J. reine angew. Math., Ahead of Print
DOI 10.1515 / crelle-2015-0024

Journal für die reine und angewandte Mathematik© De Gruyter 2015

Independence of `-adic representationsof geometric Galois groups
By Gebhard Böckle at Heidelberg, Wojciech Gajda at Poznań and
Sebastian Petersen at Kassel

Abstract. Let k be an algebraically closed field of arbitrary characteristic, let K=k bea finitely generated field extension and let X be a separated scheme of finite type over K. For
each prime `, the absolute Galois group of K acts on the `-adic étale cohomology modulesof X. We prove that this family of representations varying over ` is almost independent in thesense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of therepresentations for all ` become linearly disjoint over a finite extension of K. In doing this,we also prove a number of interesting facts on the images and on the ramification of this familyof representations.

1. Introduction
Let G be a profinite group and L0 a set of prime numbers; from the middle of Section 4
onward, L0 will denote a certain fixed set of primes. For every ` 2 L0 let G` be a profinitegroup and ` W G ! G` a continuous homomorphism. Denote byYW G !
G``2L0

the homomorphism induced by the ` . Following the notation in [35] we call the familyQ.` /`2L0 independent if .G/ D `2L0 ` .G/. The family .` /`2L0 is said to be almost indeQ
pendent if there exists an open subgroup H of G such that .H / D `2L0 ` .H /.
The main examples of such families of homomorphisms arise as follows: Let K be a field and absolute Galois groupof characteristic p  0 with algebraic closure K
Gal.K/ D Aut.K=K/:

Gebhard Böckle is supported by the DFG in the FG 1920 and by the DFG/FNR within the SPP 1489.
Wojciech Gajda was supported by the Alexander von Humboldt Foundation and by the National Centre of Sciencesof Poland under research grant UMO-2012/07/B/ST1/03541.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

2Böckle, Gajda and Petersen, Independence of `-adic representations

Let X=K be a separated algebraic scheme1) and denote by L the set of all prime numbers. For
every q 2 N and every ` 2 L X ¹pº we shall consider the representations.q/
W Gal.K/ ! AutQ` .H q .XK; Q` //
`;X

and
.q/
`;X;cW Gal.K/ ! AutQ` .Hcq .XK; Q` //q

of Gal.K/ on the étale cohomology groups H q .XK; Q` / and Hc .XK; Q` /. The following
independence result has recently been obtained.
Theorem 1.1. Let K be a finitely generated extension of Q and let X=K be a separated.q/.q/
algebraic scheme. Then the two families .`;X/`2L and .`;X;c/`2L are almost independent.
The proof of this statement in the important special case trdeg.K=Q/ D 0 is due to Serre(cf. [35]). The case trdeg.K=Q/ > 0 was worked out in [13], answering a question of Serre(cf. [33, 35]) and of Illusie [20].
The usefulness of almost independence is alluded to in Serre [35, Introduction] (cf. also
[33, Section 10]). Almost independence for a family .` W Gal.K/ ! G` /`2L over a field K
means that after a finite field extension E=K, the image of Gal.E/ under the product repQ
Q
resentation `2L ` is the product `2L ` .Gal.E// of the images. This has applications if
one has precise knowledge of the shape of the images for all `. For instance, suppose that
there exists a reductive connected algebraic subgroup G of some GLn over Q such that, after
replacing K by a finite extension K 0 , the image ` .Gal.K 0 // is open in G.Q` / \ GLn .Z` /for all ` and surjective for almost all `. Denote by G ab the torus that is the quotient of G by
its derived group G der , and assume that the induced family .`ab W Gal.K/ ! G ab .Q` //`2L has
adelically open image. Then if G der is simply connected, the almost independence of .` /`2L
implies that the image of Gal.K 0 / is adelically open, i.e., it is open in the restricted prodQ
uct 0`2L G.Q` /. For the case of general G der we refer the reader to [18], where the authors
consider adelic openness for geometric families cf. [18, Conjecture 1.1] over number fields.
The adelic openness of .`ab /`2L is in general not a consequence of almost independence.
However the case when .`ab /`2L is a compatibly system of geometric origin is well-understoodby [34], and adelic openness holds if K is a number field. The existence of a reductive group G
as above, with a priori no condition on G der , is predicted by the Mumford–Tate conjecture.q/
for a smooth projective variety X(cf. [32, C.3.3, p. 387, C.3.8, p. 389], [33, p. 390]) if ` D `;X
over a finitely generated extension K of Q.
The present article is concerned with a natural variant of Theorem 1.1 that grew out of
the study of independence of families over fields of positive characteristic. For K a finitely
generated extension of Fp it has long been known, e.g., [19] or [11], that the direct analogueof Theorem 1.1 is false: If "` W Gal.Fp / ! Zdenotes the `-adic cyclotomic character that`
describes the Galois action on `-power roots of unity, then it is elementary to see that the family."` /`2LX¹pº is not almost independent. It follows from this that for every abelian variety A=K,
if we denote by `;A W Gal.K/ ! AutQ` .T` .A// the representation of Gal.K/ on the `-adic
Tate module of A, then .`;A /`2LX¹pº is not almost independent. One is thus led to study
1)

A scheme X=K is algebraic if the structure morphism X ! Spec K is of finite type (cf. [14, Definition 6.4.1]).

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

3

independence over the compositum 
Fp K obtained from the field K by adjoining all roots ofunity. Having gone that far, it is then natural to study independence over any field K that is
finitely generated over an arbitrary algebraically closed field k. Our main result is the following
independence theorem.
Theorem 1.2 (cf. Theorem 7.7). Let k be an algebraically closed field of characteristic p  0. Let K=k be a finitely generated extension and let X=K be a separated algebraic.q/.q/jGal.K/ /`2LX¹pº and .`;X
scheme. Then the families .`;X;c jGal.K/ //`2LX¹pº are almost independent.
It will be clear that many techniques of the present article rely on [35]. Also, some of the
key results of [13] will be important. The new methods in comparison with the previous results
are the following.
(i) The analysis of the target of our Galois representations, reductive algebraic groups
over Q` , will be based on a structural result by Larsen and Pink (cf. [25]) and no longer as
for instance in [35] on extensions of results by Nori (cf. [29]). In the proof of Theorem 1.2 weuse crucially that there exists a finitely generated subfield K0 of K and a separated algebraic
scheme X0 =K0 such that kK0 D K and X0 K0 Spec.K/ D X. The group theoretical results
mentioned above facilitate greatly the passage from Gal.K0 / to Gal.K/ when studying their.q/
image under `;X;‹.
(ii) Since we also deal with cases of positive characteristic, ramification properties willplay a crucial role to obtain necessary finiteness properties of fundamental groups. The resultson alterations by de Jong (cf. [6]) will obviously be needed. However we were unable to deduce
all needed results from there, despite some known semistability results that follow from [6].
Instead we carry out a reduction to the case where K is absolutely finitely generated and where
X=K is smooth and projective (this uses again [6]).
(iii) In the latter case, we use a result by Kerz–Schmidt–Wiesend (cf. [23]) that allows one
to control ramification on X by controlling it on all smooth curves on X. By Deligne’s results.q/form a pure and strictly compatible
on the Weil conjectures, the semisimplifications of the `;X
system. On curves, we can then apply an `-independence result on tameness from [7] again
due to Deligne. Together this allows us to obtain a very clean result on a kind of semistable rami.q/
fication of .`;X/`2LX¹pº , cf. Remark 6.4.
Part (i) is carried out in Section 3. Results on fundamental groups and first results on
ramification are the theme of Section 4; there we carry out parts of (ii) and we refine someresults from [23]. Section 5 provides the basic independence criterion on which our proofof Theorem 1.2 ultimately rests. For this we introduce notions that describe ramification andsemistability in families .` /`2L . Section 6 establishes a semistability property for the families.q//`2L , for any smooth projective variety X over any field K that is finitely generated over.`;Xa perfect field of positive characteristic. This is step (iii) in the above program. Finally, in
Section 7 we complete part (ii) and we give the proof of Theorem 7.7 which is a slightly
refined form of Theorem 1.2.
We would like to point out that an alternative proof of part (ii) of our approach could
be based on recent unpublished work by Orgogozo which proves a global semistable reduction
theorem (cf. [30, Proposition 2.5.8]). When our paper was complete we were informed by Anna
Cadoret that, together with Akio Tamagawa, she has proven our Theorem 1.2 by a different
method, cf. [5].

