(F 0 K/K) → 1
we see that G(K(A6=p )/K) is finitely generated as desired.



Corollary 2.5. Let K be a finitely generated field of positive characteristic or
K be a function field over an algebraically closed field of arbitrary characteristic.
Then K has property F. In particular G(K(A6=p )/K) is finitely generated forevery abelian variety A/K.
Proof. In both cases K is a function field over a perfect field F which hasproperty F. The assertion hence follows from Proposition 2.4 
Remark 2.6. A finitely generated field K of characteristic zero does not haveproperty F. In fact, if A/K is principally polarized abelian variety, then by theexistence of the Weil pairing K(Ator ) ⊃ K(µ∞ ), and plainly G(K(µ∞ )/K) is
not finitely generated, when K is a finitely generated extension of Q.

3

Monodromy Computations

Let K be a field and A/K an abelian variety. We begin with the question
whether A[`] is a simple GK -module for sufficiently large `.
Proposition 3.1. Let A be an abelian variety over a finitely generated field K.
Assume that EndK (A) = Z. Then there is a constant `0 such that A[`] is a
simple F` [GK ]-module for all primes ` ≥ `0 .
In the cases we need to consider, this Proposition is a consequence of the following classical result (cf. [6, p. 118, p. 204], [34], [35],[22]).
Theorem 3.2. (Faltings, Zarhin) Let K be a finitely generated field and A/Kan abelian variety. Then there is a constant `0 > char(K) such that the F` [GK ]module A[`] is semisimple and the canonical map EndK (A) ⊗ F` → EndF` (A[`])is injective with image EndF` [GK ] (A[`]) for all primes ` ≥ `0 .
Proof of Proposition 3.1. By Theorem 3.2 there is a constant `0 such that A[`]is a semisimple F` [GK ]-module with EndF` [GK ] (A[`]) = F` Id for every prime` ≥ `0 . This is only possible if A[`] is a simple F` [GK ]-module for all primes` ≥ `0 .

We need some notation in order to explain a theorem of Raynaud that will beof importance later. Let E/Fp be a finite field extension with |E| = pd and
F/Fp an algebraic extension. Denote by Emb(E, Fe) the set of all embeddings

8E → Fe. If i ∈ Emb(E, Fe) is one such embedding, then there is a unique function
e : Emb(E, Fe) → {0, · · · , p − 1} such thatY
(j|E × )e(j) ,χ=e)j∈Emb(E,Fand such that e(j) < p − 1 for some j ∈ Emb(E, Fe). We define amp(χ) :=max(e(j) : j ∈ Emb(E, Fe)) to be the amplitude of the character χ. Let ρ : E × →AutFp (V ) be a representation of E × on a finite dimensional Fp -vector space V . If
V is a simple Fp [E × ]-module, then there is a finite field FV with |FV | = |V | anda structure of 1-dimensional FV -vector space on V such that ρ factors througha character χρ : E × → FV× . We then define amp(V ) := amp(ρ) := amp(χρ ).
In general we define amp(V ) := amp(ρ) := max(amp(Vi ) : i = 1, · · · , t) where{V1 , · · · , Vt } is the set of Jordan-Hölder quotients of V to be the amplitude of
the representation ρ.
Theorem 3.3. (Raynaud [24], [26, p. 277]) Let A be an abelian variety overa number field K. Let v be a place of K with residue characteristic p. Let e be
the ramification index of v|Q. Let w be an extension of v to K(A[p]). Let I be theinertia group of w|v and P the p-Sylow subgroup of I. Let C ⊂ I be a subgroupthat maps isomorphically onto I/P . Then there is a finite extension E/Fp anda surjective homomorphism E × → C such that the resulting representation
ρ : E × → C → AutFp (A[p])has amplitude amp(ρ) ≤ e.
The technical heart of our monodromy computations is the following grouptheoretical result, which can be extracted from the work of C. Hall [14], [15].
Theorem 3.4. Let ` > 2 be a prime, let (V, eV ) be a finite-dimensional symplectic space over F` and M a subgroup of Γ := GSp(V, eV ). Assume that M
contains a transvection and that V is a simple F` [M ]-module. Denote by R the
subgroup of M generated by the transvections in M .
a) Then there is a non-zero symplectic subspace W ⊂ V , which is a simple
F` [R]-module, such that the following properties hold true:i) Let HL= StabM (W ). There is a orthogonal direct sum decomposition
V = g∈M/H gW . In particular |M/H| ≤ dim(V ).
∼Q∼Qii) R =
Sp(W ) and NΓ (R) =GSp(W ) o Sym(M/H).
g∈M/H

