A variational open image theorem in positive characteristic

0.18 MB·

by  Gebhard Böckle

A VARIATIONAL OPEN IMAGE THEOREM IN POSITIVE
CHARACTERISTIC
GEBHARDBO¨CKLE,WOJCIECHGAJDAANDSEBASTIANPETERSEN
7
1 Abstract. In this note we prove a variational open adelic image theorem for the Galois
0 action on thecohomology ofsmooth proper S-schemes whereS isa smoothvariety over a
2 finitelygeneratedfieldofpositivecharacteristic. AcentraltoolisarecentresultofCadoret,
HuiandTamagawa.
n
a
J Introduction

9
1 Let k be a finitely generated infinite field of characteristic p > 0, S a smooth geometrically
connected k-variety of positive dimension and f : X → S a smooth proper morphism of
G] schemes. Let K = k(S) be the function field of S and let X/k(S) be the generic fibre of X.
Fix j ∈N. For every prime number ℓ6=p we define V :=Hj(X ,Q ) and let
A ℓ K ℓ
h. ρℓ∞ :π1(S)→GLVℓ(Qℓ)
at be the representationof π1(S) on the Qℓ-vector space Vℓ. We write
m ρ:π1(S)→ YGLVℓ(Qℓ)
[ ℓ6=p

2 for the induced adelic representation Qℓ6=pρℓ∞.
v
For every point s ∈ S with residue field k(s) we denote by s : Gal(k(s)) → π (S) the homo-
7 ∗ 1
4 morphism induced by s (well-defined up to conjugation), and for any group homomorphism
7 τ : π1(S) → H, we define τs := τ ◦ s∗ : Gal(k(s)) → H as the specialization of ρ at s.
4 Note that ρ is isomorphic to the representation of Gal(k(s)) on Hj(X ,Q ) where
ℓ∞,s s,k(s) ℓ
0 X =X × Spec(k(s)) is the special fibre of X in s, see [15, VI. Cor. 4.2].
. s S
1
Theaimofthispaperistostudythevariationofthemonodromygroupsρ (Gal(k(s)))(resp.
0 ℓ∞,s
7 ρs(Gal(k(s)))) for closed points s ∈ S in comparing them to the corresponding monodromy
1 group ρ (π (S)) (resp. ρ(π (S))) of the generic point of S.
ℓ∞ 1 1
:
v For everyprime number ℓ6=p let G(ρ ) (resp. G(ρ )) be the connected componentof the
ℓ∞,s ℓ∞
i
X Zariski closure of ρℓ∞,s(Gal(k(s))) (resp. of ρℓ∞(π1(S))) in GLVℓ/Qℓ, and define
r Sgen(ρ )={s∈S a closed point:G(ρ )=G(ρ )}
ℓ∞ ℓ∞,s ℓ∞
a
Being in Sgen(ρ ) is a prioriweaker than being ℓ-Galoisgeneric in the sense of Cadoret-Kret
ℓ∞
(see [5, 3.1], [2, 1.5.3] and [14, § 6]). By Theorem 1(c), however, the notions are equivalent.
The following result is the main theorem of the present work.
Theorem 1. (see Proposition 2.7, Lemma 2.1, Theorem 3.4)
(a) The sets Sgen(ρ ) are independent of ℓ.Let Sgen(X/S):=Sgen(ρ ) for any ℓ6=p.
ℓ∞ ℓ∞
(b) The set Sgen(X/S) is Zariski dense in S, and in particular it is infinite.
(c) The group ρ (Gal(k(s))) is open in ρ(π (S)) for every s∈Sgen(X/S).
s 1
2010 Mathematics Subject Classification. 11G10,14K15.
Key words and phrases. Compatiblesystem,adelicopenness, positivecharacteristic.
1
2 GEBHARDBO¨CKLE,WOJCIECHGAJDAANDSEBASTIANPETERSEN
The above result relies on a similar result that holds if one replaces S by its base change S
Fpk
under k →F k. This base change allows us to apply standardtools to derive (a) and (b), and
p
recent results from [4] by Cadoret, Hui and Tamagawato deduce (c). Given these results, the
proof of Theorem 1 is rather elementary.
In the case where k is a finitely generated field of zero characteristic, Cadoret established a
theoremanalogousto our main theorem in the case where X/S is an abelian scheme (cf. [2]).
