Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

0.24 MB·English

by  Sara Arias-de-Reyna

Abelian varieties over finitely generated fields and
the conjecture of Geyer and Jarden on torsion
Sara Arias-de-Reyna Wojciech Gajda
2
1 Institut fu¨r Experimentelle Department of Mathematics,
0 Mathematik, Adam Mickiewicz University,
2
45326 Essen, Germany 61614 Poznan´, Poland

n
a [email protected] [email protected]
J

1
1 Sebastian Petersen∗

] Universit¨at Kassel,
G
Fachbereich Mathematik,
A
34132 Kassel, Germany
.
h
[email protected]
t
a
m
January 12, 2012
[

2
v
4
4
Abstract
4
2
. InthispaperweprovetheGeyer-Jardenconjectureonthetorsionpart
0 oftheMordell-Weilgroupforalargeclassofabelianvarietiesdefinedover
1
finitelygeneratedfieldsofarbitrarycharacteristic. Theclassconsistsofall
0
abelian varieties with big monodromy, i.e., such that the image of Galois
1
representation on ℓ-torsion points, for almost all primes ℓ, contains the
:
v full symplectic group.
i
X
r
a
∗thecorrespondingauthor
2000 MSC:11E30,11G10, 14K15.
Key words and phrases: Abelianvariety,Galoisrepresentation, Haarmeasure.
1
Introduction
Let A be a polarized abelian variety defined over a finitely generated field K.
Denote by K (respectively, K ) the algebraic (resp., separable) closure of K.
sep
It is well known that the Mordell-Weil group A(K) is a finitely generated Z-
e
module. On the other hand A(K) is a divisible group with an infinite torsion
partA(K) andA(K)hasinfiniterank,unlessK isalgebraicoverafinitefield.
tor e
Hence, it is of fundamental interest to study the structure of the groups A(Ω)
e e
for infinite algebraic extensions Ω/K smaller than K. For example, Ribet in
[26]and Zarhinin[37] consideredthe questionof finiteness ofA(K ) , where
e ab tor
K is the maximal abelian extension of K.
ab
WedenotebyG :=G(K /K)theabsoluteGaloisgroupofK. Forapositive
K sep
integer e and for σ = (σ ,σ ,...,σ ) in the group Ge = G ×G ×...×G ,
1 2 e K K K K
wedenotebyK (σ)the subfieldinK fixedbyσ ,σ ,...,σ . Thereexistsa
sep sep 1 2 e
substantial literature on arithmetic properties of the fields K (σ). In particu-
sep
lar,theMordell-WeilgroupsA(K (σ))havebeenalreadystudied,e.g.,Larsen
sep
formulatedaconjecturein[22]ontherankofA(K (σ))(cf. [2],[12]forresults
sep
supporting the conjecture of Larsen).
In this paper we consider the torsion part of the groups A(K (σ)). In order
sep
to recall the conjecture which is mentioned in the title, we agree to say that a
propertyA(σ)holds for almost all σ ∈Ge ,ifA(σ)holdsforallσ ∈Ge ,except
K K
for asetofmeasurezerowith respectto the (unique)normalizedHaarmeasure
on the compact group Ge . In [10] Geyer and Jarden proposed the following
K
conjecture on the torsion part of A(K (σ)).
sep
Conjecture of Geyer and Jarden. Let K be a finitely generated field. Let
A be an abelian variety defined over K.
a) For almost all σ ∈ G there are infinitely many prime numbers ℓ such
K
that the group A(K (σ))[ℓ] of ℓ−division points is nonzero.
sep
b) Lete≥2. For almost all σ ∈Ge thereare only finitely manyprime num-
K
bers ℓ such that the group A(K (σ))[ℓ] of ℓ−division points is nonzero.
sep
It is known due to the work of Jacobson and Jarden [18] that for all e ≥ 1,
almost all σ ∈ Ge and all primes ℓ the group A(K (σ))[ℓ∞] is finite. This
K sep
was formerly part (c) of the conjecture. Moreover the Conjecture is known for
elliptic curves [10]. Part (b) holds true provided char(K) = 0 (see [18]). In a
very recent preprint Zywina proves part (a) in the special case where K is a
number field (cf. [?]), stengthening results of Geyer and Jarden [11].