g∈M/H

iii) R ⊂ M ⊂ NΓ (R).
Denote by ϕ : NΓ (R) → Sym(M/H) the projection.
b) Let e ∈ N. Let E/F` be a finite extension and ρ : E × → M ⊂ GSp(V, eV )a homomorphism such that the corresponding representation of E × on V
has amplitude amp(ρ) ≤ e. If ` > dim(V )e + 1, then ϕ(ρ(E × )) = {1}.
Hall’s proof in [14], [15] adresses a slightly less general situation. We will present
a self-contained proof of Theorem 3.4 in the Appendix.
9Remark 3.5. Assume that in the situation of Theorem 3.4 the module V isa simple F` [ker(ϕ) ∩ M
Q]-module. Then V is in particular a simple F` [ker(ϕ)]module and ker(ϕ) = g∈M/H GSp(W ). This is only possible if M = H, V =W and R = Sp(V, e) ⊂ M .
We now state the main result of this section.
Theorem 3.6. Let K be a finitely generated field. Let (A, λ) be a polarizedabelian variety over K of Hall type. Then (A, λ) has big monodromy.
The case where K is a global field is due to Hall (cf. [15]) and we follow his lineof proof to some extent, but we need a lot of additional arguments in order tomake things work in the more general situation. The proof will occupy almost
the rest of this section.
There is a constant `0 > max(deg(λ), char(K)) such that the following holds
true for all primes ` ≥ `0 :1. The subgroup MK (A[`]) of GSp(A[`], eλ` ) contains a transvection. Denote by R` the subgroup of MK (A[`]) generated by the transvections in
MK (A[`]) (cf. Proposition 1.2).
2. A[`] is a simple F` [GK ]-module (cf. Proposition 3.1).
Now Hall’s group theory result (cf. Theorem 3.4) gives - for every prime ` ≥ `0
- a non-zero symplectic subspace W` ⊂ A[`], which is simple as a F` [R` ]-module
such that the properties i), ii) and iii) of Theorem 3.4 are satisfied. Let H` be
the stabilizer of W` under the action of MK (A[`]). Define M` := MK (A[`])and Γ` := GSp(A[`], eλ` ). ThenY
Y
Sp(W` , eλ` ) ∼Sp(W` , eλ` ) o Sym(M` /H` ),= R` ⊂ M` ⊂ NΓ` (R` ) =M` /H`

M` /H`

and we denote by ϕ` : NΓ` (R` ) → Sym(M` /H` ) the projection. We have thefollowing property (cf. Remark 3.5):
If A[`] is a simple F` [ker(ϕ` )∩M` ]-module for some prime ` ≥ `0 , then M` = H` ,W` = A[`] and M` ⊃ Sp(A[`], eλ` ) for this prime `.
We denote by N` the fixed field inside Ksep of the preimage ρ−1A[`] (M` ∩ ker(ϕ` )),where ρA[`] : GK → Γ` is the mod-` representation attached to A. Then N`
is an intermediate field of K(A[`])/K which is Galois over K, and G(N` /K) is
isomorphic to the subgroup ϕ` (M` ) of Sym(M` /H` ). In particular
Q [N` : K] ≤(2 dim(A))! is bounded independently of `. If we denotebyN:=`≥`0 prime N`T
the corresponding composite field, then GN = `≥`0 prime GN` . Hence thefollowing property holds true.
If A[`] is simple as a F` [GN ]-module for some prime ` ≥ `0 , then M` ⊃Sp(A[`], eλ` ) for this prime `.
(∗)
10

