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BANACH CENTER PUBLICATIONS, VOLUME **INSTITUTE OF MATHEMATICSPOLISH ACADEMY OF SCIENCESWARSZAWA 201*

ON COHOMOLOGICAL SYSTEMS OF GALOISREPRESENTATIONSWOJCIECH GAJDA
Faculty of Mathematics and Computer Sciences
Adam Mickieiwcz University
Umultowska 87, 61-614 Poznań, Poland
E-mail: gajda@amu.edu.pl
SEBASTIAN PETERSEN
Universität Kassel
Fachbereich 10
Wilhelmshöher Allee 73
34119 Kassel
E-mail: petersen@mathematik.uni-kassel.de

Abstract. The paper contains an expanded version of the talk delivered by the first author
during the conference ALANT 3 in Bȩdlewo in June 2014. We survey recent results on independence of systems of Galois representations attached to `-adic cohomology of schemes. Someother topics ranging from the Mumford-Tate conjecture and the Geyer-Jarden conjecture to applications of geometric class field theory are also considered. In addition, we have highlighted avariety of open questions which can lead to interesting research in near future.

1. Geometric Galois representations. Let K be a field of characteristic p ≥ 0, K
an algebraic closure of K and X/K a smooth projective variety of dimension d. Let
L be the set of all prime numbers. Grothendieck and M. Artin defined in [4] for everyj ∈ N the étale cohomology groups H j (XK , Z/n). These groups are Z/n-modules. Unlike
singular cohomology the cohomology groups of Grothendieck and Artin can be definedeven if p > 0. In the need for cohomology groups which are vector spaces over a field of
characteristic zero one defines (cf. [4]) for every ` ∈ L the `-adic étale cohomology groups!
H j (XK , Q` ) =lim H j (XK , Z/`i )←−⊗Z` Q` .

i∈N2010 Mathematics Subject Classification: Primary 11G10; Secondary 14F20.
Key words and phrases: Galois representation, Mumford-Tate group, Abelian variety.
[1]

2

W.GAJDA AND S.PETERSEN

There is a natural representation(j)ρ`,X : GK → Aut(H j (XK , Q` ))of the absolute Galois group GK =Gal(K̄/K) on them. Main topics of the paper center(j)around the families of `-adic Galois representations (ρ`,X )`∈L and their images.
1.1. Properties of `-adic cohomology.
Basic facts. We refer the reader to [12] and to Milne’s lecture notes [24] for definitions
and proofs of the facts discussed in this section. The groups H j (XK , Q` ) behave well for` 6= char K and they constitute a cohomology theory with properties which we know very
well from algebraic topology.
• Finiteness: H j (XK , Q` ) are finite dimensional Q` -vector spaces. The groups vanishfor j > 2 dim X = 2d and the top non-zero group H 2d (XK , Q` ) identifies canonicaly
with Q` .
• Poincare duality: For i + j = 2d there exists a perfect pairing of finite dimensionalQ` -vector spaces
H i (XK , Q` ) × H j (XK , Q` ) −→ H 2d (XK , Q` ) = Q` .
In particular, H i (XK , Q` ) is dual to H 2d−i (XK , Q` ), and they have the same dimension.
• Künneth theorem: For any i > 0, and two smooth projective varieties X and Y ,there are canonical isomorphismsM∼
H i (X × Y , Q` ) =H j (X , Q` ) ⊗ H k (Y , Q` ).
K

K

K

K

j+k=i

• Lefschetz trace formula: One of the most important features of the `-adic cohomology is a fixed point formula which says that for any morphism f : X −→ X with isolated fixed points one can compute the number of fixed points of f : X(K) → X(K),counted with multiplicities, by calculating traces of Q` -linear maps induced on cohomology by the morphism f (cf. [24], Thm. 25.1): If Γ is the graph of f and ∆the diagonal of X, thenX(Γ · ∆) =(−1)i Tr[f ∗ : H i (XK , Q` ) → H i (XK , Q` )].
i

The natural action of the Galois group GK on the vector space H j (XK , Q` ) defines acontinuous homomorphism(j)ρ`,X : GK −→ AutQ` H j (XK , Q` )the `-adic geometric Galois representation attached to the variety X. The representation(j)ρ`,X encodes a lot of important information, which is very interesting even in the casewhen K is a subfield of the field of complex numbers.
Classical special cases. We remark to illustrate the usefulness of `-adic cohomology bydescribing three special cases of varieties X and their cohomology GK -modules.
Remark. If dim X = 0, then X = Spec(K[T ]/(P (T ))) for a polynomial P ∈ K[T ] andthe set of K-rational points X(K) = {α ∈ K : P (α) = 0} of X is the set of roots