As for today, for an abelian variety A of dimension ≥ 2 defined over a finitely
generated field of positive characteristic, parts (a) and (b) of the Conjecture
of Geyer and Jarden are open and part (a) is open over a finitely generated
transcendental extension of Q.
2
In this paper we provethe Conjecture of GeyerandJardenfor abelianvarieties
withbigmonodromy. Toformulateourmainresultweneedsomenotation. Let
ℓ 6= char(K) be a prime number. We denote by ρ : G −→ Aut(A[ℓ]) the
A[ℓ] K
Galois representation attached to the action of G on the ℓ-torsion points of
K
A. We defineM (A[ℓ]):=ρ (G )andcallthisgroupthe mod-ℓ monodromy
K A[ℓ] K
group of A/K. We fix a polarization and denote by e :A[ℓ]×A[ℓ] → µ the
ℓ ℓ
corresponding Weil pairing. Then M (A[ℓ]) is a subgroup of the group of
K
symplectic similitudes GSp(A[ℓ],e ) of the Weil pairing. We will say that A/K
ℓ
has big monodromy if there exists a constant ℓ such that M (A[ℓ]) contains
0 K
the symplectic group Sp(A[ℓ],e ), for every prime number ℓ ≥ ℓ . Note that
ℓ 0
the property of having big monodromy does not depend on the choice of the
polarization.
The main result of our paper is the following
Main Theorem. [cf. Thm. 3.1, Thm. 5.1] Let K be a finitely generated field
andA/K an abelian variety withbig monodromy. Then theConjectureofGeyer
and Jarden holds true for A/K.
Surprisinglyenough,themostdifficulttoproveisthecase(a)oftheConjecture
for abelian varieties with big monodromy, when char(K) > 0. The method of
ourproofreliesinthiscaseonthe Borel-CantelliLemmaofmeasuretheoryand
onadelicatecountingargumentinthegroupSp (F )whichwasmodeledafter
2g ℓ
aconstructionofsubsetsS′(ℓ)inSL (F )inSection3oftheclassicalpaper[10]
2 ℓ
of Geyer and Jarden.
It is interesting to combine the main Theorem with existing computations of
monodromy groups for certain families of abelian varieties. Certainly, the most
prominentresultofthistypeistheclassicaltheoremofSerre(cf. [28],[29]forthe
numberfieldcase;thegeneralizationtofinitelygeneratedfieldsofcharacteristic
zero is well-known): If A is an abelian variety over a finitely generated field
K of characteristic zero with End(A) = Z and dim(A) = 2,6 or odd, then
A/K has big monodromy. Here End(A) = End (A ) stands for the absolute
K˜ K˜
endomorphism ring of A.
InthispaperwefocusourattentionatabelianvarietieswithEnd(A)=Z,which
havebeenrecentlyconsideredby Chris Hallinhis openimagetheorem[16]. To
simplifynotation,wewillsaythatanabelianvarietyAoverafinitelygenerated
field K is of Hall type, if End(A)=Z and K has a discrete valuation at which
Ahassemistable reductionoftoricdimensionone. The followingresult, proven
in our paper [1], gives examples of abelian varieties with big monodromy in all
dimensions (and including the case char(K)> 0): If A is an abelian variety of
HalltypeoverafinitelygeneratedinfinitefieldK,thenA/K hasbigmonodromy.
In the special case where K is a global field this has recently been shown by
Hall(cf. [15],[16]). Thegeneralizationtoanarbitraryfinitelygeneratedground
field K is carriedoutin [1] using methods of grouptheory,finiteness properties
ofthefundamentalgroupofschemesandGaloistheoryoflargefieldextensions.