Proof of Theorem 3.6 in the special case char(K) > 0. If char(K) > 0, then the
Galois group G(K(A6=p )/K) (p := char(K)) is finitely generated, because K
then has property F by Corollary 2.5. Furthermore N` is an intermediate fieldof K(A6=p )/K which is Galois over K and with [N` : K] bounded independentlyof `. Hence N/K must be finite. In particular N is finitely generated. A second
application of the result of Faltings and Zarhin (cf. Proposition 3.1) yields a
constant `1 ≥ `0 such that A[`] is a simple F` [GN ]-module for all primes ` ≥ `0 .
Hence A has big monodromy by (∗).
To finish the proof of Theorem 3.6 we assume for the rest of the proof thatchar(K) = 0. We shall prove that N/K is finite also in that case, but the proofof this fact is more complicated, because now K is not F-finite (cf. Remark
2.6). We briefly sketch the main steps in the proof, before we go into thedetails: The first and hardest step is to show that the algebraic closure L of Q
in N is a finite extension of Q. In order to achieve this we will construct a finite
extension L0 /Q such that some L0 -rational “place” of KL0 splits up completely
into L0 -rational “places” of N` L0 for every sufficiently large prime `. We use
this to show that G(N L/KL) ∼= G(N Lsep /KLsep ) and the fact that the lattergroup can be proved to be finite, because KLsep is F-finite (unlike K itself).
This suffices to prove that N/K is finite. Once we know this, we shall proceed
as in the positive characteristic case above.
We now go into the details. Let F be the algebraic closure of Q in K. Then F
is a number field. Let S be a smooth affine F -variety with function field K suchthat A extends to an abelian scheme A over S with generic fibre A (i.e. suchthat A has good reduction along S). Let S` be the normalization of S in N`
and let S`0 be the normalization of S` in K(A[`]). Then S`0 → S` → S are finiteétale covers. (Note that char(F (s)) = 0 for every point s ∈ S.) In particular S`0
and S` are smooth F -schemes. (Compare the diagram below.)
Fix a geometric point P ∈ S(Fsep ) and denote by AP := A ×S Spec(F (P )) the
corresponding special fibre of A. Then AP is an abelian variety over the number
field F (P ). Fix for every ` ≥ `0 a geometric point Q` ∈ S` (Fsep ) over P and ageometric point Q0` ∈ S`0 (Fsep ) over Q` . Then F (Q0` )/F (Q` ) and F (Q` )/F (P )are finite extensions of number fields. Note that F (Q0` ) = F (P )(AP [`]) by
Lemma 2.2. Denote by O (resp. O` , resp. O`0 ) the integral closure of Z in F (P )(resp. in F (Q` ), resp. in F (Q0` )). For every prime ` ≥ `0 we have the following
diagram on the level of schemes

11Spec(F (P )(AP [`]))

Spec(K(A[`]))/ Spec(N` )/ Spec(K)


S`0
O


/ S`
O


/S
O

Spec(F (Q0` ))/ Spec(F (Q` ))/ Spec(F (P ))Spec(O`0 )
/ Spec(O` )f`/ Spec(O)We now study the ramification of prime ideals m ∈ Spec(O) in the extension
F (Q` )/F (P ). Let Pbad be the (finite) set of primes p ∈ Spec(O) where AP /F (P )has bad reduction.
Lemma 3.7. There is a constant `2 ≥ `0 with the following property: For every
prime number ` ≥ `2 the map f` : Spec(O` ) → Spec(O) is étale at every pointm ∈ Spec(O) outside of Pbad .
Proof. Let `2 := max(`0 , (2 dim(A))![F (P ) : Q] + 2).
Now let ` ≥ `2 be a prime number. Let m ∈ Spec(O) be an arbitrary prime
ideal with m ∈/ Pbad . We have to show that m is unramified in F (Q` ). Let
p = char(O/m) be the residue characteristic of m.
If p 6= `, then m is unramified even in F (Q0` ) = F (P )(AP [`]).
We can hence assume that p = ` . Let m` ∈ Spec(O` ) be a point over m andm0` ∈ Spec(O`0 ) a point over m` . Let D(m0` ) (resp. D(m` )) be the decompositiongroup of m0` /F (P ) (resp. of m` /F (P )) and I(m0` ) (resp. I(m` )) the corresponding inertia group. Let P (m0` ) (resp. P (m` )) be the (unique) p-Sylow subgroupof I(m0` ) (resp. I(m` )).
We have the following commutative diagram on the level of groups:

12

QM` /H` GSp(W` )

O



?

M` ∩ ker(ϕ` )G(K(A[`])/N` ) 
O



?

G(F (Q0` )/F (Q` )) 

/ / NΓ (M` )`O/ / Sym(M` /H` )
O

?
/M
`

?
/ / ϕ` (M` )/ G(K(A[`])/K)O/ / G(N` /K)O

?
/ G(F (Q0 )/F (P ))`O?
/ / G(F (Q` )/F (P ))O?
D(m0` )
O

?
/ / D(m` )O

?
I(m0` )
O

?
/ / I(m` )O?
P (m0` )?
/ / P (m` )We have to prove that the image of I(m0` ) in Sym(M` /H` ) by the maps inthe diagram is {1}. Now p = ` > (2 dim(A))! due to our choice of `2 and|Sym(M` /H` )| ≤ (2 dim(A))!, hence P (m0` ) maps to {1} in Sym(M` /H` ). In particular, P (m` ) = {1}. Consider the tame ramification group It = I(m0` )/P (m0` ).
It is a cyclic group of order prime to p. Choose a subgroup C ⊂ I(m0` ) thatmaps isomorphically onto It under the projection. It is enough to show that C
maps to {1} in Sym(M` /H` ).
By Raynaud’s theorem (cf. Theorem 3.3) there is a finite extension E/Fp andan epimorphism E × → C such that the resulting representation
E × → C → Aut(AP [`]) = Aut(A[`])
has amplitude ≤ e, where e is the ramification index of m over Q. Clearlye ≤ [F (P ) : Q]. By part b) of Theorem 3.4, the image of E × in Sym(M` /H` ) is{1}. Hence the image of C in Sym(M` /H` ) is {1} as desired.