3

INDEPENDENCE OF SYSTEMS

of P. We identify the only `-adic cohomology group H 0 (XK , Q` ) with the free Q` -module`(0)0α∈X(K) Q` with basis X(K). The representation ρ`,X : GK −→ AutQ` (H (XK , Q` ))is then induced by the permutation representation GK → Sym(X(K)) well known frombasic courses in algebra and Galois theory.
Remark. Let X be a smooth projective geometrically connected curve over K. Let
g be the genus of the curve X. Then H 0 (XK , Q` ) = Q` , H 1 (XK , Q` ) = Q2g` , andH 2 (XK , Q` ) = Q` . The absolute Galois group GK acts on H 2 (XK , Q` ) via the inverse of
the cyclotomic character ` : GK −→ Z×` defined by the action of GK on `-powers rootsof unity. The action of GK on H 1 (XK , Q` ) = Q2g` is closely related to the action of GK
on the Tate module of the Jacobian variety of X, compare the next example.
Remark. Let X be an abelian variety defined over K. Let∼ Z2g
T` (X) = lim X(K)[`k ] =
`

←

Vi

be the Tate module of X. Then H i (XK , Q` ) =H 1 (XK , Q` ) is the ith exterior power
1of the Q` -vector space H (XK , Q` ). Denote by X̂ the dual abelian variety. Then there is
an isomorphism of GK -modules
T` (X) ⊗Z Q` ∼= H 1 (X̂ , Q` )(1),K`

where for a Q` -vector space M which is a GK -module, by M (1) we denote the Tate
(1)
twist M ⊗ ` . The above isomorphisms show the importance of the representation ρ`,X
(1)

for an abelian variety X. Note that the Tate module representation σ` := ρ`,X ⊗ −1` andits deformation theory was instrumental in the proof of Shimura-Taniyama modularity
conjecture by Wiles, for a semistable elliptic curve X defined over the rationals.
1.2. Spectacular applications.
Weil conjectures. Let X be a smooth projective variety over the finite field K = Fp . Fix
a rational prime ` 6= p. Consider the congruence zeta function for X:
X
Tr
Z(X, T ) := exp(
|X(Fpr )| )rr≥1

which by definition is a formal power series in the variable T. Let Frob ∈ GK =Gal(Fp /Fp )denote the Frobenius automorphism induced by the pth power homomorphism. It follows
easily by the Lefschetz formula in `-adic cohomology that for every r ≥ 1
X(i)|X(Fpr )| =(−1)i Tr(ρ`,X (Frobr )).
i≥0(i)
We denote by Pi (T ) := det(1 − ρ`,X (Frob)T ) the characteristic polynomial of the Q` (i)linear map ρ`,X (Frob) : H i (XK , Q` ) → H i (XK , Q` ). It was proven by Grothendieck andcompleted in the important case of Riemann conjecure for X by Deligne [11] that thefollowing statements hold.
• Rationality of zeta: The congruence zeta function
P1 (T )P3 (T ) . . . P2d−1 (T )Z(X, T ) =P0 (T )P2 (T ) . . . P2d (T )

4

W.GAJDA AND S.PETERSEN

is a rational function in Q(T ) which satisfies a functional equation.
• Riemann conjecture: For every i ≥ 0 the polynomial Pi (T ) has integral coefficients,
which are independent of ` 6= p, and an analogue of Riemann conjecture holds
ifor X/Fp : The inverse roots of the polynomial Pi (T ) have absolute value p 2 .
We have stressed the independence of ` of the coefficients of the polynomial Pi (T ). It isa first manifestation we would like to point out of the independence of the prime ` ofcohomological properties of algebraic varieties.
Mordell conjecture. Let X be an abelian variety defined over a finitely generated extention K of Q. In 1983 Faltings proved in the impressive paper [14] the following crucial
properties of the Tate module representationσ` : GK −→ Aut(T` (X) ⊗ Q` ) ∼= Gl2d (Q` )of the abelian variety X, which were conjectured in 70s by Tate.
• The representation σ` is semsimple.
• The natural homomorphism
Hom(X, X) ⊗ Z` −→ HomZ` (T` (X), T` (X))GK
induced by sending any algebraic K-morphism X → X to the attached Z` -linear
map T` (X) → T` (X), is an isomorphism of Z` -modules.
Using the above properties of representations σ` Faltings in [14] proved the conjectureof Mordell from 1922 which says that every smooth projective curve C of genus g > 1
defined over a number field F has finite number of F -rational points.
2. Mumford-Tate conjecture.
2.1. On two conjectures of Serre. It is a natural and central question to what extent
the `-adic cohomology groups are independent of the auxiliary prime number `. Thereis a variety of classical results on this question. For example, if L0 = L \ {p}, then by awell-known corollary to the Riemann hypothesis for varieties over finite fields proved by
Deligne, the function
L0 → N,` 7→ dimQ` (H j (XK , Q` ))is constant.
Assume now that K is a finitely generated subfield of C. It is then natural to consider
in addition to the data involved so far the more classical singular cohomology groups:
j
j
Hsing
(X(C), Z/n) for n ∈ N, Hsing(X(C), Q) and the comparison theorem from [4] gives
isomorphismsj
H j (XK , Z/n) ∼= Hsing (X(C), Z/n)j
and H j (XK , Q` ) ∼= Hsing (X(C), Q) ⊗Q Q` .