HertheoremoffersastrongtooltoreducetheproofofconjecturesaboutGaloisrepresentations
attached to abelian varieties over finitely generated fields of zero characteristic to the number
fieldcase. Similarlyour theoremcanbe usedinorderto reducethe proofofconjecturesabout
smoothpropervarietiesoverfinitelygeneratedfieldsofpositivecharacteristictothecasewhere
the ground field is a global function field.
The results from [4] also give one a rather precise conceptual description of ρ(π (S )), as
1 Fpk
we shall explain in Section 4, and as is presumably well-known to the authors of [4]: Let D
ℓ∞
denote the connected component of the Zariski closure of ρ (π (S )), or, equivalently, the
ℓ∞ 1 Fpk
derivedgroupofG(ρ )(seeCorollary1.4). InTheorem4.1weprovethatfors∈Sgen(X/S)
ℓ∞
the group ρ (Gal(F k(s))) generates a special adelic subgroup in Dsc (Q ) in the sense
s p Qℓ6=p ℓ∞ ℓ
of Hui and Larsen [11], where Dsc →D denotes the simply connected cover.
ℓ∞ ℓ∞
After submitting our manuscript to ArXiv we were informed by Anna Cadoret that she has a
manuscript [3] with similar results, now also available on her homepage.
Acknowledgements
Part of this work was done during a stay of the three authors at Adam Mickiewicz university
in Poznan´ financed by NCN grant no. UMO-2014/15/B/ST1/00128. G.B. was supported by
theDFGwithinthe SPP1489andthe FG1920. Apreviousversionofthis workaddressedonly
the caseofabelianvarietiesoverabasevarietyinpositivecharacteristic. We thank the referee
of that version for much valuable feedback that is also relevant for the present generalization.
Notation
For a field k we denote by k an algebraic closure and by Gal(k) the absolute Galois group
of k. For a k-variety S we denote by k(S) its function field, by |S| its set of closed points,
and by π (S) the ´etale fundamental group of S with base point the geometric generic point
1
Spec(k(S))→S. If chark =p, let S denote the base change of S under k →F k.
Fpk p
Suppose that V is a finite-dimensional Q -vector space, Π is a profinite group and ρ : Π →
ℓ
GL (Q )isacontinuoushomomorphism. We denotebyG(ρ)the connectedcomponentofthe
V ℓ
Zariski closure of ρ(Π) in GL . Then G(ρ) is an algebraic group over Q . It is reductive if ρ
V ℓ
is semisimple (see [16, 22.138]). We write ρ|H for the restriction of ρ to a subgroup H of Π,
and we denote by Π+ the closed subgroup of Π generated by its pro-ℓ Sylow subgroups.
If Π=π (S), then we define the set of Galois generic points with respect to ρ as
1
Sgen(ρ):={s∈|S|:G(ρ)=G(ρ )}.
s
By L′ we denote the set of all prime numbers ℓ 6= p. For a linear algebraic group G defined
over a field, we denote by DG its derived group.
Let K be a finitely generated field of characteristic p > 0. We call a family (ρ : Gal(K) →
ℓ∞
GLVℓ(Qℓ))ℓ∈L′ of continuous homomorphisms, where the Vℓ are finite-dimensional Qℓ-vector
spaces,astrictly compatible system over K pure of weight j ifthere existsa smoothF -variety
p
T with F (T)=K such that the following properties (i) and (ii) hold:
p
(i) ρ factors through π (T) for every ℓ∈L′.
ℓ∞ 1
A VARIATIONAL OPEN IMAGE THEOREM IN POSITIVE CHARACTERISTIC 3
1
(ii) For every t∈|T|, denoting by Frt ∈Gal(Fp(t)) the arithmetic Frobenius x7→x|Fp(t)|,
the characteristic polynomial of ρ (Fr ) has coefficients in Z, it is independent of ℓ,
ℓ∞,t t
and its roots all have absolute value |Fp(t)|2j.
1. Preliminaries
In this section we collect basic results, mostly not due to the present authors, for use in later
sections. Let K be a finitely generatedinfinite fieldof characteristicp>0 and X/K a smooth
proper scheme. Fix j ∈N and define for ℓ∈L′
V (X) := Hj(X ,Q ) and
ℓ K ℓ
T (X) := Hj(X ,Z )/(Torsion).
ℓ K ℓ
ThenV (X)isafinitelygeneratedQ -vectorspace,T (X)isafinitelygeneratedfreeZ -module
ℓ ℓ ℓ ℓ
and V (X)=T (X)⊗Q for all ℓ∈L′. Let ρ be the representation of Gal(K) on V (X).