Lemma 3.8. Let L be the algebraic closure of F in N . Then L/F is a finite
extension.
Q
Proof. Let L0 := `≥`0 prime F (Q` ). For every prime ` ≥ `2 the Galois extensionof number fields F (Q` )/F (P ) is unramfied outside Pbad by Lemma 3.7. Furthermore [F (Q` ) : F (P )] ≤ (2 dim(A))! for every primeQ ` ≥ `2 . The Theorem ofHermite-Minkowski (cf. [19], p. 122) implies that `≥`2 prime F (Q` ) is a finite
extension of F (P ). This in turn implies that L0 /F is a finite extension. It isthus enough to show that L ⊂ L0 .

13

Recall that K = F (S) is the function field of the F -variety S and S` is thenormalization of S in the finite Galois extension N` /K. Denote by Ŝ the normalization of S in N and by h` : Ŝ → S` the canonical projection. The canonicalmorphism Ŝ → S is surjective, hence there is a point P̂ ∈ Ŝ(Fsep ) over P . The
point h` (P̂ ) ∈ S` (Fsep ) lies over P . Hence h` (P̂ ) is conjugate to Q` under theaction of G(N` /K). This implies that F (h` (P̂ )) = F (Q` ). For every ` ≥ `0
there is a diagram
Spec(N )OŜSpec(F (P̂ ))h`/ Spec(N` )O/ Spec(K)
O

/ S`

/S


/ Spec(F (Q` ))/ Spec(F (P ))Q
where the morphisms S` → S are étale covers and N = `≥`0 N` . It follows
Q
that F (P̂ ) = `≥`0 F (Q` ) = L0 . On the other hand L is the algebraic closure of
F in N , hence Ŝ is a scheme over L. This implies that L is a subfield of F (P̂ ).
Hence in fact L ⊂ L0 as desired.

End of the proof of Theorem 3.6 in the case char(K) = 0. We have an isomorphism G(N Lsep /KLsep ) ∼= G(N/KL), because N/L and KL/L are regularextensions. The field KLsep is F-finite by Corollary 2.5. Hence the profinitegroup G(KLsep (Ator )/KLsep ) is finitely generated. As N Lsep ⊂ KLsep (Ator ),G(NQ Lsep /KLsep ) must be finitely generated as well. Furthermore N Lsep =`≥`0 N` Lsep where [N` Lsep : KLsep ] is bounded independently from `. Hence
G(N Lsep /KLsep ) is finite and this implies that N/KL is a finite extension. On
the other hand it follows from Lemma 3.8 that KL/K is finite. Hence N/K isa finite extension. Consequently N is finitely generated, because K is finitely
generated. Proposition 3.1 yields a constant `3 > `0 such that A[`] is a simple
F` (GN )-module for every prime ` ≥ `3 . Hence A/K has big monodromy by (∗),
as desired.


4

Applications

In this section we apply our methods to prove a generalization of a result ofEllenberg, Elsholz, Hall and Kowalski on endomorphism rings and simplicity offibres in certain families of abelian varieties (cf. [5, Theorem 8]).
Proposition 4.1. Let K be a field and (A, λ) a polarized abelian variety over
K with big monodromy. Let L/K be a finite extension. Then the following
properties hold.
a) There is a constant `0 ≥ max(char(K), deg(λ)) such that ML (A[`]) ⊃Sp(A[`], eλ` ) for every prime number ` ≥ `0 .
14b) A is geometrically simple.
Proof. Part a). Let E0 be the maximal separable extension of K in L andE/K a finite Galois extension containing E0 . By our assumption there is aconstant `0 > max(deg(λ), char(K), 5) such that MK (A[`]) ⊃ Sp(A[`], eλ` ) for
every prime ` ≥ `0 . For ` ≥ `0 let K` be the fixed field of Sp(A[`], eλ` ) inK(A[`])/K. Then MK` (A[`]) = Sp(A[`], eλ` ) and MEK` (A[`]) is a normal subgroup of MK` (A[`]) of index ≤ [E : K]. Put `1 := max(`0 , [E : K] + 1). Then
|MEK` (A[`])| ≥1|Sp(A[`], eλ` )| > 2[E : K]