This can already be viewed as an `-independence result, because the Q-vector spacej
Hsing(X(C), Q) does not depend on `. To go further let us look at the images of the rep(j)resentations ρ`,X and ask for some kind of `-independence. Let us put V` = H j (XK , Q` ),5

INDEPENDENCE OF SYSTEMS(j)ρ` = ρX,` for ` ∈ L and defineG` := ρ` (GK ),G` := Zariski closure of G` in GLV` ,G◦` := connected component of the neutral element in G` .
Note that G` and G◦` are algebraic groups over Q` . Serre proved (cf. [28]) that the kernelofρ`GK −→ G` (Q` ) → G` (Q` )/G◦` (Q` )does not depend on `. This implies the existence of a finite extension Kconn /K such thatj
the Zariski closure of ρ` (G(K̄/Kconn )) equals G◦` . We define V := Hsing(X(C), Q). As◦
far as the algebraic groups G` are concerned there is the following conjecture (cf. [30]).
Conjecture 2.1. (Mumford-Tate-Serre Conjecture) There exists a (connected) reductive subgroup G/Q of GLV /Q, not depending on `, such that G◦` = G ×Q Spec(Q` ) for
every ` ∈ L.
There is a natural candidate for such an algebraic group G/Q, namely the corresponding
motivic Galois group. This “conjecture of Mumford-Tate type” is a wide open problem,
though there is some evidence due to work of Serre, Ribet, Larsen-Pink, Zarhin-Moonen,Chin, Hall, Banaszak-Gajda-Krasoń [5], [6], Vasiu, Hui, Arias-de-Reyna-Gajda-Petersen[1], [2] and many others, mainly in the case of abelian varieties.
Let us assume now that there is an algebraic group G as above for the variety X andjvalue j under consideration. Choose a Z-lattice T in V (T = Hsing(X(C), Z)/(Torsion)
is a natural choice) such that T` := T ⊗Z Z` is a GK -equivariant Z` -lattice in V` for every` ∈ L, and define G◦` (Z` ) := G◦` (Q` ) ∩ GLT` (Z` ) where the intersection takes place inside
GLV` (Q` ). One can then consider the so-called adelic representation
Y
ρ Y
i
b
ρadelic : Gal(K/Kconn ) −→ρ` (Gal(K/Kconn )) −→G◦ (Z` ) ⊂ G(Q ⊗Z Z)
`

`∈L

`∈L

where ρ is induced by the restricted representations ρ` |Gal(K/Kconn ) → GLV` (Q` ). It isnatural to ask whether ρadelic has an open image (cf. [30, 10.6 ?]).
Conjecture 2.2. (Adelic Openness) The image of ρadelic is open if the corresponding
motive is maximal (cf. [30, 10.6 ?], for the details).
The adelic openness conjecture is a wide open problem, but surprisingly there is a recent
progress on the first task in the above setup, i.e., in considering the map ρ alone andproving unconditionally that it has a large image. Serre’s recent paper [31] studied im(ρ)
in the important case where K is a number field, continuing part of his work on abelianvarieties over number fields back in the 80s. The paper [15] of Gajda and Petersen studied
im(ρ) in the case trdeg(K/Q) > 1. Together this gave a solution to conjecture [30, 10.1?]
of Serre in the case of motives attached to smooth projective varieties, and it answered aquestion of Serre and Illusie (cf. [31, Section 3] and [19, 5.5]) at the same time. Preprints ofBöckle-Gajda-Petersen [7], Cadoret-Tamagawa [8] and the habilitation thesis of Petersen
follow this line of thought begun by Serre even further.