In combination with the main Theorem we obtain the following
3
Corollary. Let A be an abelian variety over a finitely generated infinite field
K. Assume 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 Conjecture of Geyer and Jarden holds true for A/K.
Wethusobtainovereveryfinitelygeneratedinfinitefieldandforeverydimension
families ofabelianvarietiesforwhichthe ConjectureofGeyerandJardenholds
true. In the case when char(K) > 0 the Corollary offers the first evidence for
the conjecture of Geyer and Jarden on torsion going beyond the case of elliptic
curves.
We warmly thank Gerhard Frey, Dieter Geyer, Cornelius Greither and Moshe
Jarden for conversationsand useful comments on the topic of this paper.
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.
IfK isafield, thenwedenotebyK (resp. K)theseparable(resp. algebraic)
sep
closure of K and by G = G(K /K) its absolute Galois group. If G is a
K sep e
profinite(hencecompact)group,thenithasaunique normalizedHaarmeasure
µ . The expression “assertion A(σ) holds for almost all σ ∈G” means “asser-
G
tion A(σ) holds true for all σ outside a zero set with respect to µ ”. A finitely
G
generated field is by definition a field which is finitely generated over its prime
field. Let X be a scheme of finite type over a field K. For a geometric point
P ∈X(K˜) we denote by K(P)⊂K˜ the residue field at P.
For n ∈ N coprime to char(K), we let A[n] be the group of n-torsion points in
∞
A(K˜) and define A[n∞] = A[ni]. For a prime ℓ 6= char(K) we denote by
iS=1
T (A) = limA[ℓi] the ℓ-adic Tate module of A. Then A[n], A[n∞] and T (A)
ℓ ←− ℓ
i∈N
are G -modules in a natural way.
K
If M is a G -module (for example M = µ or M = A[n] where A/K is an
K n
abelian variety), then we shall denote the corresponding representation of the
Galois group G by
K
ρ :G →Aut(M)
M K
and define M (M):=ρ (G ). We define K(M):= Kker(ρM) to be the fixed
K M K sep
field in K of the kernel of ρ . Then K(M)/K is a Galois extension and
sep M
G(K(M)/K)∼=M (M).
K
4
If R is a commutative ring with 1 (usually R = F or R = Z ) and M is a
ℓ ℓ
finitely generated free R-module equipped with a non-degenerate alternating
bilinear pairing e : M ×M → R′ into a free R′-module of rank 1 (which is a
multiplicatively written R-module in our setting below), then we denote by
Sp(M,e)={f ∈Aut (M) | ∀x,y ∈M :e(f(x),f(y))=e(x,y)}
R
the corresponding symplectic group and by
GSp(M,e)={f ∈Aut (M) | ∃ε∈R× :∀x,y ∈M :e(f(x),f(y))=εe(x,y)}
R
the corresponding group of symplectic similitudes.
Letnbeanintegercoprimetochar(K)andℓbeaprimedifferentfromchar(K).
Let A/K be an abelian variety. We denote by A∨ the dual abelian variety and
leten :A[n]×A∨[n]→µn andeℓ∞ :TℓA×TℓA∨ →Zℓ(1)be the corresponding
Weilpairings. Ifλ:A→A∨ isapolarization,thenwededuceWeilpairingseλ :
n
A[n]×A[n]→µ andeλ :T A×T A→Z (1)intheobviousway. Ifℓdoesnot
n ℓ∞ ℓ ℓ ℓ
dividedeg(λ)andifniscoprimetodeg(λ),theneλ andeλ arenon-degenerate,
n ℓ∞
alternating, G -equivariant pairings. Hence we have representations
K
ρ :G →GSp(A[n],eλ),
A[n] K n
ρ :G →GSp(T A,eλ )
TℓA K ℓ ℓ∞
with images M (A[n]) ⊂ GSp(A[n],eλ) and M (T A) ⊂ GSp(T A,eλ ). We
K n K ℓ ℓ ℓ∞
shallsaythatanabelianvariety(A,λ)overafieldK hasbigmonodromy,ifthere
is aconstantℓ >max(char(K),deg(λ)) suchthatM (A[ℓ])⊃Sp(A[ℓ],eλ)for
0 K ℓ
every prime number ℓ≥ℓ .