ℓ ℓ ℓ ℓ∞ ℓ
By Deligne’s theoremonthe Weil conjectures (cf. [7,Thm. 1.6]), standardspreading-outprin-
ciples, and proper-smooth base change (cf. [15, VI. Cor. 4.2]), one has the following result.
Theorem 1.1. (Deligne) The family of representations (ρℓ∞)ℓ∈L′ is a strictly compatible sys-
tem over K pure of weight j.
The following combines results from [1], [6], [13] and [19].
Theorem 1.2. There exists a finite Galois extension K /K with the following properties.
ind
(a) For all ℓ∈L′, one has
ρ (Gal(K ))⊂G(ρ )(Q ) and ρ (Gal(F K ))⊂G(ρ |Gal(F K ))(Q ).
ℓ∞ ind ℓ∞ ℓ ℓ∞ p ind ℓ∞ p ind ℓ
(b) One has ρ (Gal(F K ))=ρ (Gal(F K))+ for all ℓ≫0 in L′.
ℓ∞ p ind ℓ∞ p
(c) If ρ:Gal(K)→ GL (Z ) is the homomorphism induced by ρ , then
Qℓ∈L′ Tℓ(X) ℓ Qℓ∈L′ ℓ∞
ρ(Gal(FpKind))= Y ρℓ∞(Gal(FpKind)).
ℓ∈L′
(d) IfK =k(S)forasmoothk-varietyS suchhatρfactors viaπ (S),thenonecanfurther
1
requirethatthereexistsaconnectedfinite´etalecoverS ofS suchthatK =k(S ).
ind ind ind
Proof. The existence of a finite Galois extension K′/K such that the first half of (a) holds is
due to Serre (see [19, 2nd letter]); see also [12, Prop. 6.14]. For the second half of (a) this
follows from [13, 2.2] by Larsen and Pink. It follows from [1, Thm. 7.7], or alternatively from
[6], that after replacing K′ by a larger finite Galois extension K of K also (b) and (c) hold.
ind
LetS be the connectedfinite´etalecoverofS correspondingto the quotientρ(Gal(K )) of
ind ind
π (S). Then (a)–(d) hold if we replace K by k(S ). (cid:3)
1 ind ind
Theorem 1.3. (Deligne, Grothendieck, Serre)
(a) The restriction ρ |Gal(F K) is semisimple.
ℓ∞ p
(b) The group G(ρ |Gal(F K)) is semisimple.
ℓ∞ p
(c) The group ρ (Gal(F K )) is open in G(ρ |Gal(F K)).
ℓ∞ p ind ℓ∞ p
Proof. Part (a) is [8, Cor. 3.4.13] and due to Deligne. Part (b) is attributed by Deligne to
Grothendieck and given in [8, Cor. 1.3.9]. Part (c) is due to Serre (cf. [17]). (cid:3)
Corollary 1.4. We have DG(ρ )=G(ρ |Gal(F K)).
ℓ∞ ℓ∞ p
4 GEBHARDBO¨CKLE,WOJCIECHGAJDAANDSEBASTIANPETERSEN
Proof. Clearly ρ (Gal(K)) normalizes ρ (Gal(F K)). This is preserved under closures, so
ℓ∞ ℓ∞ p
that G(ρ |Gal(F K)) is a normal subgroup of G(ρ ). In particular the quotient Q :=
ℓ∞ p ℓ∞
G(ρ )/G(ρ |Gal(F K)) is a connected linear algebraic group. Now Q contains as a Zariski
ℓ∞ ℓ∞ p
dense subset a finite index subgroup of ρ (Gal(K))/ρ (Gal(F K)), and the latter is a
ℓ∞ ℓ∞ p
quotient of Gal(FpK/K) ∼= Zˆ and thus Q is abelian. From the universal property of the
derived group, we deduce DG(ρ ) ⊂ G(ρ |Gal(F K)). By Theorem 1.3(b), we have
ℓ∞ ℓ∞ p
DG(ρ |Gal(F K))=G(ρ |Gal(F K)), and so also G(ρ |Gal(F K))⊂DG(ρ ). (cid:3)
ℓ∞ p ℓ∞ p ℓ∞ p ℓ∞
2. The sets of Galois generic points
In addition to the data introduced above let k be an infinite finitely generated field and S/k
a smooth geometrically connected variety with function field K. Assume that X extends to
a smooth proper scheme X over S. Then ρ factors through π (S). As recalled in the
ℓ∞ 1
introduction, for every s ∈ |S| the representation ρ is isomorphic to the representation
ℓ∞,s
of Gal(k(s)) on Hj(X ,Q ) where X = X × Spec(k(s)) is the special fibre of X in s.
s,k(s) ℓ s S
For our applications below, note that the results of Section 1 also apply to ((ρℓ∞,s)ℓ∈L′,k(s))
instead of ((ρℓ∞)ℓ∈L′,K).