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W.GAJDA AND S.PETERSEN

2.2. Mumford-Tate for abelian varieties. For abelian varieties defined over number
fields, or more generally over finitely generated fields over their prime fields, one can say
more about images of the Galois representations and the Mumford-Tate conjecture. We
illustrate this by quoting few theorems from [5], [6] and [1], where authors investigated
the image of Galois `−adic and mod ` representations associated with the Tate moduleof a simple abelian variety of type I, II and III in the Albert classification list. We use
Weil restriction functor RE/K of affine group schemes for a finite field extension E/K inorder to descibe the image of Galois representation in terms of linear algebraic groups.
Denote byσ` : GK → GL(T` (A)),σ`0 : GK → GL(V` (A))σ ` : GK → GL(A[`])the Galois representations related to the Tate module T` = T` (A) of the abelian varieties
A over a finitely generated field K, where V` (A) = T` ⊗Q` and A[`] = T` /`T` . We consider
Zariski closure G` , of the image of the representation σ`0 and the the image G(`) of therepresentation σ ` . We have a classical theorem of Serre (cf. [29]):
Theorem 2.1. Let A/K be an abelian variety over a finitely generated field of characteristic zero with End(A) = Z and dim(A) = 2, 6 or odd. Then for almost all ` the image ofσ` is as large as possible, i.e., it equals the group of Z` −points of the group of symplectic
similitiudes GSp2g .
Methods modeled on the approach of Serre and based on Lie algebra technics enabled toprove in [5] the following results.
Theorem 2.2 ([5], Thm. 6.9). Let A be a simple abelian variety over a number field K,of type I or II with the endomorphism ring D = End (A) ⊗ Q with center E of degree
e = [E : Q] and d = [D : E] such that 2g = 2hed where h is odd. Then for ` big enough
we have equalities of group schemes:Y(G` )0 =REλ /Q` (Sp2h )λ|`(G(`))0 =YRkλ /F` (Sp2h ),λ|`where G0 is the derived group scheme of G.
Theorem 2.3 ([5], Thm. 6.16). Let A be an abelian variety as in the last theorem. Thenfor all `  0, we have:Y(σ` (GK ))0 =Sp2h (kλ ),λ|`(σ` (GK ))0 =YSp2h (Oλ )λ|`where (σ` (GK ))0 denotes the closure of (σ` (GK ))0 in the natural (λ-adic in each factor)
Q
topology of the group λ|` Sp2h (Oλ ).

INDEPENDENCE OF SYSTEMS

7

Remark. In [6] similar (but more technically involved) theorems about Galois images
were obtained for a large class of abelian varieties of type III. The last two theorems
enabled verification of the Mumford-Tate for the investigated class of abelian varieties.
As a final application of the method an analogue of the open image theorem of Serre wasproven.
There is yet another way developed recently by Chris Hall [17] to compute the image of therepresentation σ` for abelian varieties which is based on group theory, instead of theoryof Lie algebras. Using Hall’s method we proved in [1] together with Sara Arias-de-Reyna
the following result which works over fields of arbitrary characterisitic. In order to state
it we need a definition. We denote by N −→ SpecOv the Néron model of A and by Nv◦
the connected component of its special fibre. We say that an abelian variety A/K has
semistable reduction of toric dimension one at a place v of K, if there is an exact sequenceof group schemes over the residue field κ(v) = Ov /mv
1 −→ T −→ Nvo −→ B −→ 0
where T /κ(v) is a torus of dimension one and B/κ(v) is an abelian variety.
Example 2.4. Let f ∈ Z[x] be a monic square free polynomial of degree deg f = n ≥ 5.
Denote by Cf the smooth projective curve with an affine part the hyperelliptic curvey 2 = f (x). Let A = Jac(Cf ) denote the Jacobian variety. It was proven by Zarhin [32]that if Gal(Spl(f )/Q) = Sn , then End A = Z. Moreover the Jacobian variety A has
semistable reduction of toric diminesion one at a rational prime p, if the image f¯ of f inFp [x] admits a factorization f¯ = f1 f2 in Fp [x], with coprime polynomials f1 , f2 ∈ Fp [x],such that f1 = (x − α)2 with α ∈ Fp , and f2 is a square-free polynomial of degree n−2.
Note that Kowalski proved in [17] using sieves that most polynomials f ∈ Z[x] have aprime p with the above properties.
Theorem 2.5 (Arias-de-Reyna, Gajda, Petersen, [1]). Let A/K be an abelian variety
over a finitely generated field of arbitrary characteristic with End(A) = Z and such that
A has semistable reduction of toric dimension one. Then A has big monodromy, i.e., foralmost all ` the image of σ` contains the group of Z` −points of the group Sp2g .
2.3. Application to Geyer-Jarden conjecture. We applied Theorem 2.5 to prove a
new result towards the Geyer-Jarden conjecture of 1978. Denote by Ksep the separable
closure of K. For a positive integer e and for σ = (σ1 , σ2 , . . . , σe ) in the group GeK =GK ×GK × . . . ×GK , we denote by Ksep (σ) the subfield in Ksep fixed by σ1 , σ2 , . . . , σe .
There exists a substantial literature on arithmetic properties of the fields Ksep (σ). In
particular, the Mordell-Weil groups A(Ksep (σ)) have been already studied. We considered
the torsion part of the groups A(Ksep (σ)). In order to recall the Geyer-Jarden conjecture,
we agree to say that a property A(σ) holds for almost all σ ∈ GeK , if A(σ) holds for allσ ∈ GeK , except for a set of measure zero with respect to the (unique) normalized Haar
measure on the compact group GeK . In [16] Geyer and Jarden proposed the following
conjecture on the torsion part of A(Ksep (σ)).
Conjecture 2.3. Let K be a finitely generated field. Let A be an abelian variety definedover K.