0
2 Properties of abelian varieties with big mon-
odromy
Let (A,λ) be a polarized abelian variety with big monodromy over a finitely
generated field K. Then Sp(A[ℓ],eλ) ⊂ M (A[ℓ]) for sufficiently large primes
ℓ K
ℓ. In this section we determine M (A[n]) completely for every “sufficiently
K
large” integer n. The main result (cf. Proposition 2.4 below) is due to Serre in
the number field case, and the general case requires only a slight adaption of
Serre’slineofreasoning. However,asthefinaloutcomeissomewhatdifferentin
positivecharacteristic,wedoinclude the details. Proposition2.4willbe crucial
for our results on the Conjecture of Geyer and Jarden.
Now let K be an arbitrary field and A/K an abelian variety. Recall that for
every algebraic extension L/K we defined M (A[n]) = ρ (G ) (n coprime
L A[n] L
to char(K)) and M (T A) = ρ (G ) (ℓ > char(K) a prime number). Fur-
L ℓ TℓA L
thermore the representations induce isomorphisms G(L(A[n])/L) ∼= M (A[n])
L
and G(L(A[ℓ∞]/L) ∼= M (T A). Note that M (T A) → M (A[ℓi]) is surjec-
L ℓ L ℓ L
tive (because G(L(A[ℓ∞])/L) → G(L(A[ℓi])/L) is surjective) for every integer
i. Clearly M (A[n]) is a subgroup of M (A[n]).
L K
5
Remark 2.1. If L/K is a Galois extension, then M (A[n]) is a normal sub-
L
group of M (A[n]) andthe quotient groupM (A[n])/M (A[n]) is isomorphic
K K L
to G(L∩K(A[n])/K).
Proposition 2.2. Let K be a field and (A,λ) a polarized abelian variety over
K with big monodromy. Let L/K be an abelian Galois extension with L ⊃
µ . Then there is a constant ℓ > max(char(K),deg(λ)) with the following
∞ 0
properties.
a) M (T A)=Sp(T A,eλ ) for all primes ℓ≥ℓ .
L ℓ ℓ ℓ∞ 0
b) Let c be the product of all prime numbers ≤ ℓ . Then M (A[n]) =
0 L
Sp(A[n],eλ) for every integer n which is coprime to c.
n
Proof. Part a). There is a constant ℓ > max(char(K),deg(λ),5) such that
0
M (A[ℓ]) ⊃ Sp(A[ℓ],eλ) for all primes ℓ ≥ ℓ , because A has big monodromy.
K ℓ 0
Letℓ≥ℓ beaprimeanddefineK :=K(µ ). ThenbasicpropertiesoftheWeil
0 ℓ ℓ
pairing imply that G(K (A[ℓ])/K )∼=M (A[ℓ])=Sp(A[ℓ],eλ). This group is
ℓ ℓ Kℓ ℓ
perfect, because ℓ ≥ 5 (cf. [33, Theorem 8.7]). As L/K is an abelian Galois
ℓ
extension, M (A[ℓ]) is a normalsubgroup of the perfect groupM (A[ℓ]) and
L Kℓ
the quotient M (A[ℓ])/M (A[ℓ]) is isomorphic to a subquotient of G(L/K)
Kℓ L
(cf. Remark 2.1), hence abelian. This implies that
M (A[ℓ])=M (A[ℓ])=Sp(A[ℓ],eλ).
L Kℓ ℓ
Denote by p : Sp(T A,eλ ) → Sp(A[ℓ],eλ) the canonical projection. Then
ℓ ℓ∞ ℓ
M (T A) is a closed subgroup of Sp(T A,eλ ) with
L ℓ ℓ ℓ∞
p(M (T A))=M (A[ℓ])=Sp(A[ℓ],eλ).
L ℓ L ℓ
Hence M (T A)=Sp(T A,eλ ) by [21, Proposition 2.6].
