Journal de Théorie des Nombresde Bordeaux 00 (XXXX), 000–000

A variational open image theorem in positive
characteristic
par Gebhard Böckle, Wojciech Gajda et Sebastian PetersenRésumé. Nous démontrons un théorème d’image adélique ouverte variationnelle pour l’action Galois sur la cohomologie d’unS-schéma propre lisse, où S est une variété lisse sur un corps
de type fini sur Fp . Un outil essentiel est un resultat récent deCadoret, Hui et Tamagawa.
Abstract. We prove a variational open adelic image theorem for
the Galois action on the cohomology of smooth proper S-schemes
where S is a smooth variety over a finitely generated field of positive characteristic. A central tool is a recent result of Cadoret,
Hui and Tamagawa.

Introduction
Let k be a finitely generated infinite field of characteristic p > 0, S asmooth geometrically connected k-variety of positive dimension and f :X → S a smooth proper morphism of schemes. Let K = k(S) be thefunction field of S and let X/k(S) be the generic fibre of X . Fix j ∈ N.
For every prime number ` 6= p we define V` := H j (XK , Q` ) and letρ`∞ : π1 (S) → GLV` (Q` )be the representation of π1 (S) on the Q` -vector space V` . We writeYρ : π1 (S) →GLV` (Q` )`6=pQ
for the induced adelic representation `6=p ρ`∞ .
For every point s ∈ S with residue field k(s) we denote bys∗ : Gal(k(s)) → π1 (S)2010 Mathematics Subject Classification. 11G10, 14K15.
Mots-clefs. Compatible system, adelic openness, positive characteristic.
Acknowledgements: Part of this work was done during a stay of the three authors at Adam
Mickiewicz university in Poznań financed by NCN grant no. UMO-2014/15/B/ST1/00128. G.B.
was supported by the DFG within the SPP1489 and the FG1920. We thank the anonymousreferees, of the present work and of an earlier version, for many comments that much improved
our exposition.

2

Gebhard Böckle, Wojciech Gajda, Sebastian Petersen

the homomorphism induced by s (well-defined up to conjugation), andfor any group homomorphism τ : π1 (S) → H, we define τs := τ ◦ s∗ :Gal(k(s)) → H as the specialization of τ at s. Note that ρ`∞ ,s is isomorphic to the representation of Gal(k(s)) on H j (Xs,k(s) , Q` ) where Xs =X ×S Spec(k(s)) is the special fibre of X in s, see [16, VI. Cor. 4.2].
The aim of this paper is to study the variation of the monodromy groupsρ`∞ ,s (Gal(k(s))) (or ρs (Gal(k(s))), resp.) for closed points s ∈ S in comparing them to the corresponding monodromy group ρ`∞ (π1 (S)) (or ρ(π1 (S)),
resp.) of the generic point of S.
For every prime number ` 6= p let G(ρ`∞ ,s ) (or G(ρ`∞ ), resp.) be
the connected component of the Zariski closure of ρ`∞ ,s (Gal(k(s))) (or ofρ`∞ (π1 (S)), resp.) in GLV` /Q` , and define
S gen (ρ`∞ ) = {s ∈ S a closed point : G(ρ`∞ ,s ) = G(ρ`∞ )}Being in S gen (ρ`∞ ) is a priori weaker than being `-Galois generic in the senseof Cadoret-Kret (see [6, 3.1], [3, 1.5.3] and [18, § 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 3.5)(a) The sets S gen (ρ`∞ ) are independent of `. Let S gen (X /S) := S gen (ρ`∞ )for any ` 6= p.
(b) The set S gen (X /S) is Zariski dense in S, and in particular it isinfinite.
(c) The group ρs (Gal(k(s))) is open in ρ(π1 (S)) for every s ∈ S gen (X /S).
The above result relies on a similar result that holds if one replaces S
by its base change SFp k under k → Fp k. This base change allows us toapply standard tools to derive (a) and (b), and recent results from [5] by
Cadoret, Hui and Tamagawa to deduce (c). Given these results, the proofof Theorem 1 is rather elementary.
In the case where k is a finitely generated field of zero characteristic,
Cadoret established a theorem analogous to our main theorem in the casewhere X /S is an abelian scheme (cf. [3]). Her theorem offers a strong tool
to reduce the proof of conjectures about Galois representations attached toabelian varieties over finitely generated fields of zero characteristic to thenumber field case. Similarly our theorem can be used in order to reduce theproof of conjectures about smooth proper varieties over finitely generated
fields of positive characteristic to the case where the ground field is a global
function field.
The results from [5] also give one a rather precise conceptual description of ρ(π1 (SFp k )), as we shall explain in Section 4, and as is presumablyA variational open image theorem