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

4Böckle, Gajda and Petersen, Independence of `-adic representations

Acknowledgement. Gebhard Böckle thanks the Fields Institute for a research stayin the spring of 2012 during which part of this work was written. He also thanks Adam
Mickiewicz University in Poznań for making possible a joint visit of the three authors in the fallof 2012. Wojciech Gajda thanks the Interdisciplinary Center for Scientific Computing (IWR)
at Heidelberg University for hospitality during research visits in January 2012 and in January
2014. Sebastian Petersen thanks the Mathematics Department at Adam Mickiewicz University
for hospitality and support during several research visits. We thank F. Orgogozo and L. Illusie
for interesting correspondence concerning this project. In addition, the authors thank the anonymous referee for a thorough review of the paper and many helpful comments and suggestions,and in particular for pointing us to [7, Théorème 9.8] that replaced earlier arguments involving
the global Langlands correspondence proven by L. Lafforgue.

2. Notation
Let G be a profinite group. A normal series in G is a sequence
G D N0 F N1 F N2 F    F Ns D ¹eºof closed subgroups such that Ni 1 is a normal subgroup of Ni for all 1  i  2. Throughout this manuscript L denotes the set of all prime numbers. From Section 4 on we define
L0 D L X ¹pº where p  0 is the characteristic of a base field k. For any ` 2 L we denoteby G`C the normal hull of the pro-` Sylow subgroups of G.
 we denote by Ks  K
 a separable closure. Then
For a field K with algebraic closure K,
Gal.K/ is equivalently defined as Gal.Ks =K/ and as Aut.K=K/, since any field automorphism If E=K is an arbitrary field extension, and if K
 isof Ks fixing K has a unique extension to K.
chosen inside E, then there is a natural isomorphism'
 K
 \ E/ !

Aut.K=
Aut.KE=E/:

Composing its inverse with the natural restriction Gal.E/ ! Aut.E K=E/
one obtains a canonical map
resE=K W Gal.E/ ! Gal.K/: then resE=K
For homomorphism W Gal.K/ ! G we denote  ı resE=K by jGal.E / . If E  K,
is injective and we identify Gal.E/ with the subgroup resE=K .Gal.E// of Gal.K/.
A K-variety X is a scheme X that is integral separated and algebraic over K. We denoteby K.X / its function field. A K-curve shall be a K-variety of dimension 1. Let S be a normal
connected scheme with function field K. A separable algebraic extension E=K is said to beunramified along S if for every finite extension F=K inside E the normalization of S in F
is étale over S . We usually consider S as a scheme equipped with the generic geometric basee ! S and denote by 1 .S / WD 1 .S; s/ the étale fundamental group of S.
point sW Spec.K/
If  denotes the maximal extension of K in Ks which is unramified along S , then 1 .S / canbe identified with the Galois group Gal.=K/. A continuous homomorphism W Gal.K/ ! H
ker./
is said to be unramified along S if the fixed field Ks
is contained in , i.e., if  factors
through the quotient 1 .S / of Gal.K/. In fact, we shall identify continuous homomorphisms
1 .S/ ! H with continuous homomorphisms Gal.K/ ! H which are unramified along S.
If S is a variety defined over a field k, then by a compactification of S we mean a proper
k-variety S containing S as an open subscheme.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

5

3. Concepts from group theory
In this section, we prove a structural result for compact profinite subgroups of linear ` (cf. Theorem 3.6) that will be crucial for the proof of the main
algebraic groups over Q
theorem of this article. It is a consequence of a variant (cf. Proposition 3.10) of a theorem of
Larsen and Pink (cf. [25, Theorem 0.2, p. 1106]). The proof of Proposition 3.10 makes strong
use of the results and methods in [25], and in particular does not depend on the classificationof finite simple groups.
Definition 3.1. For c 2 N and ` 2 L we denote by †` .c/ the class of profinite groups M
which possess a normal series by open subgroups(1)

M F I F P F ¹1ºsuch that M=I is a finite product of finite simple groups of Lie type in characteristic `, thegroup I =P is finite abelian of order prime to ` and index ŒI W P   c, and P is a pro-` group.
We observe that if M lies in †` .c/, then the normal series (1) is uniquely determinedby M . In fact, P is then the maximal normal pro-` subgroup of M and I is the maximal
normal pro-solvable subgroup of M . In particular, P and I are characteristic subgroups of M .
Note also that for any group M in †` .c/, the quotient M=M`C is abelian of order at most c.
Definition 3.2. For d 2 N and ` 2 L we denote by Jor` .d / the class of finite groups H
which possess a normal abelian subgroup N of order prime to ` and of index ŒH W N   d . We
define Jor.d / as the union of the Jor` .d / over all ` 2 L.
The following lemma records a useful permanence property of groups in the classes †` .c/
and Jor` .d /.
Lemma 3.3. Fix c; d 2 N. Then for any ` 2 L the following holds:(a) If H 0 G H is a normal subgroup of some H 2 Jor` .d /, then H 0 and H=H 0 lie in Jor` .d /.
(b) If M 0 G M is a closed normal subgroup of some M 2 †` .c/, then M 0 and M=M 0 lie
in †` .c/.
If M 0 in part (b) of the lemma was a non-normal closed subgroup of M; then clearly M 0need not lie in †` .c/ again.
Proof. We only give the proof of (b), the proof of (a) being similar but simpler. Let M
be in †` .c/ and consider a normal series
M F I F P F ¹1º
as in Definition 3.1. Then L WD M=I is isomorphic to a product L1      Ls for certain finite
simple groups of Lie type Li in characteristic `. Suppose M 0 is a closed normal subgroup of M
and define M 0 D M 0 I =I . By Goursat’s lemma the groups M 0 and L=M 0 are products of someof the Li . From this it is straightforward to see that both M 0 and M=M 0 lie in †` .c/.
The following corollary is immediate from Lemma 3.3 (b).