8

W.GAJDA AND S.PETERSENa) For almost all σ ∈ GK there are infinitely many prime numbers ` such that thegroup A(Ksep (σ))[`] of `−division points is nonzero.
b) Let e ≥ 2. For almost all σ ∈ GeK there are only finitely many prime numbers `such that the group A(Ksep (σ))[`] of `−division points is nonzero.
It is known due to the work of Jacobson and Jarden [20] that for all e ≥ 1, almost allσ ∈ GeK and all primes ` the group A(Ksep (σ))[`∞ ] is finite. This was formerly part (c) of
the conjecture. Moreover Conjecture 2.3 is known for elliptic curves [16]. Part (b) holds
true provided char(K) = 0 (see [20]). Zywina [33] proves part (a) in the special casewhere K is a number field, strengthening results of Geyer and Jarden. 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 Conjecture 2.3 are open and part (a) is open over afinitely generated transcendental extension of Q. We prove the Geyer-Jarden conjecture
for abelian varieties of Theorem 2.5.
Theorem 2.6 (Arias-de-Reyna, Gajda, Petersen, [2]). Let K be a finitely generated fieldand A/K an abelian variety with big monodromy. Then the Conjecture 2.3 of Geyer andJarden holds true for A/K.
Surprisingly enough, the most difficult part in the proof of Theorem 2.6 is the case (a)of the Conjecture 2.3 for abelian varieties with big monodromy, when char(K) > 0. The
method of our proof relies in this case on the Borel-Cantelli Lemma of measure theory
and on a delicate counting argument in the group Sp2g (F` ).
3. Main results.
3.1. Statements. Our main results concern research initiated by Serre [31] and continued by us in [15], by Cadoret-Tamagawa [8], Böckle-Gajda-Petersen [7] and by the second
author in his habilitation thesis [26]. In the sequel we call a family of homomorphisms(ρ` : G → G` )`∈L0 of locally compact topological groups with index set L0 ⊂ L almostQ
independent if there exists an open subgroup H ⊂ G such that ρ(H) = `∈L0 ρ` (H)Q
where ρ : G → `∈L0 G` ist the homomorphism induced by the ρ` . The main results are
as follows.
Theorem 3.1 (Serre, Gajda-Petersen, cf. [31], [15]). Let K be a finitely generated fieldof characteristic zero. Let X/K be a separated algebraic scheme and j ∈ N. For ` ∈ L let
V` = H j (XK , Q` ) and letρ` : Gal(K/K) → GLV` (Q` )be the corresponding Galois representation. Then the family (ρ` )`∈L is almost independent.
The following independence theorem concerns the case where the ground field is a geometric function field, i.e., a field K containing an algebraically closed subfield k such thatK/k is finitely generated. It was obtained independently by Cadoret-Tamagawa [8] andBöckle-Gajda-Petersen [7]. Part b) carries a considerable amount of information about
the images of the representations.