3well-known to the authors of [5]: Let D`∞ denote the connected component of the Zariski closure of ρ`∞ (π1 (SFp k )), or, equivalently, the derived group of G(ρ`∞ ) (see Theorem 1.4). In Theorem 4.1 we prove thatfor s ∈ S gen (XQ /S) the group ρs (Gal(Fp k(s))) generates a special adelic
subgroup in `6=p Dsc`∞ (Q` ) in the sense of Hui and Larsen [12], where→D
Dscdenotesthe simply connected cover.
`∞`∞
After submitting our manuscript to ArXiv we were informed by Anna
Cadoret that she has a manuscript [4] with similar results, now also available on her homepage.
Notation
For a field k we denote by k an algebraic closure and by Gal(k) theabsolute 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 π1 (S) the étale fundamentalgroup of S with base point the geometric generic point Spec(k(S)) → S. Ifchar k = p, let SFp k denote the base change of S under k → Fp k.
Suppose that V is a finite-dimensional Q` -vector space, Π is a profinitegroup and ρ : Π → GLV (Q` ) is a continuous homomorphism. We denoteby G(ρ) the connected component of the Zariski closure of ρ(Π) in GLV .
Then G(ρ) is an algebraic group over Q` . It is reductive if ρ is semisimple(see [17, 22.138]). We write ρ|H for the restriction of ρ to a subgroup H ofΠ, and we denote by Π+ the closed subgroup of Π generated by its pro-`
Sylow subgroups.
If Π = π1 (S), then we define the set of Galois generic points with respectto ρ asS gen (ρ) := {s ∈ |S| : G(ρ) = G(ρs )}.
Let K be a finitely generated field of characteristic p > 0. By L0 wedenote the set of all prime numbers ` 6= p. We call a family (ρ`∞ :Gal(K) → GLV` (Q` ))`∈L0 of continuous homomorphisms, where the V` are
finite-dimensional Q` -vector spaces, a strictly compatible system over K
pure of weight j if there exists a smooth Fp -variety T with Fp (T ) = K suchthat the following properties (i) and (ii) hold:(i) ρ`∞ factors through π1 (T ) for every ` ∈ L0 .
(ii) For every t ∈ |T |, denoting by Frt ∈ Gal(Fp (t)) the arithmetic Frobe1

nius x 7→ x |Fp (t)| , the characteristic polynomial of ρ`∞ ,t (Frt ) has coefficients in Z, it is independent of `, and its roots all have absolutejvalue |Fp (t)| 2 .
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 generated infinite