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM6Böckle, Gajda and Petersen, Independence of `-adic representations

Corollary 3.4. Fix a constant c 2 N. Let G be a profinite group, and for each ` 2 L let` W G ! G` be a homomorphism of profinite groups such that ` .G/ 2 †` .c/ for all ` 2 L.
Then for any closed normal subgroup N G G one has ` .N / 2 †` .c/ for all ` 2 L.
Definition 3.5. A profinite group G is called n-bounded at ` if there exist closed com ` / such that G1 is normal in G2 and G Š G2 =G1 .
pact subgroups G1  G2  GLn .Q
The following is the main result of this section.
Theorem 3.6. For every n 2 N there exists a constant J 0 .n/ (independent of `) suchthat the following holds: Any group G` that is n-bounded at some ` 2 L lies in a short exact
sequence
1 ! M` ! G` ! H` ! 1
such that M` is open normal in G` and lies in †` .2n 1 / and H` lies in Jor` .J 0 .n//.
We state an immediate corollary:
Corollary 3.7. Let ` > J 0 .n/ and let G` be a profinite group which is n-bounded at `.
With notation as in Theorem 3.6 and in Section 2, G`C is an open normal subgroup of M` ofindex at most 2n 1 .
In the remainder of this section we shall give a proof of Theorem 3.6. The content of thefollowing lemma is presumably well known.
Lemma 3.8. For every r 2 N, every algebraically closed field F and every semisimple
algebraic group G of rank r the center Z of G satisfies jZ.F /j  2r .
Proof. Lacking a precise reference, we include a proof for the reader’s convenience.
Observe first that the center Z is a finite (cf. [27, I.6.20, p. 43]) diagonalizable algebraic group.
Let T be a maximal torus of G. Denote by X.T / D Hom.T; Gm / the character group of T
and by ˆ  X.T / the set of roots of G. Then R D .X.T / ˝ R; ˆ/ is a root system. Let
P D Zˆ be the root lattice and Q the weight lattice of this root system. Then P  X.T /  Q.
The center Z of G is the kernel of the adjoint representation (cf. [27, I.7.12, p. 49]). HenceT
Z D 2ˆ ker./ and there is an exact sequence
Y
0!Z!T !
Gm
2ˆwhere the right hand map is induced by the characters W T ! Gm ( 2 ˆ). We apply thefunctor Hom. ; Gm / and obtain an exact sequenceY
Z ! X.T / ! Hom.Z; Gm / ! 0:2ˆThe cokernel of the left hand map is X.T /=P . Thus jZ.F /j  ŒX.T / W P   ŒQ W P .
Furthermore, the root system R decomposes into a direct sum
RDsM.Ei ; ˆi /i D1Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

7of indecomposable root systems Ri WD .Ei ; ˆi /. Let ri D dim.Ei / be the rank of Ri . Let PiL
be the root lattice and Qi the weight lattice of Ri . Note that by definition we have P D i Pi
L
and Q D i Qi . It follows from the classification of indecomposable root systems thatjQi =Pi j  2ri (cf. [27, Table 9.2, p. 72]) for all i . Hence jZ.F /j  jQ=P j  2r1 2r2    2rs D 2r
as desired.
Remark 3.9. The semisimple algebraic group .SL2;C /r has rank r and its center .2 /r
has exactly 2r C-rational points. Hence the bound of Lemma 3.8 cannot be improved.
The following result is an adaption of the main result of [25] by Larsen and Pink.
Proposition 3.10. For every n 2 N, there exists a constant J 0 .n/ such that for every
field F of positive characteristic ` and every finite subgroup  of GLn .F /, there exist normal
subgroups L; M; I and P of  forming a normal series
 F L F M F I F P F ¹1ºwith the following properties:(i) Œ W L  J 0 .n/.
(ii) The group L=M is abelian of order prime to `.
(iii) The group M=I is a finite product of finite simple groups of Lie type in characteristic `.
(iv) The group I =P is abelian of order prime to ` and ŒI W P   2n 1 .
(v) P is an `-group.
Furthermore, the constant J 0 .n/ is the same as in [25, Theorem 0.2, p. 1106].
Proof. We can assume that the field F is algebraically closed. Let J 0 .n/ be the constant from [25, Theorem 0.2, p. 1106]. Larsen and Pink construct in the proof of their theorem([25, Theorem 0.2, pp. 1155–1156]) normal subgroups i of  such that there is a normal series F 1 F 2 F 3 F ¹1º
and such that Œ W 1   J 0 .n/, 1 = 2 is a product of finite simple groups of Lie type in characteristic `, 2 = 3 is abelian of order prime to ` and 3 is an `-group. The construction of
the groups i in [25, Theorem 0.2, pp. 1155–1156] shows that there exists a smooth algebraic
group G over F containing  such that, if we denote by R the unipotent radical of the connected component G ı of G and by Z the center of the reductive group G WD G ı =R, then
1 G G ı .F /, 3 D  \R.F / and 2 = 3 is contained in Z.F /. Let D D ŒG; G be the derived
group of G and D D ŒG ı ; G ı R.
Now define L D 1 , M D 1 \ D.F /, I D 2 \ D.F / and P D 3 . These groups arenormal in , because D.F / is characteristic in G ı .F / and because 1 ; 2 ; 3 are normal in .
The group L=M is a subgroup of the abelian group G ı .F /=D.F /. As G ı =D is isomorphic
to the torus G=D, it follows that the order of L=M is prime to `. The group M=I is a normal
subgroup of 1 = 2 , hence it is a product of finite simple groups of Lie type in characteristic `.
The group I =P is a subgroup of 2 = 3 , hence I =P is abelian of order prime to `. Furthermore, I =P D I = 3 is a subgroup of G.F / which lies in D.F / and in Z.F /. Thus I =P lies

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM8Böckle, Gajda and Petersen, Independence of `-adic representationsin the center Z.F / \ D.F / of the semisimple group D.F /. It follows by Lemma 3.8 thatŒI W P   2rk.D/ .
It remains to show that rk.D/  n 1. Let T be a maximal torus of D and denoteby W G ı ! G the canonical projection. Note that  induces an epimorphism ŒG ı ; G ı  ! D.
The algebraic group B WD  1 .T / \ ŒG ı ; G ı  sits in an exact sequence
0 ! R \ ŒG ı ; G ı  ! B ! T ! 0
and B is connected smooth and solvable, because R and T have these properties. The above
exact sequence splits (cf. [10, XVII.5.1]); hence B contains a copy of T . This copy is containedin a maximal torus T 0 of SLn;F because B is a subgroup of SLn;F . Thusn

1 D dim.T 0 /  dim.T / D rk.D/

as desired.
Proof of Theorem 3.6. Suppose the group G` is n-bounded at `, so that it is a quotient ` /. By Lemma 3.3, it will suffice to prove the theorem in the
G2;` =G1;` with Gi;`  GLn .Q ` /. By
case G` D G2;` . Thus we assume that G` is a compact profinite subgroup of GLn .Q
the compactness of G` and a Baire category type argument (cf. [12, proof of Corollary 5]) thegroup G` is contained in GLn .E/ for some finite extension E of Q` . Let OE be the ring ofintegers of the local field E. Again by compactness of G` one can then find an OE -lattice in E nthat is stable under G` . Hence we may assume that G` is a closed subgroup of GLn .OE /.
Let p be the maximal ideal of the local ring OE and let F D OE =p be its residue
field. The kernel K of the canonical map pW GLn .OE / ! GLn .F / is a pro-` group. HenceQ` D K \ G` is pro-` and open normal in G` . We now apply Proposition 3.10 to the finite
subgroup G` =Q` of GLn .F /  GLn .F / with F D e
F ŠeF ` . This yields normal subgroups L` ,M` , I` and P` of G` such that there is a normal seriesG` F L` F M` F I` F P` F Q` F ¹1ºwith the following properties: The group G` =M` lies in Jor` .J 0 .n//, and the group M` liesin †` .2n 1 / – for the latter use that Q` is pro-` and normal in G` and P` =Q` is a finite`-group.

4. Fundamental groups: Finiteness properties and ramification
The purpose of this section is to recall some finiteness properties of fundamental groups
and to provide some basic results on ramification. Regarding the latter we draw from results byKerz, Schmidt and Wiesend (cf. [23]).
We begin with a finiteness result of which a key part is from [13].
Proposition 4.1. Suppose that either k is a finite field and S is a smooth proper k-varietyor that k is a number field and S is a smooth k-variety, and denote by K D k.S / the function
e such thatfield of S. For d 2 N, let Md be the set of all finite Galois extensions E=K inside K
Gal.E=K/ satisfies Jor.d / and such that E is unramified along S . Then there exists a finite
Q 0 for every E 2 Md .
Galois extension K 0 =K which is unramified along S such that E  kK

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

9

Proof. For every E 2 Md the group Gal.E=K/ satisfies Jor.d / and hence there isa finite Galois extension E 0 =K inside E such that ŒE 0 W K  d and such that E=E 0 is abelian.
Consider the composite fields
YY
0 D
E0   D
E:
E 2Md

E 2Md

Then =0 is abelian. Let k0 (resp.  0 , resp. ) be the algebraic closure of k in K (resp. in 0 ,resp. in ),
K
0


k

k0

0

.