L ℓ ℓ ℓ∞
Part b). Consider the map
ρ:G → M (T A)= Sp(T A,eλ )
L L ℓ ℓ ℓ∞
ℓY≥ℓ0 ℓY≥ℓ0
inducedbytherepresentationsρ anddenotebyX :=ρ(G )itsimage. Then
TℓA L
X isaclosedsubgroupof Sp(T A,eλ ). Ifpr denotestheℓ-thprojection
ℓ≥ℓ0 ℓ ℓ∞ ℓ
of the product, then pr (XQ) = Sp(T A,eλ ). Hence [31, Section 7, Lemme 2]
ℓ ℓ ℓ∞
implies that X = Sp(T A,eλ ), i.e. that ρ is surjective.
ℓ≥ℓ0 ℓ ℓ∞
Q
Let c be the product of all prime numbers ≤ ℓ . Let n be an integer coprime
0
to c. Then n = ℓ|n primeℓvℓ for certain integers vℓ ≥ 1. The canonical map
r :ML(A[n])→ Qℓ|n primeML(A[ℓvℓ]) is injective. Consider the diagram
Q
G ρ′ // M (T A) Sp(T A,eλ )
L ℓ|n L ℓ ℓ|n ℓ ℓ∞
Q Q
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
ML(A[n])(cid:31)(cid:127) r / ℓ|nML(A[ℓvℓ])(cid:31)(cid:127) / ℓ|nSp(A[ℓvℓ],eλℓvℓ).
Q Q
6
The vertical maps are surjective. The horizontal map ρ′ is surjective as well,
because ρ is surjective. This implies, that the lower horizontal map
ML(A[n])→ Sp(A[ℓvℓ],eλℓvℓ)
Y
ℓ|n
is in fact bijective. It follows from the Chinese Remainder Theorem that the
canonical map
Sp(A[ℓvℓ],eλℓvℓ)→Sp(A[n],eλn)
Y
ℓ|n
is bijective as well. Assertion b) follows from that. (cid:3)
Corollary 2.3. Let K be a field and (A,λ) a polarized abelian variety over
K with big monodromy. Then there is a constant c coprime to deg(λ) and
to char(K), if char(K) is positive, with the following property: M (A[n]) ⊃
K
Sp(A[n],eλ) for every integer n coprime to c.
n
Proof. LetL=K bethemaximalabelianextension. Thenthereisaconstant
ab
c as above, such that M (A[n]) = Sp(A[n],eλ) for every n coprime to c by
L n
Proposition 2.2. Furthermore M (A[n])⊂M (A[n]) by the discussion before
L K
Remark 2.1. (cid:3)
Let K be a field and (A,λ) a polarized abelian variety over K with big mon-
odromy. Thereisaconstantc(divisiblebydeg(λ)andbychar(K),ifchar(K)6=
0) such that
Sp(A[n],eλ)⊂M (A[n])⊂GSp(A[n],eλ)
n K n
for all n ∈ N coprime to c (cf. Corollary 2.3). One can easily determine
M (A[n]) completely, if K is finitely generated. Let K :=K(A[n]). There is
K n
a commutative diagram
// // // //
0 G(K /K(µ )) G(K /K) G(K(µ )/K) 0
n n n n
ρA[n] ρµn
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //Sp(A[n],eλ) // GSp(A[n],eλ) ε //(Z/n)× //0
n n
withexactrowsandinjectiveverticalmaps,whereρ isthecyclotomiccharac-
µn
terandεisthemultiplicatormap. Thelefthandverticalmapisanisomorphism
for every n∈N coprime to c. Hence
M (A[n])={f ∈GSp(A[n],eλ) | ε(f)∈im(ρ )}.
K n µn
Assume fromnow onthatK is finitely generated. Then the image ofthe cyclo-
tomic character involved above has a well known explicit description. Denote
by F the algebraicclosureofthe prime fieldof K inK anddefine q :=q(K):=
|F|∈N∪{∞}. Then, after possibly replacing c by a larger constant, we have
hqi char(K)6=0,
im(ρ )=
µn (cid:26) (Z/n)× char(K)=0.