4

Gebhard Böckle, Wojciech Gajda, Sebastian Petersenfield of characteristic p > 0 and X/K a smooth proper scheme. Fix j ∈ N
and define for ` ∈ L0
V` (X) := H j (XK , Q` ) andT` (X) := H j (XK , Z` )/(Torsion).
Then V` (X) is a finitely generated Q` -vector space, T` (X) is a finitely generated free Z` -module and V` (X) = T` (X) ⊗ Q` for all ` ∈ L0 . Let ρ`∞ be
the representation of Gal(K) on V` (X).
By Deligne’s theorem on the Weil conjectures (cf. [8, Thm. 1.6]), standard spreading-out principles, and proper-smooth base change (cf. [16,VI. Cor. 4.2]), one has the following results.
Theorem 1.1 (Deligne). The family of representations (ρ`∞ )`∈L0 is a strictly compatible system over K pure of weight j.
Theorem 1.2 (Deligne, Grothendieck).
(a) The restriction ρ`∞ | Gal(Fp K) is semisimple.
(b) The group G(ρ`∞ | Gal(Fp K)) is semisimple.
Proof. Part (a) is [9, Cor. 3.4.13] and due to Deligne. Part (b) is attributedby Deligne to Grothendieck and given in [9, Cor. 1.3.9].
For a linear algebraic group G defined over a field, we denote by DG its
derived group.
Corollary 1.3. We have DG(ρ`∞ ) = G(ρ`∞ | Gal(Fp K)).
Proof. Clearly ρ`∞ (Gal(K)) normalizes ρ`∞ (Gal(Fp K)). This is preserved
under closures, so that G(ρ`∞ | Gal(Fp K)) is a normal subgroup of G(ρ`∞ ).
In particular the quotient Q := G(ρ`∞ )/G(ρ`∞ | Gal(Fp K)) is a connected
linear algebraic group. Now Q contains as a Zariski dense subset a finite index subgroup of ρ`∞ (Gal(K))/ρ`∞ (Gal(Fp K)), and the latter is a quotientof Gal(Fp K/K) ∼= Ẑ and thus Q is abelian. From the universal propertyof the derived group, we deduce DG(ρ`∞ ) ⊂ G(ρ`∞ | Gal(Fp K)). By Theorem 1.2(b), we have DG(ρ`∞ | Gal(Fp K)) = G(ρ`∞ | Gal(Fp K)), and so alsoG(ρ`∞ | Gal(Fp K)) ⊂ DG(ρ`∞ ).

The following combines results from [1], [7], [19] and [20], and extends a
result from [15].
Theorem 1.4. There exists a finite Galois extension Kind /K with thefollowing properties.
(a) For all ` ∈ L0 , one has ρ`∞ (Gal(Kind )) ⊂ G(ρ`∞ ).
(b) One has ρ`∞ (Gal(Fp Kind )) = ρ`∞ (Gal(Fp Kind ))+ for all `  0 in L0 .

A variational open image theorem(c) If ρ : Gal(K) →Q`∈L0 ρ`∞ , thenQ

5`∈L0 GLT` (X) (Z` ) is the homomorphism induced byρ(Gal(Fp Kind )) =Yρ`∞ (Gal(Fp Kind )).

`∈L0(d) For all ` ∈ L0 the group ρ`∞ (Gal(Fp Kind )) is an open subgroup ofG(ρ`∞ | Gal(Fp Kind )).
(e) If K = k(S) for a smooth k-variety S such that ρ factors via π1 (S),then one can further require that there exists a connected finite étale
cover Sind of S such that Kind = k(Sind ).
Proof. The existence of a finite Galois extension Kind /K such that (a) holds
true is due to Serre (see [20, 2nd letter]); see also [14, Prop. 6.14]. It follows
from [1, Thm. 7.7], or alternatively from [7], that after replacing Kind by alarger finite Galois extension of K also (b) and (c) hold. If K is a global
field, then (d) follows from [15, 2.2] by Larsen and Pink. We now explain
an independent proof of (d) for any finitely generated K.
By [1, Def. 5.1. and Cor. 7.4] and by (a)–(c), there exists a finite Galois
extension Kind of K satisfying the properties in (a), (b) and (c), and asmooth Fp -variety V such that Fp (V ) = Kind and such that the restrictionρ`∞ | Gal(Kind ) factors through the fundamental group π1 (V ) and such thatfor each ` ∈ L0 the restriction ρ`∞ | Gal(Kind ) is `-tame (cf. [1, Def. 4.2]).
After making an alteration on V and replacing Kind accordingly we canassume that V admits a smooth compactification V such that V \ V is anormal crossing divisor (cf. [2, Thm. 1.2]). It follows that ρ`∞ | Gal(Kind )factors through the tame fundamental group π1t (V ) for every ` ∈ L0 , where
π1t (V ) is defined as in [13]. Consider the homomorphismYρ : π1t (V ) →(G(ρ`∞ )/DG(ρ`∞ ))(Q` )`∈L0induced by the ρ`∞ . Because ρ has an abelian image, it factors through theabelianization π1t,ab (V ) of π1t (V ). Let F0 be the largest algebraic extensionof Fp in Kind . Then F0 /Fp is finite, V /F0 is geometrically connected andby [13, Thm. 7.5] the kernel Ker(π1t,ab (V ) → Gal(F0 )) is finite, and thusρ(Gal(Fp Kind )) is finite. Therefore we can replace Kind by a finite Galois
extension of K, so that ρ(Gal(Fp Kind )) = {e}. By Corollary 1.3, this
implies the containment ρ`∞ (Gal(Fp Kind )) ⊂ DG(ρ`∞ ) asserted in (d) for
every ` ∈ L0 . The openness claimed in (d) follows from [19, Prop. 2 and itsCor.].
Concerning (e), let Sind be the connected finite étale cover of S corresponding to the image of Gal(Kind ) of π1 (S). Then (a)–(e) hold if we
replace Kind by k(Sind ).
6Gebhard Böckle, Wojciech Gajda, Sebastian Petersen