INDEPENDENCE OF SYSTEMS

9Theorem 3.2 (Böckle-Gajda-Petersen, Tamagawa-Cadoret, cf. [7], [8]). Let K be a geometric function field of characteristic p ≥ 0 and L0 = L \ {p}. Let X/K be a separated
algebraic scheme. For ` ∈ L0 let ρ` be the representation of Gal(K/K) on H j (XK , Q` ).
There exists a finite Galois extension K 0 /K and a constant c ∈ N with the following
properties.
a) The family (ρ` )`∈L0 is almost independent.
b) For every ` ∈ L0 the group H` := ρ` (Gal(K/K 0 )) is generated by its `-Sylow subgroups and it has a normal seriesH` . N` . P` . {e}with the following composition factors.
i) The group H` /N` is a finite product of finite simple groups of Lie type incharacteristic `.
ii) The group N` /P` is abelian of order prime to ` and |N` /P` | ≤ c.
iii) The group P` is pro-`.
Illusie suggested some time ago to use the theory developed so far, combined with uniform
constructability and semistability results recently proved by Orgogozo [25], in order toprove a relative version for families of sheaves over arithmetic schemes. Let FT D be thecategory of all separated reduced and irreducible Spec(Z)-schemes S which are of finite
type and dominant over Spec(Z). Let S ∈ FT D, let K = R(S) be the field of rational
functions on S and let x̄ : Spec(K) → S be the geometric generic point of S afforded by
the choice of algebraic closure of K. Let F` be an étale sheaf of F` -vector spaces on S for
every ` ∈ L. We call such a family (F` )`∈L almost independent, if the family (F`,¯§ )`∈Lof stalks, viewed as a family of representations of Gal(K/K), is almost independent in
the above sense. We define a certain condition (Ram) for families (F` )`∈L as above which
is roughly a “potential uniform constructibility” and “potential semistability” condition.
in a certain sense. The family (F` )`∈L of constant sheaves on S does satisfy condition(Ram).
Theorem 3.3 (Petersen, cf. [26]). Let S ∈ FT D. For every ` ∈ L let F` be an étale
sheaf of F` -vector spaces on S.
a) If the family (F` )`∈L satisfies condition (Ram), then (F` )`∈L is almost independent.
b) If the family (F` )`∈L satisfies condition (Ram) and f : S → S 0 is a morphism
in FT D, then for every j ∈ N the families (Rj f∗ F` )`∈L and (Rj f! F` )`∈L satisfy
condition (Ram).
Remark. The main merit of Theorem 3.3 is to give more flexibility on the level ofcoefficients than Theorem 3.1. For example, if f : S → S 0 and g : S 0 → S 00 are twomorphisms in FT D, then for i, j ∈ N the family of E2 -terms of the Leray spectral
sequence(Ri g∗ Rj f∗ F` )`∈Lis almost independent by Theorem 3.3. It is an interesting question to prove a version ofTheorem 3.3 for families of Q` -sheaves instead of families of F` -sheaves. The thesis [26]

10

W.GAJDA AND S.PETERSEN

restricted attention to the technically simpler case of finite coefficients only. Furthermore
it is desirable to prove a version that does not assume S to be dominant over Z; forexample S should well be allowed to be an Fp -variety. The final result of this type should
contain Theorems 3.1 and 3.2 as a special case.
Remark. The independence results mentioned so far have applications in field arithmetic. Some such applications have been worked out in [26], and it is possible that one
can work out more. For example it seems promising to apply the new independence results on part a) of the Geyer-Jarden conjecture about torsion of abelian varieties (cf.
[16]). Part a) of this conjecture has been solved by Zywina [33] for abelian varieties overnumber fields. It seems plausible that one can prove part a) in the case of abelian varieties
over finitely generated fields of characteristic zero combining the methods of Zywina [33]with Theorem 3.2. One can speculate that Theorem 3.2 might help to solve Part a) of
the conjecture in positive characteristic.
3.2. Proof sketch of Theorem 3.2.
Group theory. Let ` be a rational prime and let d be a positive integer. We define theset Jor(d) of Jordanian groups as the set of finite groups H that have a normal abeliansubgroup N of index at most d. The subset of Jor(d) of those H for which N has orderprime to ` is denoted by Jor` (d). For a positive integer c define the set Σ` (c) of profinitegroups M such that M has normal open subgroups P ⊂ I ⊂ M such that P is a pro-`group, I/P is a finite abelian group of order prime to ` with the index [I : P ] ≤ c, and
M/I is a product of finite simple groups of Lie type in characteristic `. As the starting
point for the proof of Theorem 3.2 we use the following description of closed subquotientsof Gln (Q` ), which we deduce from the main result of the paper [23] of Larsen and Pink.
Proposition 3.4. For any positive integer n there exists a constant J(n) such that for
all rational primes ` and any closed subquotient G` of Gln (Q` ) there is a short exact
sequence of profinite groups
1 −→ M` −→ G` −→ H` −→ 1
n−1such that M` ∈ Σ` (2) and H` ∈ Jor(J(n)).