7
foralln∈Ncoprimeto c. Herehqiis the subgroupof(Z/n)× generatedbythe
residue class q of q modulo n, provided q is finite. If q is finite, then we define
GSp(q)(A[n],eλ)={f ∈GSp(A[n],eλ) | ε(f)∈hqi}.
n n
Finally we put GSp(∞)(A[n],eλ)=GSp(A[n],eλ). We have shown:
n n
Proposition 2.4. Let K be a finitely generated field and (A,λ) a polarized
abelian variety over K with big monodromy. Let q = q(K). Then there is
a constant c (divisible by deg(λ) and by char(K), if char(K) 6= 0) such that
M (A[n])=GSp(q)(A[n],eλ) for all n∈N coprime to c.
K n
3 Proof of the Conjecture of Geyer and Jarden,
part b)
Let (A,λ) be a polarized abelian variety of dimension g over a field K. In this
section we will use the notation K := K(A[ℓ]) and G := G(K /K) for every
ℓ ℓ ℓ
prime ℓ6=char(K). Our main result in this section is the following theorem.
Theorem 3.1. If (A,λ) has big monodromy, then for all e≥2 and almost all
σ ∈Ge (in the sense of the Haar measure) there are only finitely many primes
K
ℓ such that A(K (σ))[ℓ]6=0.
sep
The following Lemma 3.2 is due to Oskar Villareal (private communication).
We thank him for his kind permission to include it into our manuscript.
Lemma 3.2. Assume that A has big monodromy. Then there is a constant ℓ
0
such that [K(P) : K]−1 ≤ [Kℓ : K]−21g for all primes ℓ ≥ ℓ0 and all P ∈ A[ℓ],
where K(P) denotes the residue field of the point P.
Proof. By assumption on A, there is a constant ℓ such that Sp(A[ℓ],eλ) ⊂
0 ℓ
M (A[ℓ]) for all primes ℓ≥ℓ . Let ℓ≥ℓ be a prime and P ∈A[ℓ]. Then the
K 0 0
F -vectorspacegeneratedinside A[ℓ]bythe orbitX :={f(P):f ∈M (A[ℓ])}
ℓ K
is the whole of A[ℓ], because A[ℓ] is a simple F [Sp(A[ℓ],eλ)]-module. Thus
ℓ ℓ
we can choose an F -basis (P ,··· ,P ) of A[ℓ] with P = P in such a way
ℓ 1 2g 1
that each P ∈ X. Then each P is conjugate to P under the action of G
i i K
and [K(P) : K] = [K(P ) : K] for all i. The field K is the composite field
i ℓ
K =K(P )···K(P ). It follows that
ℓ 1 2g
[K :K]≤[K(P ):K]···[K(P ):K]=[K(P):K]2g.
ℓ 1 2g
The desired inequality follows from that. (cid:3)
The following notation will be used in the sequel: For sequences (x ) and
n n
(y ) of positive real numbers we shall write x ∼ y , provided the sequence
n n n n
(xn) converges to a positive real number. If x ∼ y and x < ∞, then
yn n n n
y <∞. P
n
P
The proof of Theorem 3.1 will make heavy use of the following classical fact.
8
Lemma 3.3. (Borel-Cantelli, [9, 18.3.5]) Let (A ,A ,···) be a sequence of
1 2
measurable subsets of a profinite group G. Let
∞ ∞
A:= A ={x∈G:x belongs to infinitely many A }.
i i
n\=1i[=n
a) If ∞ µ (A )<∞, then µ (A)=0.
i=1 G i G
P
b) If ∞i=1µG(Ai) = ∞ and (Ai)i∈N is a µG-independent sequence (i.e.
forPevery finite set I ⊂ N we have µG( i∈IAi) = i∈IµG(Ai)), then
µG(A)=1. T Q
Proof of Theorem 3.1. Assume that A/K has big monodromy and let ℓ be a
0
constant as in the definition of the term “big monodromy”. We may assume
that ℓ ≥char(K). Let e≥2 and define
0
X :={σ ∈Ge :A(K (σ))[ℓ]6=0}
ℓ K sep
for every prime ℓ. Let µ be the normalized Haar measure on Ge . Theorem 3.1
K
follows from Claim 1 below, because Claim 1 together with the Borel-Cantelli
Lemma 3.3 implies that
X
ℓ
n\∈Nℓ≥n[prime
has measure zero.