for all primes ` ≥ `1 . On the other hand the only proper normal subgroups ofSp(A[`], eλ` ) are {±1} and the trivial group (cf. [30, p. 53]). Hence
ME0 (A[`]) ⊃ ME (A[`]) ⊃ MEK` (A[`]) = Sp(A[`], eλ` )for all primes ` ≥ `1 . As L/E0 is purely inseparable, we find
ML (AL [`]) = ME0 (A[`]) ⊃ Sp(A[`], eλ` )for all primes ` ≥ `1 as desired.
e be abelian varieties and f : A e → A1 × A2 an isogeny.
Part b). Let A1 , A2 /K
K
Then A1 , A2 and f are defined over some finite extension L/K. Hence there isan F` [GL ]-module isomorphism A[`] ∼= A1 [`] × A2 [`] for every prime ` > deg(f ).
By part a) ML (A[`]) ⊃ Sp(A[`]), eλ` ) for all sufficiently large primes `. Hence
A[`] is a simple F` [ML (A[`])]-module and in particular a simple F` (GL )-module
for all sufficiently large primes `. This is only possible if A1 = 0 or A2 = 0. 
Let F be a finitely generated field and K/F a finitely generated transcendental
field extension and A/K an abelian variety. We say that A/K is weakly isotrivialewith respect to F , if there is an abelian variety B/Fe and a K-isogeny
BKe → AKe .
Proposition 4.2. Let F be a finitely generated field, K/F a finitely generated
separable transcendental field extension and (A, λ) a polarized abelian variety
over K. Assume that A/K has big monodromy and that A/K is not weakly
isotrivial with respect to F . Define K 0 := Fsep K. Then there is a constant`0 ≥ max(char(K), deg(λ)) such that MK 0 (A[`]) = Sp(A[`], eλ` ) for every prime
number ` ≥ `0 .
Proof. Let `0 ≥ max(deg(λ), char(K), 5) be a constant such that MK (A[`]) ⊃Sp(A[`], eλ` ) for every prime ` ≥ `0 . Let ` ≥ `0 be a prime number. Then wehave
MK 0 (A[`]) ⊂ Sp(A[`], eλ` ) ⊂ MK (A[`]),
because K 0 contains µ` . Furthermore MK 0 (A[`]) a normal subgroup of MK (A[`]),
because K 0 /K is Galois. It follows that MK 0 (A[`]) is normal in Sp(A[`], eλ` ).
The only proper normal subgroups in Sp(A[`], eλ` ) are {1} and {±1} (cf. [30, p.
53]), because ` ≥ 5. Hence either MK 0 (A[`]) = Sp(A[`], eλ` ) or |MK 0 (A[`])| ≤ 2.
15

Let Λ be the set of prime numbers ` ≥ `0 where |MK 0 (A[`])| ≤ 2. We claimthat Λ is finite.
For every ` ∈ Λ we have [K 0 (A[`]) : K 0 ] ≤ 2. Furthermore G(K 0 (A6=p )/K 0 ) is
profinitely generated, where p = char(K). To see this note that
G(K 0 (A6=p )/K 0 ) = G(FeK 0 (A6=p )/FeK 0 )because Fe/Fsep is purely inseparable and use Corollary 2.5. Hence N :=Q00`∈Λ K (A[`]) is a finite extension of K . In particular N/Fsep is a finitely
generated regular extension. A/K must be geometrically simple by our assumption that A/K has big monodromy (cf. Proposition 4.1). In particular
AN is simple. Hence assumption that A is not weakly isotrivial with respect toF implies that the Chow trace TrN/Fsep (AN ) is zero. It follows by the MordellLang-Néron theorem (cf. [3, Theorem 2.1]) that A(N ) is a finitely generated
Z-module. In particular the torsion group A(N )tor is finite. On the other hand,A(N ) contains a non-trivial `-torsion point for every ` ∈ Λ. It follows that Λ isin fact finite.
Thus, after replacing `0 by a bigger constant, we see that MK 0 (A[`]) = Sp(A[`], eλ` )for all primes ` ≥ `0 .
Corollary 4.3. Let F be a finitely generated field and K = F (t) the function
field of P1 /F . Let A/K be a polarized abelian variety. Let U ⊂ P1 be an open
subscheme such that A extends to an abelian scheme A/U . For u ∈ U (F )denote by Au /F the corresponding special fiber of A. Assume that A is not
weakly isotrivial with respect to F and that either condition i) or ii) is satisfied.
i) A is of Hall type.
ii) char(K) = 0, End(A) = Z and dim(A) = 2, 6 or odd.
Then the sets:
X1 := {u ∈ U (F ) |End(Au ) 6= Z}

andX2 := {u ∈ U (F ) |Au /F is not geometrically simple}

are finite.
Proof. The abelian variety A/K has big monodromy. In case i) this follows by
Theorem 3.6. In case ii) this is a well-known theorem of Serre, cf. [27], [28].)
Define K 0 := Fsep K. As A/K is not weakly isotrivial with respect to F byassumption, Proposition 4.2 implies that there is a constant `0 > char(K) suchthat MK 0 (A[`]) = Sp(A[`], eλ` ) for all primes ` ≥ `0 . Hence AK 0 /K 0 has big
monodromy. Now Propositions 4 and 7 of [5] imply the assertion. Note thatthe notion of “big monodromy” in the paper [5] is slightly different from ours.