Bounding ramification. In the proof of Theorem 3.2 we need deep results on the ramification behavior of the representations ρ` . The following proposition summarizes theinformation we need.
Proposition 3.5. Let k be a perfect field of characteristic p ≥ 0, S/k a smooth k-variety,
K the function field of S and X/K a separated algebraic scheme. For ` ∈ L \ {p} let ρ`
be the representation of Gal(K) on H j (XK , Q` ). There exists a dense open subscheme
U of S, a smooth proper k-variety V , a dense open subscheme V of V and an alteration
V → U such that, if we denote by E the function field of V , the following holds true.
a) For all ` ∈ L \ {p} the representation ρ` factors through π1 (U ) (and consequentlyρ` |Gal(E) factors through π1 (V )).
b) For every ` ∈ L \ {p} and every discrete valuation v of K which is localized at a
codimension 1 point of V the image ρ` (I(v)) of the inertia group I(v) of v is a

INDEPENDENCE OF SYSTEMS

11

pro-` group.
This proposition is established in [7] making use of the base change theorems, de Jong’s
alteration technique, the different notions of tameness from Kerz-Schmidt [22] and two
results of Deligne: from [13] and from [11].
Remark 3.6. If in the situation of Proposition 3.5 ρ` (Gal(K)) is of order prime to ` forevery ` ∈ L \ {p}, then ρ` |Gal(E) factors through π1 (V ) because of the purity of the
branch locus.
Finiteness results for fundamental groups. Let us call a (possibly infinite) Galois extension Ω/K Jordanian if there exists an integer d and a family (Ki )i∈I of Galois extensions
Qof K inside Ω such that Gal(Ki /K) ∈ Jor(d) and Ω = i∈I Ki . We shall make use of
the following finiteness theorem for Jordanian extensions of function fields with restricted
ramification.
Proposition 3.7. Let k be a prime field. Let S/k be a smooth variety. If k is finite,assume that S/k is proper. Let K/k be the function field of S. Let Ω/K be a Jordanianextension which is unramified along S. Then kΩ/kK is a finite extension.
The proof of Proposition 3.7 makes use of the fact that π1 (Sk ) is a finitely generated
profinite group and of central results of Katz and Lang in geometric class field theory.
Putting things together. We shall now give a tentative sketch of proof for Theorem 3.2
in the special case where K is a finitely generated extension of Fp . This special case isparticularly important. Theorem 3.2 in the case p > 0 is derived from this special casein [7], and the proof in the case p = 0 follows similar lines being in fact easier. In thespecial case under consideration there exists a finitely generated extension K0 /Fp and aseparated algebraic K0 -scheme X0 such that K = Fp K0 and X = X0 ⊗ Spec(K). Henceρ` extends to a representation of Gal(K0 ) on H j (XK , Q` ) and we denote this extension
again by ρ` .
By Proposition 3.5 there exists, after replacing K0 by a finite extension, a finite extension
k/Fp inside K0 , a smooth proper geometrically connected k-variety V with function field
K0 and a dense open subscheme V in V such that ρ` factors through π1 (V ) and suchthat for every ` ∈ L \ {p} and every discrete valuation v of K which is localized at a
codimension 1 point of V the group ρ` (I(v)) is a pro-` group.
Let G` = ρ` (Gal(K0 )) and let K` be the fixed field of G` inside K. By Proposition 3.4
there exist constants J, c ∈ N (not depending on `) such that for every ` ∈ L \ {p} there
exists a normal subgroup M` in G` such that M` ∈ Σ` (c) and G` /M` ∈ Jor(J). For everyQ` > J the extension K`M` /K0 is unramified along V . Let Ω = `∈L\{p} K`M` . It follows
from Proposition 3.7 that K 0 := kΩ is a finite extension of K. By construction we seethatρ` (Gal(K 0 )) / M` ∈ Σ` (c) for all ` ∈ L0 .
Using Goursat’s Lemma, one can prove that the class Σ` (c) is closed under taking normal
subgroups, hence ρ` (Gal(K 0 )) ∈ Σ` (c), for all ` ∈ L0 .