In this section we group together various results about the sets Sgen(ρ ) and some conse-
ℓ∞
quences. The following result is due to Serre (cf. [20, § 6.10]). We outline the argument.
Lemma 2.1. For any ℓ∈L′ the set Sgen(ρ ) is Zariski-dense in S.
ℓ∞
Proof. LetℓbeinL′ andletΦ bethe FrattinisubgroupofG:=ρ (π (S)), i.e.,theintersec-
ℓ ℓ∞ 1
tionof allmaximalclosedsubgroupsof G. Clearly G is a compactsubgroupof GL (Q ), and
Vℓ ℓ
so by [9, Thm. 8.33., p. 201] there is an open pro-ℓ subgroup of G of finite rank. We deduce
from [20, § 6.10 Prop.] that Φ is open in G.
ℓ
Consider now the composite homomorphism:
ρ : Gal(K)−ρℓ→∞ G→G/Φ .
ℓ ℓ
Bothρ andρ factorviaπ (S),andbytheuniversalpropertyofthe Frattinigroup,wehave
ℓ∞ ℓ 1
(1) {s∈|S|:ρ (Gal(k(s)))=G}={s∈|S|:ρ (Gal(k(s)))=G/Φ }.
ℓ∞,s ℓ,s ℓ
LetM denotetherighthandset. BecauseG/ΦisfiniteandkisHilbertian,thesetM isZariski
dense in S. This completes the proof, because Sgen(ρ ) contains the left hand side of (1). (cid:3)
ℓ∞
For the rest of this paper we define, for s∈|S| and ℓ∈L′, the semisimple groups
D :=G(ρ |Gal(F K)) and D :=G(ρ |Gal(F k(s)))
ℓ∞ ℓ∞ p ℓ∞,s ℓ∞,s p
over Q and recall that D =DG(ρ ) and D =DG(ρ ).
ℓ ℓ∞ ℓ∞ ℓ∞,s ℓ∞,s
Because of Theorem 1.1, the following result follows from [13, Thm. 2.4].
Theorem 2.2. (Larsen and Pink) If trdeg K = 1, then the functions ℓ 7→ dimD and
Fp ℓ∞
ℓ7→dimD on L′ are both constant.
ℓ∞,s
As an application of Lemma 2.1 we extend Theorem 2.2 to all fields K considered here.
Corollary 2.3. The functions ℓ7→dimD and ℓ7→dimD on L′ are both constant.
ℓ∞ ℓ∞,s
Proof. Note that it suffices to prove the assertion on ℓ 7→ dimD , since for the second
ℓ∞
assertion one may take k(s) for K and ρ for ρ . Let now K be a finitely generated field
ℓ∞,s ℓ∞
over F . We choose a subfield κ ⊂ K such that trdeg κ = 1 and K/κ is a regular extension
p F
p
of fields. Next we choose a geometrically connected smooth κ-variety B with κ(B) = K and
a smooth proper morphism X → B with generic fibre X/K. Let ℓ ∈ L′ be such that
B 0
A VARIATIONAL OPEN IMAGE THEOREM IN POSITIVE CHARACTERISTIC 5
dim(Dℓ∞0 ) = maxℓdim(Dℓ∞). By Lemma 2.1 there exists a point b ∈ Bgen(ρℓ∞0 ); note that
trdeg κ(b)=1. Then for any ℓ∈L′ we have
F
p
dim(Dℓ∞)≥dim(DG(ρℓ∞,b))Thm=.2.2dim(DG(ρℓ∞0 ,b))choi=ceofbdim(Dℓ∞0 ),
and it follows from the choice of ℓ that ℓ7→dimD is constant. (cid:3)
0 ℓ∞
Remark 2.4. Intheaboveproof,forthereductionfromK totranscendencedegree1,onecould
also use results on “space filling curves”, as for instance [10, Rem. 2.18(ii)], cf. [6, Ex. 3.1].