16

5

Appendix. Proof of Theorem 3.4

The aim of this Appendix is to provide a selfcontained proof of Theorem 3.4,
which was first proven in the papers [14] and [15]. We have also taken advantageof the exposition in [18].
Let ` > 2 be a prime number, let (V, e) be a finite-dimensional symplectic spaceover F` and Γ = GSp(V, e). In what follows M will be a subgroup of Γ whichcontains a transvection, such that V is a simple F` [M ]-module.
Remark 5.1.
• For a set U ⊂ V , we will denote by hU i the vector space
generated by U in V .
• For a vector u ∈ V and a scalar λ ∈ F` , we denote by Tu [λ] ∈ Γ themorphism v 7→ v + λe(v, u)u. For each transvection τ ∈ Γ there exist
u 6= 0, λ 6= 0 such that τ = Tu [λ], and hui = ker(τ − Id). If this is thecase we will say that hui is the direction of τ . Each nonzero vector in hui
shall be called a direction vector of τ .
• Given a group G ⊂ Γ, we will denote by L(G) the set of vectors u ∈ Vsuch that there exists a transvection in G with direction vector u.
• We will say that a group G ⊂ Γ fixes a vector space W if {g(w) : g ∈G, w ∈ W } ⊂ W .
The proof of Part iii) of Theorem 3.4 is quite simple and is based at the following
observation.
Lemma 5.2. Let G ⊆ GSp(V ) be a subgroup and R the subgroup of G generated
by the transvections in G. Then for all g ∈ G, r ∈ R, grg −1 ∈ R.
Proof. Note that if T = Tv [λ] ∈ G is a transvection, then gTv [λ]g −1 = Tgv [λ]is also a transvection, which belongs to G, therefore also to R. Now if wehave an element of R, say T1 ◦ · · · ◦ Tk for certain transvections T1 , . . . , Tk , theng(T1 ◦· · ·◦Tk )g −1 = (gT1 g −1 )◦· · ·◦(gTk g −1 ) is the composition of transvectionsof G, therefore an element of R.
Part i) of Theorem 3.4 is essentially Lemma 3.2 of [14]. Before proceeding toprove it, note the following elementary facts.
Lemma 5.3. Let G be a group P
that acts irreducibly on V , and let W ⊂ V anonzero vector space. Then V = g∈G gW .
Proof. Let S be the set S = {g(w) : g ∈ G, w ∈ W }. Consider the vector space
hSi. This vector space is fixed by G, hence since G acts irreducibly on V it
must coincide with V .
Lemma 5.4. Let W be a vector subspace of V , and assume that it is fixed by atransvection T = Tu [λ]. Then either u ∈ W or u ∈ W ⊥ .

17

Proof. Recall that, for all v ∈ V , T (v) = v + λe(v, u)u. If u 6∈ W , the only way
for T to fix W is that e(w, u) = 0 for all w ∈ W .

Proof of Theorem 3.4, i)
Consider the action of R on V . The first step is to fix one simple nonzero Rsubmodule W contained in V (This always exists because V is finite-dimensional
as an F` -vector space).
P
By Lemma 5.3, we know that V = g∈M gW . Moreover, for g1 , g2 ∈ M it
holdsP
that g1 W = g2 W if and only if g1 H = g2 H. Therefore we can write
V = g∈M/H gW , where H is the stabilizer of W in M . The proof of i) boils
down to prove that the sum is direct and orthogonal, that is, if g1 H 6= g2 H,
then g1 W ∩ g2 W = 0 and g1 W ⊂ (g2 W )⊥ . Equivalently, we will prove that for
any g ∈ M , if gW 6= W , then gW ∩ W = 0 and gW ⊥ W .
The first claim, namely gW 6= W implies gW ∩ W = 0 is easy. The key point is
to note that for each g ∈ M , gW is also fixed by R. Take r ∈ R, gw ∈ gW . Thenrgw = g(g −1 rg)w ∈ gW since g −1 rg ∈ R by Lemma 5.2 and hence fixes W .
Now it follows that W ∩ gW is fixed by R, and thus is an R-subrepresentationof W . But W is an simple R-module, hence since W ∩ gW 6= W , it must followthat gW ∩ W = 0.
To prove that gW 6= W implies gW ⊥ W , we need to make first the following
very important observation.
Claim 5.5. The set L(M ) ∩ W generates W .
Proof of Claim 5.5. First let us see that L(M ) ∩ W is nontrivial. Since anytransvection in M fixes W by definition of W , it follows by Lemma 5.4 that either
its direction vector belongs to W , or else it is orthogonal to W , in which case thetransvection acts trivially on W . But it cannot happen that all transvections in
M act trivially on W . For, if a transvection T acts trivially on W , then for allg ∈ M , gT g −1 acts trivially on gW . But since R = gRg −1 (because of Lemma5.2), thenP
if all R acts trivially on W , it also acts trivially on gW . Now recall
that V = g∈M gW . Then R would act trivially on V . But R contains at leasta transvection, and this does not act trivially on V . We have a contradiction.
Hence L(M )∩W is non zero. But now observe that this set is fixed by the actionof R, since the elements of M bring direction vectors into direction vectors.
Therefore the vector space hL(M ) ∩ W i ⊂ W is fixed by the action of R. Since
we are assuming W is an simple R-module, it follows that hL(M ) ∩ W i = W 
Now we are able to prove that if gW 6= W , then gW ⊂ W ⊥ . Because of theprevious claim, it suffices to show that, for any nonzero vector w ∈ W whichis the direction vector of a transvection in M , say T , w ∈ (gW )⊥ . Now recall
that, since T fixes gW , by Lemma 5.4 either w ∈ gW or w ∈ (gW )⊥ . But
gW ∩ W = 0, so w ∈ (gW )⊥ .