It suffices to prove the following:
Claim. The extension =K is finite.
In fact, once this is shown, it follows that the finite separable extension =K has a primitive element !. Then  D K.!/ and K.!/=K is a finite separable extension. Let K 0 be theQ 0  K 0  K.!/ D  as desired.
normal closure of K.!/=K in . Then kK
In the case where k is a number field the claim has been shown in [13, Proposition 2.2].
Assume from now on that k is finite. It remains to prove the claim in that case. The structure
morphism S ! Spec.k/ of the smooth scheme S factors through Spec.k0 / and S is a geometQ is topologically finitely
rically connected k0 -variety. The profinite group 1 .S k0 Spec.k//Ogenerated (cf. [16, Theorem X.2.9]) and Gal.k0 / Š Z. Thus it follows by the exact sequence(cf. [16, Theorem IX.6.1])
Q ! 1 .S / ! Gal.k0 /! 1
1 ! 1 .S k0 Spec.k//that 1 .S / is topologically finitely generated. Thus there are only finitely many extensions of K
in KQ of degree  d which are unramified along S . It follows that 0 =K is a finite extension.
Thus  0 is a finite field. If we denote by S 0 the normalization of S in 0 , then S 0 ! S is finiteand étale, hence S 0 is a smooth proper geometrically connected  0 -variety. Furthermore, =0
is abelian and unramified along S 0 . Hence =0 is finite by Katz–Lang (cf. [21, Theorem 2,
p. 306]). As 0 =K is finite, it follows that =K is finite.
To introduce below a notion of tameness that is inspired by [23] and applies to coverings of general schemes, we require further notation. For a Galois extension E=K of fields,
a discrete valuation vW K  ! Z of K and an extension w of v to E we define IE=K .w/ (resp.
IE=K .v/) to be the inertia group of w (resp. of v) in the extension E=K.2) Note that IE=K .v/
is well-defined only up to conjugation. We put I.v/ D IKs =K .v/. In the special case where v
is the trivial valuation, the valuation w must be trivial as well and IE=K .v/ is the trivial group.
Now let p be the residue characteristic of v and let ƒ  L X ¹pº. The extension E=K shallbe called ƒ-tame at v if the order of IE=K .v/ (viewed as a supernatural number) is divisible2)

If E=K is infinite, then w need no longer be discrete but its restriction to any finite Galois subextensionof E=K is so. For any E=K, the group IE=K .w/ is the inverse limit over ramification groups of finite extensions.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM10Böckle, Gajda and Petersen, Independence of `-adic representations

only by primes in ƒ. Note that E=K is L X ¹pº-tame at v if and only if IE=K .v/ is a groupof order prime to p, i.e., if and only if E=K is tame at v in the usual sense.3) For us the casewhere ƒ D ¹`º for a single prime number ` ¤ p will be particularly important, and in that casewe speak of `-tameness rather than of ¹`º-tameness. If K=k is a finitely generated extensionof fields, then we will denote by VK=k the set of all discrete valuations K  ! Z which aretrivial on k.
For the rest of this section let k be a field of characteristic p  0, L0 D L X ¹pº andƒ  L0 . Furthermore, let S be a regular variety over k and K D k.S / its function field. Let G
be a locally compact topological group and W 1 .S / ! G a continuous homomorphism. Let E
be the fixed field of ker./ in Ks .
Recall that we identify continuous homomorphisms W 1 .S / ! G with continuous
homomorphisms W Gal.K/ ! G which are unramified along S .
Definition 4.2. Let v 2 VK=k . The homomorphism  is said to be ƒ-tame at v if theorder of the profinite group .I.v// (viewed as a supernatural number) is divisible only byprime numbers in ƒ. The homomorphism  is called ƒ-tame if it is ƒ-tame at every v 2 VK=k .
Note that the homomorphism  is ƒ-tame at v if and only if the extension E=K is ƒ-tame
at v.
Lemma 4.3. Let v 2 VK=k . Then there exists a normal compactification S of S anda codimension 1 point s 2 S such that v D vs is the discrete valuation of K attached to s.
Proof. Let S 0 be a normal compactification of S , which exists by the theorem ofNagata [26]. By [37, Proposition 6.4], there exists a blow-up S of S 0 with center outside S
such that v is the valuation of a codimension 1 point s 2 S. By normalization, we may further
assume that S is normal. Both operations, blow-up and normalization, do not affect S , andso there exist a normal compactification S of S that contains a codimension 1 point s withvaluation v D vs .
Remark 4.4. As an immediate consequence of Lemma 4.3 we see that the following
statements are equivalent.
(a) The homomorphism  is ƒ-tame.
(b) For every normal compactification S of S and every codimension 1 point s 2 S the extension E=K is ƒ-tame in the discrete valuation vs of K attached to s.
In particular,  is L0 -tame if and only if E=K is divisor tame in the sense of [23].
For a morphism f W S 0 ! S , we denote by f W 1 .S 0 / ! 1 .S / the induced continuous
homomorphism of fundamental groups. The following base change property of ƒ-tameness isquite useful.
3)Note that if the residue field extension at the valuation v for E=K is inseparable, then p will divide theorder of IE=K .v/.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

11

Lemma 4.5. Let k 0 =k be an arbitrary field extension and S 0 a regular k 0 -variety. Let
K 0 D k 0 .S 0 / and recall that K D k.S /. Assume that there is a diagramfS0

Spec.k 0 //S/ Spec.k/where f is dominant.
(a) If W 1 .S / ! G is ƒ-tame, then the composite homomorphism ı f W 1 .S 0 / ! 1 .S / ! Gis ƒ-tame.
(b) If f is finite, K 0 =K is purely inseparable and  ı f is ƒ-tame, then  is ƒ-tame.
Proof. Recall that E is the fixed field of ker./. Let E 0 be the fixed field of ker. ı f /.
Then E 0 D EK 0 in some separable closure Ks0  Ks of K 0 , and we have a diagram of fields
E

E0

K

K0

where E=K and E 0 =K 0 are Galois. Let v 0 2 VK 0 =k 0 and v D v 0 jK. The restriction map
rW Gal.E 0 =K 0 / ! Gal.E=K/;

 7! jE

is injective because E 0 D EK 0 . It is easy to check that r.IE 0 =K 0 .v 0 // is conjugate to a closed
subgroup of IE=K .v/. If  is ƒ-tame, then the order of IE=K .v/ is divisible only by primes inƒ, and thus the order of IE 0 =K 0 .v 0 / is divisible only by prime numbers in ƒ; hence E 0 =K 0 isthen ƒ-tame at v 0 as desired. This proves part (a).
To prove (b) assume that f is finite, that K 0 =K is purely inseparable and that  ı f
is ƒ-tame. Then E 0 =K 0 is ƒ-tame at v 0 . As K 0 =K is purely inseparable, the map r is an
isomorphism and r.IE 0 =K 0 .v 0 // is conjugate IE=K .v/. Thus the order of IE=K .v/ is divisible
only by primes in ƒ and it follows that  is ƒ-tame. This completes the proof of part (b).
The following proposition is a useful criterion to establish ƒ-tameness for a given homomorphism 1 .S / ! G. It is a variant of parts of [23, Theorem 4.4].
Proposition 4.6. Assume that for every regular curve C =k and for every morphism
f W C ! S the homomorphism ı f W 1 .C / ! 1 .S / ! Gis ƒ-tame. Then  is ƒ-tame.
Proof. We can assume that G is finite and  is surjective. Let v 2 VK=k and let w be
an extension of v to E. Let I D I.w/ and J D .I /. Then J is solvable. We have to provethat the order of J is divisible only by primes in ƒ. Assume to the contrary that there exists