We also need an analog of [5, 3.2.3] for Sgen, as defined here.
Lemma 2.5. (a) For s¯∈|S | denote by s¯ the closed point of S under s¯. Then
Fpk S
Sgen(ρ )={s¯ |s¯∈Sgen(ρ |π (S ))}.
ℓ∞ S Fpk ℓ∞ 1 Fpk
(b) If S′ is a finite ´etale cover of S and for s′ ∈ |S′| denote by s′ the closed point of S
S
under s′. Then Sgen(ρ )={s′ |s′ ∈(S′)gen(ρ |π (S′))}.
ℓ∞ S ℓ∞ 1
Proof. We only prove (a), the proof of (b) being elementary. There is a bijection between
points in |S| and orbits under Gal(F k/k) in |S |. So let s be in |S| and denote by s¯a point
p Fpk
in |S | above it. Consider the commutative diagram
Fpk
(cid:31)(cid:127)
Dℓ∞(cid:127)_,s //G(ρℓ∞(cid:127)_ ,s)
(cid:15)(cid:15) (cid:31)(cid:127) (cid:15)(cid:15)
D //G(ρ ).
ℓ∞ ℓ∞
IfsisGaloisgeneric,thentherightverticalinclusionisanisomorphism. HencebyCorollary1.4
the same holds for the left inclusion, and this means that s¯is Galois generic. Conversely,let s¯
beGaloisgeneric,sothattheleftverticalinclusionisanisomorphism,andwehaveaninduced
monomorphism
ι : G(ρ )/D ֒→G(ρ )/D
s ℓ∞,s ℓ∞,s ℓ∞ ℓ∞
of algebraic groups (see the proof of Corollary 1.4). Now there exists a finite Galois exten-
sion k′/k(s) such that the image of the group Gal(F k′/k′) is Zariski dense in the connected
p
group G(ρ )/D . Because Gal(F k(s)/k(s)) ֒→ Gal(F k/k) is of finite index, the group
ℓ∞,s ℓ∞,s p p
Gal(F k′/k′)is alsoZariskidense inthe connectedgroupG(ρ )/D . Hence ι is anisomor-
p ℓ∞ ℓ∞ s
phism, and it follows that s is Galois generic. (cid:3)
Remark 2.6. Let ℓ be in L′ and denote by ρss the semisimplification of ρ . Then from
ℓ∞ ℓ∞
Lemma 2.5, Theorem 1.3 and Corollary 1.4 it is immediate that Sgen(ρss )=Sgen(ρ ).
ℓ∞ ℓ∞
Proposition 2.7. For any two primes ℓ1,ℓ2 ∈L′ we have Sgen(ρℓ∞)=Sgen(ρℓ∞).
1 2
Proof. By Lemma 2.5 it suffices to show
(2) SFgpenk(ρℓ∞1 |Gal(Fp)K)=SFgpenk(ρℓ∞2 |Gal(FpK)).
For this let s¯ be in |S |. Observe that for any ℓ ∈ L′ we have the obvious assertion (a )
Fpk ℓ
that D ֒→D is an inclusion of connected semisimple groups and from Corollary2.3 the
ℓ∞,s ℓ∞
assertion(b)thatbothfunctionsℓ7→dim(D )andℓ7→dim(D )onL′ areconstant. From
ℓ∞ ℓ∞,s
6 GEBHARDBO¨CKLE,WOJCIECHGAJDAANDSEBASTIANPETERSEN
these one deduces the following chain of equivalences
s¯∈SFgpenk(ρℓ∞1 |Gal(FpK)) ⇐(aℓ⇒1) dimDℓ∞1 ,s =dimDℓ∞1
(b)
⇐⇒ dimD =dimD
ℓ∞,s ℓ∞
2 2
⇐(aℓ⇒2) s¯∈SFgpenk(ρℓ∞2 |Gal(FpK)).
(cid:3)
We define Sgen(X/S) = Sgen(ρ ) for any ℓ ∈ L′, and we call Sgen(X/S) the set of Galois
ℓ∞
generic points of X/S.
3. Adelic openness
Throughout this section, we shall assume K =K , cf. Theorem 1.2, since the main result of
ind
this section is the proof of Theorem 1(c) which may be proved over K by Lemma 2.5(b).
ind
Lemma 3.1. Let s be in Sgen(X/S). Then for all ℓ ∈ L′ the group ρ (Gal(F k(s))) is
ℓ∞,s p
open in ρ (Gal(F K)).