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 withfunction field K. Assume that X extends to a smooth proper scheme X
over S. Then ρ`∞ factors through π1 (S). As recalled in the introduction,for every s ∈ |S| the representation ρ`∞ ,s is isomorphic to the representationof Gal(k(s)) on H j (Xs,k(s) , Q` ) where Xs = X ×S Spec(k(s)) is the special
fibre of X in s. For our applications below, note that the results of Section 1
also apply to ((ρ`∞ ,s )`∈L0 , k(s)) instead of ((ρ`∞ )`∈L0 , K).
In this section we group together various results about the sets S gen (ρ`∞ )and some consequences. The following result is due to Serre (cf. [21, § 10.6]).
We outline the argument.
Lemma 2.1. For any ` ∈ L0 the set S gen (ρ`∞ ) is Zariski-dense in S.
Proof. Let ` be in L0 and let Φ` be the Frattini subgroup of G :=ρ`∞ (π1 (S)), i.e., the intersection of all maximal closed subgroups of G.
Clearly G is a compact subgroup of GLV` (Q` ), and so by [10, Thm. 8.33.,
p. 201] there is an open pro-` subgroup of G of finite rank. We deducefrom [21, § 10.6 Prop.] that Φ` is open in G.
Consider now the composite homomorphism:
ρ∞`G → G/Φ` .
ρ` : Gal(K) −→Both ρ`∞ and ρ` factor via π1 (S), and by the universal property of the
Frattini group, we have
(2.1)
{s ∈ |S| : ρ`∞ ,s (Gal(k(s))) = G} = {s ∈ |S| : ρ`,s (Gal(k(s))) = G/Φ` }.
Let M denote the right hand set. Because G/Φ` is finite and k is Hilbertian,the set M is Zariski dense in S. This completes the proof, because S gen (ρ`∞ )contains the left hand side of (2.1).
For the rest of this paper we define, for s ∈ |S| and ` ∈ L0 , the semisimple
groupsD`∞ := G(ρ`∞ | Gal(Fp K)) and D`∞ ,s := G(ρ`∞ ,s | Gal(Fp k(s)))
over Q` and recall from Corollary 1.3 that D`∞ = DG(ρ`∞ ) and D`∞ ,s =DG(ρ`∞ ,s ).
Because of Theorem 1.1, the following result follows from [15, Thm. 2.4].
Theorem 2.2 (Larsen and Pink). If trdegFp K = 1, then the functions` 7→ dim D`∞ and ` 7→ dim D`∞ ,s on L0 are both constant.
As an application of Lemma 2.1 we extend Theorem 2.2 to all fields K
considered here.