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM12Böckle, Gajda and Petersen, Independence of `-adic representations

a prime number `0 2 L X ƒ such that jJ j is divisible by `0 . Then, by the solvability of J ,there exists a subgroup J1 of J and a normal subgroup J2 of J1 such that J1 =J2 Š Z=`0 .
For i 2 ¹1; 2º let Ki be the fixed field of  1 .Ji /, let Si be the normalization of S in Ki
and wi the restriction of w to Ki . Then w2 is totaly ramified in K2 =K1 , and restricting to 1 .S1 / yields an epimorphism 1 .S1 / ! J1 ! Z=`0 . By Lemma 4.3 there exists a normalcompactification S 1 of S1 such that w1 is the discrete valuation attached to some codimension 1
point s1 of S 1 . As S 1 is regular in codimension 1, it follows that the maximal regular open
subscheme W1 of S 1 contains s1 . Furthermore, S1  W1 .
Now let C1 =k be an arbitrary regular curve and let f W C1 ! W1 be a non-constantmorphism with f .C1 / \ S1 ¤ ¿. Let D1 D f 1 .S1 /. For every discrete valuation u on k.C1 /the composite homomorphism
0 W 1 .D1 / ! 1 .S1 / ! J1 =J2 Š Z=`0
maps the inertia group I.u/ to zero, because 0 .I.u// is of order divisible only by primesin ƒ and a subgroup of Z=`0 at the same time. In particular, 0 factors through 1 .C1 /. Thisimplies that S2 S1 D1 ! D1 extends to a not necessarily connected étale cover of C1 . Now
by [23, Proposition 4.1], which can be paraphrased as curve-unramifiedness implies unramifiedness over a regular base, it follows that the normalization W2 of W1 in K2 is étale over W1 .
But then K2 =K1 is étale along w, a contradiction.
Remark 4.7. Combining notions in [23] with our notion of ƒ-tameness, it is straightforward to define a notion of ƒ-curve-tameness. Then Proposition 4.6 asserts that ƒ-curvetameness implies ƒ-tameness. Following [23] one can show that in fact the two notions areequivalent.

5. An independence criterion
Throughout this section let k be a field of characteristic p  0 and L0 D L X ¹pº.
Let S=k be a regular k-variety with function field K D k.S /. For every ` 2 L0 let G` be
a locally compact topological group and ` W Gal.K/ ! G` a continuous homomorphism.
If for all ` 2 L0 the groups ` .Gal.K// are n-bounded at `, then by Theorem 3.6 we havea short exact sequence
1 ! M` ! ` .Gal.K// ! H` ! 1
with H` 2 Jor` .J 0 .n// and M` 2 †` .2n 1 /. In this section, we shall show in Proposition 5.5
and Theorem 5.8 how to control H` and M` in a uniform manner, if one has a uniform control
on ramification. We begin by introducing the necessary concepts and then give the result.
Recall from [6, Section 2.20] that a morphism f W V ! U between k-varieties is analteration if it is proper and surjective and there exists a dense open subscheme U 0 of U suchthat f 1 .U 0 / ! U 0 is finite.
Definition 5.1. The family .` /`2L0 satisfies(i) condition R.S=k/ if there exists a dense open subscheme U of S such that for every` 2 L0 the homomorphism ` factors through 1 .U /.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

13(ii) condition S.S=k/ if there exists a dense open subscheme U of S , a regular k-variety V
and an alteration f W V ! U such that for every ` 2 L0 the homomorphism ` factors
through 1 .U / and such that for every ` 2 L0 the composite homomorphism` ı f W 1 .V / ! 1 .U / ! G`is `-tame. Such a triple .U; V; f /, or simply f W V ! U , is called a witness of the condition S.S=k/, or we say that it witnesses the condition S.S=k/.
Note that condition S.S=k/ implies condition R.S=k/. The condition R.S=k/ is a uniform constructibility condition; S.S=k/ is a uniform semistability condition. Example 5.3
shows that both conditions are satisfied for the family of `-adic representations attached toan abelian variety A over the function field K of S .
Lemma 5.2. Assume that .` /`2L0 satisfies condition S.S=k/.
(a) If f W V ! U witnesses condition S.S=k/, then for any alteration g W W ! V with W
regular, the composition f ı gW W ! U witnesses condition S.S=k/.
(b) There exists a witness .U 0 ; V 0 ; f 0 / of condition S.S=k/ such that f 0 is finite étale.
Part (b) is useful in Proposition 5.5 when studying the property S.S=k/ under basechange: being finite étale is preserved under base change, while being an alteration is not.
Proof. The hypothesis in (a) means that ` factors through 1 .U / and the composition` ı f W 1 .V / ! G` is `-tame, for any ` in L0 . Lemma 4.5 now implies that ` ı f ı g is`-tame because g is dominant. Since the composition of alterations is an alteration, part (a)holds true.
To prove (b), let f be as in (a) and let E be the maximal separable extension of K
inside k.V /. Then E=K is separable and k.V /=E is purely inseparable. Let T be the normalization of U in E. There exists a dense open subscheme U 0 of U such that the restriction of f
to a morphism V 0 WD f 1 .U / ! U 0 is finite. By generic smoothness, after shrinking U 0
further, we can assume that the canonical morphism f 0 W T 0 WD U 0 U T ! U 0 is finite andétale. Let hW V 0 ! T 0 be the canonical morphism. We know by assumption that the homomorphism` ı f0 ı h D ` ı .f jV 0 / W 1 .V 0 / ! G`is `-tame for all ` 2 L0 . Part (b) of Lemma 4.5 now shows that ` ı f0 W 1 .T 0 / ! G` is `-tamefor all ` 2 L0 . Hence f 0 witnesses condition S.S=k/ as desired.
Example 5.3. Let A=K be an abelian variety. For every ` 2 L0 denote by`;A W Gal.K/ ! AutZ` .T` .A//
the representation of Gal.K/ on the `-adic Tate module
T` .A/ D lim AŒ`i :
i 2N

By the spreading-out principles of [15] there exists a non-empty open subscheme U of S andAuthenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM14Böckle, Gajda and Petersen, Independence of `-adic representations

an abelian scheme A over U with generic fiber A. This implies (cf. [17, IX.2.2.9]) that `;A isunramified along U , i.e., that `;A factors through 1 .U / for every ` 2 L0 . Hence the family.`;A /`2L0 satisfies condition R.S=k/.
In order to obtain also condition S.S=k/ from Definition 5.1, we choose an odd prime`0 2 L0 , and we define K 0 D K.AŒ`0 /. After shrinking U accordingly we can assume thatthe normalization U 0 of U in K 0 is étale over U . Now let v 0 2 VK 0 =k be a non-trivial discrete
valuation and Rv0 the discrete valuation ring of v 0 . Let Nv0 = Spec.Rv0 / be the Néron modelof A over Rv0 . The condition K 0  K.AŒ`0 / forces Nv0 to be semistable (cf. [17, IX.4.7]).
This in turn implies that `;A jI.v 0 / is unipotent (and hence `;A .I.v 0 // is pro-`) for every` 2 L0 (cf. [17, IX.3.5]). It follows that the family .`;A /`2L0 satisfies condition S.S=k/.
Recall that for a separated algebraic scheme X=K, for q 2 N and for every ` 2 L X ¹pº
we consider the representations.q/`;X
W Gal.K/ ! AutQ` .H q .XK; Q` //and.q/`;X;cW Gal.K/ ! AutQ` .Hcq .XK; Q` //q

of Gal.K/ on the étale cohomology groups H q .XK; Q` / and Hc .XK; Q` /.
.q/
Proposition 5.4. For a separated algebraic scheme X=K, the two families .`;X
/`2L0
.q/
and .`;X;c /`2L0 both satisfy condition R.S=k/.