ℓ∞ p
Proof. Because K = K , the group ρ (Gal(F K)) lies in D (Q ), and hence so does
ind ℓ∞ p ℓ∞ ℓ
ρ (Gal(F k(s))) ⊂ ρ (Gal(F K)). By our choice of s we have D = D , and thus
ℓ∞,s p ℓ∞ p ℓ∞ ℓ∞,s
by Theorem 1.3(c), both ρ (Gal(F K)) and ρ (Gal(F k(s))) are open in D (Q ). This
ℓ∞ p ℓ∞,s p ℓ∞ ℓ
implies the asserted openness and completes the proof. (cid:3)
Note that ρ has its image in GL (Z ). Let D /Z be the Zariski closure of D in
ℓ∞ Tℓ(X) ℓ ℓ∞ ℓ ℓ∞
GL . The following result is powered by two theorems from a recent paper of Cadoret,
Tℓ(X)
Hui and Tamagawa (cf. [4]).
Theorem 3.2. For all ℓ≫0 we have:
(a) The group scheme D /Z is semisimple.
ℓ∞ ℓ
(b) If s∈Sgen(X/S), then we have
ρ (Gal(F k(s)))+ =ρ (Gal(F K))=D (Z )+.
ℓ∞,s p ℓ∞ p ℓ∞ ℓ
Proof. Part (a) is immediate from [4, Thm. 1.2] and [4, Cor. 7.5]. For part (b), let s ∈
Sgen(X/S), so that D =D . Then D is also the Zariski closure of ρ (Gal(F k(s))
ℓ∞,s ℓ∞ ℓ∞ ℓ∞,s p
in GL . We now apply [4, 7.3] twice in order to get
Tℓ(X)
D (Z )+ = ρ (Gal(F K))+ = ρ (Gal(F K)) and
ℓ∞ ℓ ℓ∞ p ℓ∞ p
D (Z )+ = ρ (Gal(F k(s)))+.
ℓ∞ ℓ ℓ∞ p
(cid:3)
Corollary 3.3. Consider the adelic representation
ρ:Gal(K)→ Y GLTℓ(A)(Zℓ).
ℓ∈L′
For every s∈Sgen(X/S) the group ρ (Gal(F k(s))) is open in ρ(Gal(F K)).
s p p
Proof. Let G=Gal(F K) and G =Gal(F k(s)). Then
p s p
ρ(G)= Y ρℓ∞(G)
ℓ∈L′
A VARIATIONAL OPEN IMAGE THEOREM IN POSITIVE CHARACTERISTIC 7
by Theorem 1.2 because K =K . Furthermore, again by Theorem 1.2, there exists an open
ind
normal subgroup H =Gal(F k(s) ) of G such that
s p ind s
ρs(Hs)= Y ρℓ∞,s(Hs).
ℓ∈L′
For every prime number ℓ∈L′ the group ρ (H ) is open in ρ (G ) (because H is open
ℓ∞,s s ℓ∞,s s s
in G ), and ρ (G ) is open in ρ (G) by Lemma 3.1. It follows that ρ (H ) is open in
s ℓ∞,s s ℓ∞ ℓ∞,s s
ρ (G) for all ℓ∈L′.
ℓ∞
By Theorem 3.2 we have ρ (G )+ = ρ (G) for all ℓ ≫ 0 in L′ because K = K . From
ℓ∞,s s ℓ∞ ind
our construction of H via Theorem 1.2, we deduce ρ (H ) = ρ (G )+ for all ℓ ≫ 0 in
s ℓ∞,s s ℓ∞,s s
L′, and hence ρ (H )= ρ (G) for these ℓ. By the definition of the product topology, the
ℓ∞,s s ℓ∞
group ρ (H ) is open in ρ(G). As ρ (H )⊂ρ (G )⊂ρ(G), the assertion follows. (cid:3)
s s s s s s
Theorem 3.4. For s∈Sgen(X/S) the group ρ (Gal(k(s))) is open in ρ(Gal(K)).