Claim 1. The series µ(X ) converges.
ℓ prime ℓ
P
Let ℓ≥ℓ be a prime number. Note that
0
X = {σ ∈Ge | σ (P)=P for all i}= Ge .
ℓ K i K(P)
[ [
P∈A[ℓ]\{0} P∈A[ℓ]\{0}
Let P(A[ℓ]) = (A[ℓ]\{0})/F× be the projective space of lines in the F -vector
ℓ ℓ
space A[ℓ]. It is a projective space of dimension 2g−1. For P ∈ A[ℓ]\{0} we
denote by P :=F×P the equivalence classofP inP(A[ℓ]). For P ∈P(A[ℓ]) and
ℓ
P ,P ∈ P there is an a ∈ F× such that P = aP and P = a−1P , and this
1 2 ℓ 1 2 2 1
implies K(P )=K(P ). It follows that we can write
1 2
X = Ge .
ℓ K(P)
P∈P[(A[ℓ])
Hence
µ(X )≤ µ(Ge )= [K(P):K]−e,
ℓ K(P)
P∈XP(A[ℓ]) P∈XP(A[ℓ])
and Lemma 3.2 implies
ℓ2g−1 ℓ2g−1
µ(X )≤ [K :K]−e/2g = [K :K]−e/2g = |G |−e/2g.
ℓ ℓ ℓ ℓ
ℓ−1 ℓ−1
P∈XP(A[ℓ])
9
But G contains Sp (F ) and
ℓ 2g ℓ
g
s :=|Sp (F )|=ℓg2 (ℓ2i−1)
ℓ 2g ℓ
iY=1
(cf. [33]). It is thus enough to prove the following
Claim 2. The series ℓ2g−1s−e/2g converges.
ℓ≥ℓ0 prime ℓ−1 ℓ
P
But s ∼ℓg2+2+4+···+2g =ℓ2g2+g and ℓ2g−1 ∼ℓ2g−1, hence
ℓ ℓ−1
ℓ2g−1
s−e/2g ∼ℓ2g−1ℓ−e(g+21) =ℓ(2−e)g−(1+2e) ≤ℓ−2,
ℓ−1 ℓ
because e≥2. Claim 2 follows from that. (cid:3)
4 Special sets of symplectic matrices
Thissectioncontainsaconstructionofcertainspecialsetsofsymplecticmatrices
(cf. Theorem 4.6 below) that will play a crucial role in the proof of part a) of
the Conjecture of Geyer and Jarden.
Let g ≥2, and let V be a vector space of dimension 2g over a prime finite field
F , endowed with a symplectic form e : V ×V → F . Fix a symplectic basis
ℓ ℓ
E ={e ,...,e } of V such that the symplectic form is given by the matrix
1 2g
J
1
 J1  0 1
Jg = ...  where J1 =(cid:18)−1 0(cid:19).
 
 J 
1
 
For each A ∈ GSp (F ) there is an element λ ∈ F× such that e(Av,Aw) =
2g ℓ ℓ
λe(v,w)forallv,w ∈V. Wewillsaythatthevalueλ=ε(A)ofthemultiplicator
map ε is the multiplier of A, and we will denote by GSp (F )[λ] the set of
2g ℓ
matrices in GSp (F ) with multiplier λ.
2g ℓ
Remark 4.1. Here we collect some notation. Let p be a prime, q a power of p
and n∈N.
• For n not divisible by p, we will denote by ord q the order of q modulo n.
n
(q)
• Denote by GSp (Z/nZ) the set of matrices in GSp (Z/nZ) with multi-
2g 2g
plier equal to a power of q modulo n.
• Denote by GSp(∞)(Z/nZ):=GSp (Z/nZ).
2g 2g
• Let α ,α ,...,α ,β ∈F . Call u =e +α e +···+α e . We denote
3 4 2g ℓ α 2 3 3 2g 2g
by T [β] the morphism v 7→ v+βe(v,u )u (which is a transvection if
uα α α
β 6=0).
10