Proof. There exists a separated morphism f W X ! S of finite type with generic fiber X.
Next there exists a dense open subscheme U of S such that for every ` 2 L0 the sheaves
Rq f .Q` /jU and Rq fŠ .Q` /jU are lisse (cf. [22, Theorems 3.1.2–3.1.3], [20, Corollary 2.6]).
 ! U be the geometric generic point of U afforded by the choice of K.
 ThenLet W Spec.K/qqqq
the stalk R f .Q` / (resp. R fŠ .Q` / ) is H .XK; Q` / (resp. Hc .XK; Q` /), cf. [2, Théo.q/.q/
and `;X;cfactor through 1 .U / forrème VIII.5.5.2, p. 386]. Thus the representations `;X
0
every ` 2 L .
.q/.q/
For p > 0, we shall treat condition S.S=k/ for the families .`;X/`2L0 and .`;X;c/`2L0
in Corollary 7.4. Both conditions R.S=k/ and S.S=k/ behave well under base change in thefollowing sense.

Proposition 5.5. Let k 0 =k be an arbitrary field extension and S 0 a regular k 0 -variety.
Assume that there is a diagram
S0


Spec.k 0 /f/S/ Spec.k/where f is dominant. Let K 0 D k 0 .S 0 / and for ` 2 L0 let `0 D ` jGal.K 0 / . If .` /`2L0 satisfies condition R.S=k/ (resp. condition S.S=k/), then the family .`0 /`2L0 satisfies condition
R.S 0 =k 0 / (resp. condition S.S 0 =k 0 /). Moreover, if ` factors via 1 .S / and is `-tame, then `0factors via 1 .S 0 / and is `-tame.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

15

Proof. Assume that the family .` /`2L0 satisfies condition R.S=k/. Then there exists
a dense open subscheme U  S such that each ` factors through 1 .U /. Let U 0 D U S S 0 .
From the commutative diagram
Gal.K 0 // Gal.K// 1 .U /1 .U 0 /we see that each `0 factors through 1 .U 0 /, i.e., that .`0 /`2L0 satisfies condition R.S 0 =k 0 /.
Assume from now on that .` /`2L0 satisfies condition S.S=k/. By Lemma 5.2 (b) we canfind a witness hW V ! U of condition S.S=k/ such that h is finite étale – we may need toshrink the above U . Let V 0 be a connected component of V U U 0 . Then V 0 is a connectedfinite étale cover of U 0 . Let gW V 0 ! V be the canonical map. It is enough to prove that thecomposition ` ı h ı g W 1 .V 0 / ! G` is `-tame for every ` 2 L0 . But this is immediate from
Lemma 4.5 (a), as is the last assertion.
The following lemma describes a situation in which a family .` W Gal.K/ ! G` /`2L0
becomes everywhere unramified after a finite base change. In its application, all G` will be
finite.
Lemma 5.6. Assume that .` /`2L0 satisfies condition S.S=k/. Then there exists a finite
extension k 0 over k, a smooth projective k 0 variety W , and a witness gW W ! U of condition
S.S=k/ such that W is a dense open subscheme of W . In particular, if G` is of order prime
to `, then ` jGal.k.W // factors through 1 .W /.
Proof. Let f W V ! U be a witness of condition S.S=k/ for the family .` /`2L0 . By
de Jong’s theorem [6] there exists a finite extension k 0 =k, a smooth projective k 0 -variety W ,a dense open subscheme W of W and an alteration hW W ! V . Then the first assertion follows
from Lemma 5.2 (a) for g WD f ı h. Fix now ` 2 L0 and assume that ` does not divide theorder of G` . If v denotes the discrete valuation of any codimension 1 point of W , then thegroup ` ı g .Iv / is trivial because it is pro-` and of order prime to ` at the same time. By the
purity of the branch locus it follows that ` ı g factors through 1 .W / as desired.
Combining ramification properties with finiteness properties of fundamental groups, we
obtain the following criterion for a family .` W Gal.K/ ! G` /`2L0 to become trivial overGal.kK 0 / for some finite K 0 =K, provided certain finiteness conditions on ` .Gal.K// hold.
Proposition 5.7. Assume that the family .` W Gal.K/ ! G` /`2L0 satisfies condition
R.S=k/. If p > 0, then assume .` W Gal.K/ ! G` /`2L0 satisfies S.S=k/. Under either ofthe following two conditions there exists a finite Galois extension K 0 of K such that for all
Q 0 // D ¹1º.
` 2 L0 we have ` .Gal.kK(a) The field k is finite or k is a number field, and there exists a constant d 2 N such that foreach ` 2 L0 the group ` .Gal.K// lies in Jor` .d /.
(b) The field k is algebraically closed and there exists a constant c 2 N such that for each` 2 L0 the group ` .Gal.K// is of order at most c.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM16

Böckle, Gajda and Petersen, Independence of `-adic representations

Proof. Because of R.S=k/ there exists a dense open subscheme U of S such thatQ
each ` factors through 1 .U /. Let K` be the fixed field of ker.` / and let E D `2L0 K` .
Q kK
Q is finite.
Then K` =K is unramified along U . We have to prove that kE=
Q kK
Q is finite. In
Assume p D 0. In case (a) Proposition 4.1 yields that the extension kE=
Qcase (b) we have k D k and thus the (geometric) fundamental group 1 .U / is finitely generated(cf. [16, Théorème X.2.9]). Hence, independently of `, there are only finitely many possibilities
for the fields K` , and so E=K is finite in case (b), as well.
Assume from now on that p > 0. Note that in both cases (a) and (b) the order of the finitegroup G` is prime to ` for all but finitely many ` 2 L0 . By Lemma 5.6 there exists a finite
extension k 0 =k and a finite extension F=K and a smooth projective k 0 -variety W with function
field F such that the extension K` F=F is unramified along W for almost all ` 2 L0 . In case (a)Q
Q is finite. Hence kE=
Q kK
Q must be finite. Finally, in case (b)Proposition 4.1 yields that kEF=
kF
the group 1 .W / is finitely generated (cf. [17, II.2.3.1]), and thus E=K must be finite.
The following independence criterion is the main result of this section:
Theorem 5.8. Assume that k is algebraically closed. Assume that the following conditions (a) and (b) are satisfied.
(a) The family .` /`2L0 satisfies R.S=k/, and it satisfies S.S=k/ if p > 0.
(b) There exists a constant c 2 N and a finite Galois extension K 0 =K such that for all ` 2 L0
one has ` .Gal.K 0 // 2 †` .c/.
Then there exists a finite Galois extension E=K containing K 0 such that Gal.E=K 0 / is abelianand such that the following holds true.
(i) For every ` 2 L0 the group ` .Gal.E// lies in †` .c/ and is generated by its `-Sylow
subgroups; if ` > c, then the group ` .Gal.E// is generated by the `-Sylow subgroupsof ` .Gal.K//.
(ii) The restricted family .` jGal.E / /`2L0 X¹2;3º is independent and .` /`2L0 is almost independent.
Proof. Let G` D ` .Gal.K 0 // for all ` 2 L0 . The group G` WD G` =G`C is finite and oforder prime to `. Denote by ` W G` ! G` the natural projection. Let K`0 be the fixed field in Ksof the kernel of the composite morphism``Gal.K 0 / ! G` ! G` :As G` lies in †` .c/, so does its quotient G ` by Lemma 3.3 (b). Now any group in †` .c/ oforder prime to ` is abelian of order at most c, and thus the latter holds for G ` . Thus K`0 =K 0 is
an abelian Galois extension of degree prime to ` and  c. Moreover, as G`C is a characteristic
subgroup of G` , it follows that the finite extension K`0 =K is Galois. Thus the compositumQ
E D `2L0 K`0 is Galois over K, and Gal.E=K 0 / is an abelian group annihilated by cŠ. Let S 0
denote the normalization of S in K 0 and S 00 a dense regular open subscheme of S 0 . Then` jGal.K 0 / satisfies condition R.S 00 =k/ and it satisfies condition S.S 00 =K/ if p > 0 (cf. Proposition 5.5). From Proposition 5.7 (b) we conclude that E=K is finite.
We turn to the proof of (i). For every ` 2 L0 , the group ` .Gal.E// is normal in G` ,and hence it lies in †` .c/ by Lemma 3.3. Let M` D ` .Gal.E//. By construction, M` G G`C ,Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