Before proving Part ii) of Theorem 3.4, we will introduce some notation.
Definition 5.6. Let g ∈ M . We will denote by Rg the subgroup of R generated18

by the transvections that act non-trivially on gW .
The following lemma is Lemma 7 of [15].
Lemma 5.7. Let g1 , g2 ∈ M with g1 H 6= g2 H. Then the commutator [Rg1 , Rg2 ]is trivial.
Proof. For i = 1, 2, let Ti ∈ Rgi be a transvection. We will see that they
commute. By Lemma 5.4 applied to gi W , either Ti acts trivially on gi W or
its direction vector, say ui , belongs to gi W . By definition of Rgi we have thesecond possibility. But because of Part i) of Theorem 3.4, for each g ∈ M suchthat gi W 6= gW , gi W ∩ gW = 0, hence ui 6∈ gW . Therefore again by Lemma
5.4 applied now to gW , it follows that Ti acts trivially on gW . Therefore T1
and T2 commuteon each gW , since at least one of them acts trivially on it.
L
Since V = g∈M/H gW , it follows that they commute on all V .
Proof of Theorem 3.4, ii).
Let M/H = {g1 H, . . . , gs H}, with g1 = Id. Define the map
P :

s
Y

Rgi → Ri=1

(r1 , r2 , . . . , rs ) 7→ r1 · r2 · · · · · rs .
Since by Lemma 5.7 elements from the different Rgi commute, this map is agroup homomorphism. Let us see that it is also an isomorphism.
Assume that r1 · r2 · · · · rs = Id, and that there is a certain rj which is not
the identity matrix. Then rj must act nontrivially on a certain vector v ∈ V .
SinceLthe elements of Rgj act trivially on the elements of gi W for i 6= j ands
V = i=1 gi W , we can assume that v ∈ gj W . But then the remaining ri with
i 6= j act trivially on v and on rj (v). Therefore Id(v) = r1 ·· · ··rs (v) = rj (v) 6= v,
which is a contradiction. To prove surjectivity, it suffices to note that eachtransvection T of M belongs to one of the Rgi , (hence each element of R canbe generated by elements of ∪i Rgi ). And this holds because, since T fixes all
the gi W , the direction vector of T must either belong to gi W or be orthogonal
to it because of Lemma 5.4, and since V Q= ⊕si=1 gi W it cannot be orthogonal tos
all the gi W . Therefore we get that R ' i=1 Rgi .
Now we are going to apply the following result [33, Main Theorem]:
Theorem 5.8. Suppose G ⊂ GL(n, k) is an irreducible group generated bytransvections. Suppose also that k is a finite field of characteristic ` > 2, andthat n > 2. Then G is conjugate in GL(n, k) to one of the groups SL(n, k0 ),Sp(n, k0 ) or SU(n, k0 ), where k0 is a subfield of k.
Note that, if n = 2, the result is also true and well known (cf. [4, Section 252]).
Now Rg1 is generated by transvections, and acts irreducibly on W (because R
acts irreducibly on W , and Rg1 is the group generated by all those transvections
19