A variational open image theorem

7

Corollary 2.3. The functions ` 7→ dim D`∞ and ` 7→ dim D`∞ ,s on L0 are
both constant.
Proof. Note that it suffices to prove the assertion on ` 7→ dim D`∞ , sincefor the second assertion one may take k(s) for K and ρ`∞ ,s for ρ`∞ . Let
now K be a finitely generated field over Fp . We choose a subfield κ ⊂ Ksuch that trdegFp κ = 1 and K/κ is a regular extension of fields. Next wechoose a geometrically connected smooth κ-variety B with κ(B) = K anda smooth proper morphism XB → B with generic fibre X/K. Let `0 ∈ L0
be such that dim(D`∞) = max` dim(D`∞ ). By Lemma 2.1 there exists a0
gen∞point b ∈ B (ρ`0 ); note that trdegFp κ(b) = 1. Then for any ` ∈ L0 wehavedim(D`∞ ) ≥ dim(DG(ρ`∞ ,b ))Thm. 2.2=

dim(DG(ρ`∞))
0 ,bchoice of b=dim(D`∞),
0and it follows from the choice of `0 that ` 7→ dim D`∞ is constant.

Remark 2.4. In the above proof, for the reduction from K to transcendence
degree 1, one could also use results on “space filling curves”, as for instance[11, Rem. 2.18(ii)], cf. [7, Ex. 3.1].
We also need an analog of [6, 3.2.3] for S gen , as defined here.
Lemma 2.5.
s̄. Then(a) For s̄ ∈ |SFp k | denote by s̄S the closed point of S under
S gen (ρ`∞ ) = {s̄S | s̄ ∈ SFgenk (ρ`∞ |π1 (SFp k ))}.
p(b) If S 0 is a finite étale cover of S and for s0 ∈ |S 0 | denote by s0S
closed point of S under s0 . Thenthe

S gen (ρ`∞ ) = {s0S | s0 ∈ (S 0 )gen (ρ`∞ |π1 (S 0 ))}.
Proof. We only prove (a), the proof of (b) being elementary. There is abijection between points in |S| and orbits under Gal(Fp k/k) in |SFp k |. So
let s be in |S| and denote by s̄ a point in |SFp k | above it. Consider thecommutative diagramD`∞ ,s _D `∞ 



/ G(ρ`∞ ,s )_
/ G(ρ`∞ ).

If s is Galois generic, then the right vertical inclusion is an isomorphism.
Hence by Corollary 1.3 the same holds for the left inclusion, and this meansthat s̄ is Galois generic. Conversely, let s̄ be Galois generic, so that the left
vertical inclusion is an isomorphism, and we have an induced monomorphism
ιs : G(ρ`∞ ,s )/D`∞ ,s ,→ G(ρ`∞ )/D`∞

8

Gebhard Böckle, Wojciech Gajda, Sebastian Petersenof commutative algebraic groups (see the proof of Theorem 1.4). Now theimage of some open subgroup of Gal(Fp k/k) is Zariski dense in G(ρ`∞ )/D`∞ ,and moreover there exists a finite extension k 0 /k(s) such that the image ofGal(Fp k 0 /k 0 ) is Zariski dense in G(ρ`∞ ,s )/D`∞ ,s . But clearly Gal(Fp k 0 /k 0 ) ,→Gal(Fp k/k) is of finite index, and thus the map ιs is an isomorphism, andit follows that s is Galois generic.