17

and G` =M` is abelian and killed by cŠ because it is a quotient of Gal.E=K 0 /. Thus G`C =M` isan abelian `-group which is killed by cŠ; if ` > c then this implies G`C D M` . To establish (i)it now suffices to prove that M` D M`C for all ` 2 L0 with `  c. Clearly, M` =M`C is abelian,and hence G`C =M`C is a finite solvable group that is generated by its `-Sylow subgroups. In
addition, the group G`C =M`C lies in †` .c/, and therefore it must be an `-group. Thus M` =M`C
is an `-group as well, and by the definition of M`C , we deduce M` D M`C . Hence part (i)holds true.
We now prove (ii). Denote by „` the class of those finite groups which are either a finite
simple group of Lie type in characteristic ` or isomorphic to Z=`. The conditions in (i) imply
that every simple quotient of ` .Gal.E// lies in „` . But now for any `; `0  5 such that ` ¤ `0
one has „` \ „`0 D ¿ (cf. [35, Theorem 5], [1], [24]). The first part of (ii) now follows from[35, Lemme 2]. The second part follows from the first, the definition of almost independence
and from [35, Lemme 3].
Remark 5.9. We would like to point out that hypothesis (a) in the proof of Theorem 5.8 can be weakened considerably. For this we denote for a continuous homomorphism` W Gal.K/ ! G` by Q` the maximal normal pro-` subgroup of ` .Gal.K//, and by ` thecomposite homomorphism
`

Gal.K/ ! ` .Gal.K// ! ` .Gal.K//=Q` :If ` is an `-adic representation, then ` is simply the semisimplification of the mod ` reductionof ` . The proof of Theorem 5.8 only needs that the family .` /`2L0 satisfies condition R.S=k/or condition S.S=k/, if p > 0, respectively, because this weaker hypothesis suffices for thefiniteness of E=K.
We chose to work with conditions R.S=k/ and S.S=k/ as introduced in Definition 5.1,
since they seem most natural for the motivic families we consider in Theorem 1.2. These conditions are established in Proposition 5.4 and Corollary 7.4. For other purposes, the variantof Definition 5.1 using .` /`2L0 instead might be useful: There are infinitely ramified `-adic
representations of curves over finite fields, that can be constructed following [31]. Families ofsuch will never satisfy R.S=k/. Also, if a family .` /`2L0 satisfies R.S=k/ and if ` is anextension of ` by itself where the extension class is ramified at a divisor depending on `, then
R.S=k/ might fail for .` /`2L0 .
.q/

6. Effective semistability of families .`;X /`2L0 for p > 0
Let k be a perfect field of characteristic p > 0, S a separated algebraic scheme over kwith function field K D k.S /, and let X=K be a smooth projective variety. Let q 2 N. The
main result of this section, Corollary 6.3, gives an effective proof of condition S.S=k/ for.q//`2L0 . We shall describe explicit finite Galois extensions K 0 of K, such thatthe family .`;X.q/jGal.K 0 / is `-tame, cf. also Remark 6.4. Our proof uses
for all ` 2 L0 the representation `;X
a reduction to k D Fp in which case we can apply the Weil conjectures and an `-independence
result on tameness from [7], both due to Deligne. Our method sheds no light on the existenceof a semistable geometric model of X=K over some smooth proper scheme S=k 0 with function
field K 0 . Such an approach is given in [30]. However, in [30] it might be hard to find an effective
description of K 0 .

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

18

Böckle, Gajda and Petersen, Independence of `-adic representations

Throughout this section we let p > 0 be a prime number, and we set L0 D L X ¹pº.
We use the subscript 0 for fields and separated algebraic schemes that are finitely generated
over Fp . In particular, S0 =Fp will be a smooth variety with function field K0 D Fp .S0 /.
For every open subscheme U0 of S0 and every closed point u 2 U0 , let k.u/ be the (finite)
residue field of u, and let D.u/  1 .U0 / be the corresponding decomposition group (defined
only up to conjugation). Denote by Fru 2 D.u/ the preimage under the canonical isomorphismD.u/ ! Gal.k.u// of the arithmetic Frobeniuseeu W k.u/ ! k.u/;

1

x 7! x jk.u/j :Note that within 1 .U0 /, the automorphism Fru is also defined only up to conjugation. The
following proposition is an immediate consequence of the Weil conjectures proved by Deligne.
Proposition 6.1. Let X0 =K0 be a smooth projective variety. There exists a dense open.q/
subscheme U0 of S0 such that for every q 2 N and every ` 2 L0 the representation `;X
factors0.q/
through 1 .U0 / and such that the family of representations .`;X0 /`2L0 is strictly compatible
and pure of weight q, that is: For every closed point u 2 U0 the characteristic polynomial.q/.Fru / has integral coefficients, is independent of ` 2 L0 , and the roots of pu .T /pu .T / of `;X0
all have absolute value jk.u/jq=2 .
Proof. There exists a dense open subscheme U0 of S0 and a projective U0 -schemef W X0 ! U0 such that X0 U0 Spec.K0 / D X0 (cf. [15, 8.8.2] and [15, 8.10.5 (v) and (xiii)]).
By the theorem of generic smoothness, after shrinking U0 and X0 , we can assume that f issmooth.
Let q  0, u 2 U0 . Define k WD k.u/ and Xu WD X0 U0 Spec.k/. Then for every ` 2 L0
the étale sheaf Rq f Z` is lisse and compatible with any base change (cf. [28, VI.2, VI.4]). Thus.q/
`;X
factors through 1 .U0 /, and furthermore it follows that H q .X0;K; Q` / can be identified
0with H q .Xu;kQ ; Q` / in a way compatible with the Galois actions. The assertion now follows by
Deligne’s theorem on the Weil conjectures (cf. [8, Theorem 1.6]).
Lemma 6.2. Let X0 =K0 be a smooth projective variety, and let q be in Z. Supposethat for some prime `0  3 in L0 there is a Z`0 -lattice ƒ of the Q`0 -representation space
underlying `.q/that is stabilized by Gal.K0 / and such that Gal.K0 / acts trivially on ƒ=`0 ƒ.
0 ;X0
.q/
is `-tame.
Then for all ` 2 L0 , the representation `;X
0
Proof. Let C be any smooth curve over Fp and let 'W C ! U0 be any non-constantmorphism. Denote by P .C / a smooth projective model of C , and by ' W 1 .C / ! 1 .U0 / thehomomorphism on fundamental groups induced by '. We claim that the representation(2).q/ı ' /ss.`;X0

is semistable4) at all places of P .C / X C and for all ` 2 L0 . Having shown this for all pairs.C; '/, the assertion of the lemma is deduced as follows: Passing from an `-adic representation
4)