12

W.GAJDA AND S.PETERSEN

After replacing K 0 by a finite extension we can assume in addition that ρ` (Gal(K 0 )) ispro-` for all ` 6∈ {2, 3}. An additional argument, based on the fact that π1 (Sk ) is finitely
generated and some group theory proves, that we can also assume that ρ` (Gal(K 0 )) is
generated by its `-Sylow subgroups. This proves part (b) of Theorem 3.2 in the special
case under consideration. Part (a) of this theorem follows then from its part (b) with thehelp of the following classical theorem of E. Artin from group theory. Let Lie` denotethe class of all finite simple groups of Lie type in characteristic `. For example the list oforders of groups in Lie5 begins with60, 7800, 126000, 976500, . . .
and the list for Lie7 begins with
168, 58800, 1876896, 56633616, . . .
cf. [31], Section 6.
Theorem 3.8 (E.Artin, [3], [21]). If `1 and `2 are two distinct rational primes bigger
than 3, then the classes Lie`1 and Lie`2 are disjoint.
3.3. Questions: Openness in families. Another line of investigation concerns thevariation of the images of the corresponding representations in families of varieties. Let k
be a Hilbertian field and S/k a (base) variety with function field K = k(X). Let X → S
be a family of algebraic schemes. For s ∈ S we denote by Xs the fibre of X over S. Fix
j ∈ N. Let ρ`,s be the representation of Gk(s) on H j (Xs,k(s) , Q` ) where Gk(s) is the Galois
group of the residue field k(s). Let ξ be the generic point of S, X = Xξ and ρ` = ρ`,ξ . It
is natural to compare the cohomology of the fibres Xs to the cohomology of the genericfibre X; the generic base change theorem (Katz-Laumon, Illusie) tells us that there exists
an open subset U ⊂ S such that, Q` ) ∼H j (X= H j (X , Q` )s,k(s)

K

for every ` ∈ L0 and every s ∈ U . Having this information at hand one may look forequivariant versions and compare the image of ρ`,s with the image of ρ` . We can viewρ`,s (Gk(s) ) as a subgroup of ρ` (GK ). We define the “good loci” for the specialization tobe
S good (X , `) = {s ∈ S : ρ`,s (Gk(s) ) has finite index in ρ` (GK )}S good (X ) = {s ∈ S : ρ`,s (Gk(s) ) has finite index in ρ` (GK ) for all `}.
A classical Frattini argument sketched by Serre in [27] (in a special case, but the argument
works in general) shows that S good (X , `) is a Hilbertian set. The motivic philosophy
predics that S good (X , `) should be independent of `. As a consequence one would expect,that S good (X ) is Hilbertian as well. One may even look at the good locus of specialization
for the adelic representations
Y
Y
S ad (X ) = {s ∈ S :ρ`,s (Gk(s) ) is open inρ` (GK )}.
``The following catalogue of question about variational adelic openness seems very natural.
Question 3.1. Assume that K is an infinite finitely generated field. Let S (d) be the setof all closed points of S of degree ≤ d.

INDEPENDENCE OF SYSTEMS

13a) Can we prove that the good locus S good (X , `) is big in the sense that S (d) \ S good (X , `)is finite for all d ?
b) Is it true that S good (X , `) does not depend on ` ∈ L0 ? This would in particular imply
that S (d) \ S good (X ) is finite for all d.
c) What can we say about S ad (X ) ?
Remark. Important work of Cadoret and Tamagawa [9] answers question a) affirmatively if char(k) = 0 and dim(S) = 1. Question b), which may seem surprising at afirst glance, has an affirmative answer if char(k) = 0, dim(S) = 1 and X is an abelianscheme due to work of Hui (cf. [18]). Recent work of Cadoret (cf. [10]) settles questionc) if char(k) = 0, dim(S) = 1 and X is an abelian scheme. One would like to establish
analogous results in the special case where char(k) > 0, and it seems plausible to us that
this might be possible. The main problem in this area is to find a good replacement for
the Mumford-Tate group in case when the base field is of positive characteristic. This is
an interesting path of research which shall be followed in future.
Acknowledgments. W.G. thanks the Interdisciplinary Center for Scientific Computing (IWR) at Heidelberg University for hospitality during a research visit in January andFebruary 2015 when part of this work was written. He was supported by the Alexander von Humboldt Foundation and by research grants UMO-2012/07/B/ST1/03541 andUMO-2014/15/B/ST1/00128 of the National Centre of Sciences of Poland. S.P. thanks
the Mathematics Department at Adam Mickiewicz University for hospitality and support
during several research visits. The authors thank the organizers of the meeting ALANT 3
in Bȩdlewo for the invitation. Finally we want to thank an anonymous referee for carefulreading of the manuscript and for helpful comments.

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