Remark 2.6. Let ` be in L0 and denote by ρss`∞ the semisimplification ofρ`∞ . Then from Lemma 2.5, Theorem 1.2 and Theorem 1.4 it is immediategen (ρ ∞ ).
that S gen (ρss``∞ ) = S
Proposition 2.7. For any two primes `1 , `2 ∈ L0 we have S gen (ρ`∞) =1S gen (ρ`∞).
2
Proof. By Lemma 2.5 it suffices to show
(2.2)

| Gal(Fp K)) = SFgenk (ρ`∞| Gal(Fp K)).
SFgenk (ρ`∞
1
2
p

p

For this let s̄ be in |SFp k |. Observe that for any ` ∈ L0 we have the obvious
assertion (a` ) that D`∞ ,s ,→ D`∞ is an inclusion of connected semisimple groups and from Corollary 2.3 the assertion (b) that both functions` 7→ dim(D`∞ ) and ` 7→ dim(D`∞ ,s ) on L0 are constant. ¿From these one
deduces the following chain of equivalences(a` )1| Gal(Fp K)) ⇐⇒= dim D`∞s̄ ∈ SFgenk (ρ`∞dim D`∞
1
1 ,s
1
p

(b)

⇐⇒ dim D`∞= dim D`∞
2 ,s
2
(a` )

2
⇐⇒
s̄ ∈ SFgenk (ρ`∞| Gal(Fp K)).
2p
We define S gen (X /S) = S gen (ρ`∞ ) for any ` ∈ L0 , and we call S gen (X /S)
the set of Galois generic points of X /S.
3. Adelic openness
Lemma 3.1. If Theorem 1(c) holds over Kind (see Theorem 1.4), then it
holds over K.
Proof. Let Sind be the cover of S given in Theorem 1.4(e). Let s be in|S| and let s0 ∈ |Sind | be above s. Then by Lemma 2.5(b), we have s ∈S gen (ρ`∞ ) if and only if s0 ∈ S gen (ρ`∞ |π1 (Sind )). Proposition 2.7 therefore
implies that s ∈ S gen (X /S) if and only if s0 ∈ S gen (X ×S Sind /Sind ). Now
suppose that Theorem 1(c) holds over Kind . Then ρs0 (Gal(k(s0 ))) is open inρ(π1 (Sind )) for every s0 ∈ S gen (X ×S Sind /Sind ), and since ρs0 (Gal(k(s0 ))) ⊂ρs (Gal(k(s))) and ρ(π1 (Sind )) ⊂ ρ(π1 (S)) are open, the lemma is proved.
A variational open image theorem

9

For the remainder of this section, we shall assume K = Kind , cf. Theorem 1.4.
Lemma 3.2. Let s be in S gen (X /S). Then for all ` ∈ L0 the groupρ`∞ ,s (Gal(Fp k(s))) is open in ρ`∞ (Gal(Fp K)).
Proof. Because K = Kind , the group ρ`∞ (Gal(Fp K)) lies in D`∞ (Q` ), andhence so does ρ`∞ ,s (Gal(Fp k(s))) ⊂ ρ`∞ (Gal(Fp K)). By our choice of s
we have D`∞ = D`∞ ,s , and thus by Theorem 1.4(d), both ρ`∞ (Gal(Fp K))and ρ`∞ ,s (Gal(Fp k(s))) are open in D`∞ (Q` ). This implies the asserted
openness and completes the proof.

Note that ρ`∞ has its image in GLT` (X) (Z` ). Let D `∞ /Z` be the Zariski
closure of D`∞ in GLT` (X) . The following result is powered by two theorems
from a recent paper of Cadoret, Hui and Tamagawa (cf. [5]).
Theorem 3.3. (a) For all `  0 the group scheme D `∞ /Z` is semisimple.
(b) For every s ∈ S gen (X /S) we have the equalitiesρ`∞ ,s (Gal(Fp k(s)))+ = ρ`∞ (Gal(Fp K)) = D `∞ (Z` )+for all `  0 (depending on s).
Proof. Part (a) is immediate from [5, Thm. 1.2] and [5, Cor. 7.5]. For
part (b), let s ∈ S gen (X /S), so that D`∞ ,s = D`∞ . Then D `∞ is also
the Zariski closure of ρ`∞ ,s (Gal(Fp k(s)) in GLT` (X) . We now apply [5, 7.3]
twice in order to get
D `∞ (Z` )+ = ρ`∞ (Gal(Fp K))+ = ρ`∞ (Gal(Fp K)) andD `∞ (Z` )+ = ρ`∞ (Gal(Fp k(s)))+ .
Corollary 3.4. Consider the adelic representationYρ : Gal(K) →GLT` (A) (Z` ).
`∈L0