s
Proof. Let s¯ ∈ |S | be above s ∈ Sgen(X/S), and consider the following commutative
Fpk
diagram with exact rows, where ρ is induced from ρ:
1 // ρ(π (S )) //ρ(π (S)) // ρ(π (S))/ρ(π (S )) //1
1 OO Fpk 1OO 1 OO 1 Fpk
ρ ρ ρ
1 //π (S ) //π (S) // Gal(F k/k) //1
1 OOFpk 1OO OOp
s¯∗ s∗ i
1 // Gal(F k(s)) //Gal(k(s)) // Gal(F k(s)/k(s)) //1
p p
Nowρ (Gal(F k(s)))isopeninρ(π (S ))byCorollary3.3. Furthermoreρ(Gal(F k(s)/k(s)))
s p 1 Fpk p
is open in ρ(π (S))/ρ(π (S )), because k(s)/k is finite and ρ is surjective. It follows that
1 1 Fpk
ρ (Gal(k(s))) is open in ρ(Gal(K)). (cid:3)
s
4. Largeness in the sense of Hui-Larsen
Throughout this section we assume K =K and fix a Galois generic point s ∈Sgen(X/S).
ind
Let p : Dsc → D be the simply connected cover of the semisimple Q -group D .
ℓ∞ ℓ∞ ℓ∞ ℓ ℓ∞
Consider the groups
D := D (Q ),
A Q ℓ∞ ℓ
ℓ∈L′
Dsc := Dsc (Q ),
A Q ℓ∞ ℓ
ℓ∈L′
Γ := ρ(Gal(F K)⊂D ,
p A
Γs := ρs(Gal(Fpk(s)))⊂DA.
The aimofthis sectionis to provethat Γs is a subgroupofDA whichis “large”in the sense of
Hui and Larsen(cf. [11, Section 2]). This shows that an analogue of conjecture [11, Conj. 1.3]
for abelian varieties in positive characteristic holds. Denote by κ : D ×D →Dsc the
ℓ∞ ℓ∞ ℓ∞ ℓ∞
commutator map and by κ:D ×D →Dsc the map derived from the κ .
A A A ℓ∞
Theorem4.1. κ(Γ ,Γ )(resp. κ(Γ,Γ))generatesaspecialadelic(cf.[11,Section2])subgroup
s s
of Dsc which is equal to κ(Γ ,Γ )u (resp. to κ(Γ,Γ)u) for some u. Moreover κ(Γ ,Γ )2 (resp.
A s s s s
κ(Γ,Γ)2) contains a special adelic subgroup of DsAc.
Proof. Denote by pr : D → D the projection on the ℓ-th factor of the product. Note
ℓ∞ A ℓ∞
thatpr (Γ )=ρ (Gal(F k(s))) is ZariskidenseinD foreachℓ∈L′ because sis Galois
ℓ∞ s ℓ∞,s p ℓ∞
8 GEBHARDBO¨CKLE,WOJCIECHGAJDAANDSEBASTIANPETERSEN
generic. By [1, Thm. 1.2] there exists a finite extension F/F k(s) such that
p
ρs(Gal(F))= Y ρℓ∞,s(Gal(F)).
ℓ∈L′
Thus, if we define Γ := ρ (Gal(F)), then Γ ⊂ Γ ⊂ Γ. For each ℓ ∈ L′
ℓ∞,s ℓ∞,s Qℓ∈L′ ℓ∞,s s
the group Γ is open in D (Q ) by Lemma 3.1. Furthermore, as D is semisimple
ℓ∞,s ℓ∞ ℓ ℓ∞
(cf. Theorem 3.2) for ℓ≫0, [4, Cor. 8.2] implies that the grouppr−1(Γ ) is a hyperspecial
ℓ∞ ℓ∞,s
maximal compact subgroup of Dsc (Q ) for ℓ ≫ 0. Thus the assertion follows by [11, Thm.
ℓ∞ ℓ
3.8]. (cid:3)
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Gebhard Bo¨ckle
Computational Arithmetic Geometry
IWR (Interdisciplinary Center for Scientific Computing)
University of Heidelberg
Im Neuenheimer Feld 368
69120 Heidelberg, Germany
E-mail address: [email protected]
Wojciech Gajda
Faculty of Mathematics and Computer Science
Adam Mickiewicz University
Umultowska 87
61614 Poznan´, Poland
E-mail adress: [email protected]
A VARIATIONAL OPEN IMAGE THEOREM IN POSITIVE CHARACTERISTIC 9
Sebastian Petersen
Universita¨t Kassel
Fachbereich 10
Wilhelmsho¨her Allee 73
34121 Kassel, Germany
E-mail address: [email protected]