in M that act nontrivially on W ). Therefore Rg1 is conjugated to Sp(W ). Since
all Rgi are conjugated
Qs to Rg1 , the same holds for them. Therefore we have theisomorphism R ' i=1 Sp(W ).
Qs
Qs
Finally, we can view H1 = i=1 GSp(W ) ' i=1 GSp(gi W ) as the subgroupof Γ fixing each gi W and, fixing a symplectic basis on each gi W , we can view
H2 = Sym(M/H) as the subgroup of Γ that permutes the gi W by bringing the
fixed symplectic basis of each gi W into the fixed symplectic basis of another
gj W . The group generated by H1 and H2 inside Γ, which is the group ofelements of Γ that permute the gi W , is the semidirect product H1 o H2 .
Recall that NΓ (R) = {g ∈ Γ : gRg −1 = R}. Note that g ∈ NΓ (R) if andonly if for all transvections T ∈ M , gT g −1 ∈ R. Now, if T = Tv [λ], it holds
that gT g −1 = Tg(v) [λ], and this transvection belongs to R if and only if it isa transvection of M , that is to say, if and only if g(v) ∈ L(M ). Therefore
gQ ∈ NΓ (R) if and only if g(L(M )) = L(M ). Now since R is isomorphic tos
i=1 Sp(gi W ), L(M ) is the disjoint union of the gi W . And moreover, if W isan R-module and g ∈ NΓ (R), then R fixes gW . Therefore, if W is an simple
R-module, then gW 6= W implies that gW ∩ W =Thus if g ∈ NΓ (R), thenQ0.
sg permutes the gi W . In otherQwords, NΓ (R) ⊂ i=1 GSp(W ) o Sym(M/H).
s
Reciprocally, each element
S
S of i=1 GSp(W ) o Sym(G/H) carries elements ofi gi W in elements ofi gi W , that is to say, carries L(M ) into L(M ), andtherefore belongs to NΓ (R).
This completes the proof of Part a) of Theorem 3.4.
Proof of Part b) of Theorem 3.4. Recall that (V, e) is a symplectic space overF` and M a subgroup of Γ := GSp(V, e). M contains a transvection and V isa simple F` [M ]-module by assumption. Furthermore R is the subgroup of M
generated by the transvections in M , 0 6= W ⊂ V is a simple F` [R]-moduleand H = StabM (W ). WeLalready proved that there is a orthogonal
directQ∼
sum decomposition V =gW.
Furthermore
R
Sp(W
),
=
g∈M/H
g∈M/H
Q
NΓ (R) ∼GSp(W)
oSym(M/H)
and
R
⊂
M
⊂
N
(R).
Denoteby= g∈M/HΓ
ϕ : NΓ (R) → Sym(M/H) the projection.
Let E/F` be a finite extension and ρ : E × → M ⊂ GL(V ) a representation of
amplitude amp(ρ) ≤ e. Assume that ` > e dim(V ) + 1. We have to prove thatϕ(ρ(E × )) = {1}.
Define S := ker(ϕ ◦ ρ) ⊂ E × . Then [E × : S] ≤ |M/H| ≤ dim(V ), and this
implies e[E × : S] < ` − 1. FurthermoreYρ(S) ⊂ ker(ϕ) ∼GSp(gW ).
=g∈M/HObviously ρ(S) commutes with the centerY
Z(ker(ρ)) ∼F×=` IdgWg∈M/Hof ker(ρ). Now by [15, Lemma 3] ρ(E × ) commutes with Z(ker(ρ)), becausee[E × : S] < ` − 1. It can easily be seen that the centralizer of Z(ker(ρ)) in20

NΓ (R) is equal to ker(ϕ) ∼=this implies ϕ ◦ ρ(E × ) = {1}.

Qg∈M/H GSp(gW ).

Hence ρ(E × ) ⊂ ker(ϕ) and

Acknowledgements. S. A. worked on this project as a research fellow of theAlexander von Humboldt Foundation. S. A. was partially supported by the Ministerio de Educación y Ciencia grant MTM2009-07024. S. A. wants to thank theHausdorff Research Institute for Mathematics in Bonn, the Centre de Recerca
Matemàtica in Bellaterra and the Mathematics Department of Adam Mickiewicz
University in Poznań for their support and hospitality while she worked on thisproject. W.G. and S.P. were partially supported by the Deutsche Forschungsgemeinschaft research grant GR 998/5-1. W.G. was partially supported by theAlexander von Humboldt Research Fellowship and by MNiSzW grant N N201
607440 “Arithmetic Geometry”. W.G. thanks Centre Recerca Matemàtica in
Bellaterra and the Max Planck Institut für Mathematik in Bonn for support andhospitality during visits in 2010, when he worked on this project. S.P. gratefully
acknowledges the hospitality of Mathematics Department of Adam Mickiewicz
University in Poznań and of the Minkowski center at Tel Aviv University during
several research visits.
The authors would like to thank the anonymous referee for a careful reading of
the paper and for his helpful remarks and suggestions.

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