For every s ∈ S gen (X /S) the group ρs (Gal(Fp k(s))) is an open subgroupof ρ(Gal(Fp K)).
Proof. Let G = Gal(Fp K) and Gs = Gal(Fp k(s)). ThenY
ρ(G) =
ρ`∞ (G)`∈L0

by Theorem 1.4 (c) because K = Kind . Furthermore, again by Theorem 1.4(c), there exists an open normal subgroup Hs = Gal(Fp k(s)ind ) of Gs suchGebhard Böckle, Wojciech Gajda, Sebastian Petersen

10

that
ρs (Hs ) =Yρ`∞ ,s (Hs ).

`∈L0

For every prime number ` ∈ L0 the group ρ`∞ ,s (Hs ) is open in ρ`∞ ,s (Gs )(because Hs is open in Gs ), and ρ`∞ ,s (Gs ) is open in ρ`∞ (G) by Lemma 3.2.
It follows that ρ`∞ ,s (Hs ) is open in ρ`∞ (G) for all ` ∈ L0 .
By Theorem 3.3 we have ρ`∞ ,s (Gs )+ = ρ`∞ (G) for all `  0 in L0 because
K = Kind . From our construction of Hs via Theorem 1.4, we deduceρ`∞ ,s (Hs ) = ρ`∞ ,s (Gs )+ for all `  0 in L0 , and hence ρ`∞ ,s (Hs ) = ρ`∞ (G)for these `. By the definition of the product topology, the group ρs (Hs ) is
open in ρ(G). As ρs (Hs ) ⊂ ρs (Gs ) ⊂ ρ(G), the assertion follows.

Theorem 3.5. For s ∈ S gen (X /S) the group ρs (Gal(k(s))) is open inρ(Gal(K)).
Proof. Let s̄ ∈ |SFp k | be above s ∈ S gen (X /S), and consider the following
commutative diagram with exact rows, where ρ is induced from ρ:1/ ρ(π1 (S ))
F k
O p
ρ

1

/ π1 (S )
F k
O p
s̄∗

1

/ Gal(Fp k(s))/ ρ(π1 (S))
Oρ/ ρ(π1 (S))/ρ(π1 (S ))
Fp k
O

/1

ρ

/ π1 (S)O/ Gal(Fp k/k)
O

s∗

/1

i

/ Gal(k(s))/ Gal(Fp k(s)/k(s))/1Now ρs (Gal(Fp k(s))) is open in ρ(π1 (SFp k )) by Corollary 3.4. Furthermore
ρ(Gal(Fp k(s)/k(s))) is open in ρ(π1 (S))/ρ(π1 (SFp k )), because k(s)/k is finite and ρ is surjective. It follows that ρs (Gal(k(s))) is open in ρ(Gal(K)).

4. Largeness in the sense of Hui-Larsen
Throughout this section we assume K = Kind and fix a Galois genericpoint s ∈ S gen (X /S). Consider the restricted direct product
Y0
DA :=D`∞ (Q` )0`∈Lwith respect to the compact open subgroups D `∞ (Z` ) ⊂ D`∞ (Q` ) for `  0from Theorem 3.3, so thatΓs := ρs (Gal(Fp k(s))) ⊂ Γ := ρ(Gal(Fp K)) ⊂ DA .
It is tempting to expect that Γs is open in DA . But by Theorem 3.3,
this adelic openness statement can only hold if almost all D `∞ are simplyA variational open image theorem

11connected since only then D `∞ (Z` )+ = D `∞ (Z` ). In [12], Hui and Larsen
suggest a reformulation that allows one to state a meaningful adelic openness conjecture, [12, Conj. 1.3], which they formulate in the case that K isa number field. Below we prove the analogue of their conjecture for compatible systems arising in the cohomology of a smooth projective variety
over a finite type base in positive characteristic.
sc
Let p`∞ : Dsc`∞ → D `∞ (and D `∞ → D `∞ , resp.) be the simply connected
cover of the semisimple Q` -group D`∞ (and the Z` -group D `∞ , resp.).
sc
Since p`∞ is a central isogeny, the commutator morphism Dsc`∞ × D `∞ →−1 −1 factors through a morphism κ ∞ : D
Dsc``∞ × D `∞ →`∞ , (x, y) 7→ xyx yscD`∞ . Furthermore let
Y0
Dsc
Dsc
A :=
`∞ (Q` )0`∈L

be the restricted direct product with respect to the D sc`∞ (Z` ) for `  0, andletκ : DA × DA → Dsc
A
be the map derived from the κ`∞ . Because the groups D sc`∞ (Z` ) for `  0 are
hyperspecial maximal compact, as they are the Z` -points of a semisimple
group scheme over Z` , the compact open subgroups of Dsc
A are precisely
the special adelic groups as defined in [12, Section 2]. For a subset M of agroup H and u ∈ N we define the set M u := {s1 · . . . · su | s1 , . . . , su ∈ M }.
Theorem 4.1 (Analog of [12, Conj. 1.3]). Let M be in {κ(Γs , Γs ), κ(Γ, Γ)}.
Then the set M generates a compact open subgroup of Dsc
A which is equal
to M u for some u ∈ N. Moreover M 2 contains a compact open subgroup of
DscA.
Proof. Denote by pr`∞ : DA → D`∞ the projection on the `-th factor of
the product. Note that pr`∞ (Γs ) = ρ`∞ ,s (Gal(Fp k(s))) is Zariski dense inD`∞ for each ` ∈ L0 because s is Galois generic. By [1, Thm. 1.2] there
exists a finite extension F/Fp k(s) such thatY
ρs (Gal(F )) =
ρ`∞ ,s (Gal(F )).
`∈L0

Q
Thus, if we define Γ`∞ ,s := ρ`∞ ,s (Gal(F )), then `∈L0 Γ`∞ ,s ⊂ Γs ⊂ Γ.
For each ` ∈ L0 the group Γ`∞ ,s is open in D`∞ (Q` ) by Lemma 3.2 andTheorem 1.4(d). Moreover, as D `∞ is semisimple and Γ`∞ ,s = D `∞ (Z` )+sc∞
for `  0 (cf. Theorem 3.3), by [5, Cor. 8.2] we have pr−1`∞ (Γ` ,s ) = D `∞ (Z` )for `  0. 
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Gebhard Böckle, Wojciech Gajda, Sebastian Petersen[2] Pierre Berthelot, Altération de variétés algébriques. Sém. Bourbaki no. 815 (1995-96),
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[6] Anna Cadoret and Arno Kret, Galois generic points on Shimura varieties. Algebra andNumber Theory Vol. 10 (2016), 1893–1934.
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Lecture notes available at www.jmilne.org[18] Richard Pink, A Combination of the conjectures of Mordell-Lang and André-Oort. Math.
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[19] Jean-Pierre Serre, Sur les groupes de Galois attachés aux groups p-divisible. In: Proceedings on a conference on local fields, Springer, 1967
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IV, Springer, 2000
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Viehweg, Braunschweig, 1989
Gebhard Böckle
Computational Arithmetic Geometry
IWR (Interdisciplinary Center for Scientific Computing)
University of Heidelberg
Im Neuenheimer Feld 368
69120 Heidelberg, Germany
E-mail : gebhard.boeckle@iwr.uni-heidelberg.de
Wojciech Gajda
Faculty of Mathematics and Computer Science
Adam Mickiewicz University
Umultowska 87
61614 Poznań, Poland
E-mail : gajda@amu.edu.pl
Sebastian Petersen
Universität Kassel
Fachbereich 10

A variational open image theorem
Wilhelmshöher Allee 73
34121 Kassel, Germany
E-mail : petersen@mathematik.uni-kassel.de

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