We call a representation of 1 .C / semistable at v if the Frobenius semisimplified Weil–Deligne representation at v is unramified when restricted to the Weil group.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

19.q/
ı ' will be `-tame for all `.
to its semisimplification does not affect `-tameness, and so `;X
0
The present lemma now is immediate from this and Proposition 4.6, which is a variant ofa result of Kerz–Schmidt–Wiesend.
It remains to prove the claim. By Proposition 6.1, the representations (2) are pure ofweight q and semisimple for all ` 2 L0 , and they form a strictly compatible family of representation of 1 .C /. By [7, Théorème 9.8], it follows that in fact for any v 2 P .C / the representation rv;` underlying the Weil–Deligne representation .rv;` ; Nv;` / of the restriction of (2)to a decomposition group at v is independent of `. Thus it will suffice to show that rv;`0 isunramified for all v 2 P .C /.
We consider the representations (2) for ` D `0 as an action of 1 .C / on the lattice ƒ thatis trivial modulo `0 ƒ. Any filtration of ƒ ˝Z`0 Q`0 that is preserved by the action of 1 .C /induces a filtration of ƒ. Denote by ƒC the induced lattice for (2). Then it follows that theinduced action of 1 .C / on ƒC =`0 ƒC is trivial. Let n D rank ƒ. In the following we frequently identify AutZ`0 .ƒ/ with GLn .Z`0 /. We define

GL1n .Z`0 / WD ker.Aut.ƒC / ! Aut.ƒC =`0 ƒC //:Since `0 > 2, we have pairwise inverse `0 -adic exponential and logarithm maps
exp`0 W `0 Mn .Z`0 / ! GL1n .Z`0 / andlog`0 W GL1n .Z`0 / ! `0 Mn .Z`0 /;given by the usual power series expressions
X 7!

X 1
Xm
mŠ

m0

and .1 C A/ 7!

X . 1/mC1
Am
m

m1

(cf. [4, III.7.2], [4, III.7.6]). It follows that log`0 converges on the image of (2). This image
being in GL1n .Z`0 /, following the standard argument (cf. [36, Corollary 4.2.2]) shows that rv;`0
is unramified: By the continuity of rv;`0 (with the target carrying the discrete topology), anyelement A in the image of inertia at v has finite order. Thus A 2 GL1n .Z`0 / satisfies Am D 1
for some m 2 N. One deduces 0 D log`0 Am D m log`0 A which gives log`0 A D 0 and thus
A D exp`0 .log`0 A/ D 1n .
For a smooth projective variety X=k we have the following effective `-tameness result.q//`2L0 .
for the family .`;X
Corollary 6.3. Let k be perfect of characteristic p > 0, let S=k be a smooth variety
with function field K, and let X=K be a smooth projective variety. Choose a prime `0  3
and a Gal.K/-invariant lattice ƒ of `.q/, and denote by K 0 the fixed field of the kernel
0 ;X.q/of `0 ;X .mod `0 ƒ/. Then there exists a regular k-scheme U 0 with function field K 0 such that.q/
for all ` 2 L0 the family .`;X
jGal.K 0 / /`2L0 factors via 1 .U 0 / and is `-tame.
.q/
In particular, .`;X /`2L0 satisfies condition S.S=k/
Proof. Suppose first that k D Fp is finite, so that X=K D X0 =K0 in the notation above.
.q/
Let U0  S be as in Proposition 6.1 for .`;X/`2L0 , let U00 be its normalization in K 0 . By generic0étaleness, we may shrink U0 such that U0 ! U0 is étale, and hence U00 =k is smooth. Using
Lemma 6.2, the second assertion follows for U 0 D U00 . The first is clear from the construction.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

20

Böckle, Gajda and Petersen, Independence of `-adic representations

Let now k be any perfect field of characteristic p, and let ƒ and K 0 be as in the corollary.
By standard arguments from algebraic geometry, there exist an absolutely finitely generated
subfield K0 of K and a smooth projective scheme X0 over K0 such thatX D X0 Spec K0 Spec K:.q/.q/
Then .`;X/`2L0 D .`;Xj/ 0 for the canonical map Gal.K/ ! Gal.K0 /. Since ƒ is
0 Gal.K/ `2L
stabilized by Gal.K/, we may assume by passing from K0 to a finite extension contained in K
that ƒ is stabilized by Gal.K0 /. If we furthermore define K00  K0 as the fixed field of thekernel of `.q/.mod `0 ƒ/, then K00  K 0 .
0 ;X0
Let S0 be a smooth Fp -variety with function field K0 . Let U0  S0 and U00 be as in
the first paragraph of the proof, and assume without loss of generality that both are affine
U0 D Spec A0 and U00 D Spec A00 . Then we have that kA0  K and kA00  K 0 are finitely
generated k-algebras. Because k is perfect, they are generically smooth over k and kA0  kA00
is generically étale. We can choose U  Spec.kA0 / dense open such that U  S and suchthat the normalization U 0 of U in K 0 is étale over U ; note that U 0 is an open subschemeof Spec.kA00 /. The corollary now follows from Proposition 5.5 applied to U 0 ! U00 overSpec k ! Spec Fp .

Remark 6.4. We would like to point out the parallel between Corollary 6.3 andExample 5.3. In both cases we select a prime number `0  3. In the effective part of Corollary 6.3 we enlarged K to K 0 so that Gal.K 0 / acts trivially on ƒ=`0 ƒ via the representation
for a Gal.K/-stable lattice ƒ. Then we could use the uniformity provided by automorphic
`.q/
0 ;X
representations (after restricting ourselves to the case where k is finite by standard arguments,.q/and after restricting the `;X
to any curve) from Lemma 6.2, to deduce that for all `, all ramification of `;X was `-tame over K 0 . The semistability of an automorphic representation, andhence the `-tameness of the associated compatible system of `-adic Galois representations,spreads from a single `0 to all `.
In Example 5.3 we set K 0 D K.AŒ`0 /, which again is the minimal choice so that Gal.K 0 /acts trivially on the quotient T`0 .A/=`0 T`0 .A/ for the lattice T`0 .A/ from the Tate-module.
Then we use the semistability of the Néron model N of the abelian variety A over K 0 overany discrete valuation ring, that is implied by a condition at a single prime `0 : We deduce the`-tameness of T` .A/ over K 0 , uniformly for each prime `. In both cases, the field K 0 is definedin the same way. For general X as in Corollary 6.3, it seems unlikely that X always acquires
some geometric semistability over K 0 , as it is the case for X an abelian variety.

7. Reduction steps and the proof of Theorem 1.2
Throughout this section let k be a field of characteristic p  0, let L0 D L X ¹pº andlet S=k be a smooth variety with function field K. By X=K we denote a separated algebraic
Lqscheme, and by `;X the representation of Gal.K/ on q0 .Hc .XK; Q` / ˚ H q .XK; Q` //.
In this section, we give the necessary reduction steps to deduce hypotheses (a) and (b) of
Theorem 5.8 for the family .`;X /`2L0 . Thereby we shall complete the proof of Theorem 1.2.
Recall that hypothesis (a) is condition S.S=k/, and that hypothesis (b) is the uniform containment `;X .Gal.K 0 // 2 †` .c/ for all ` for some finite extension K 0  K.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

21

As a preparation to establish S.S=k/ we introduce the following notion.
Definition 7.1. For a given representation ` W Gal.K/ ! GLn .Q` / we define its strictsemisimplification `sss as the direct sum over the irreducible subquotients of ` where eachisomorphism type occurs with multiplicity one.
Note that `sss .Gal.K// D `ss .Gal.K// where `ss denotes the usual semisimplificationof ` .
Lemma 7.2. For every ` 2 L0 let ` and `0 be representations Gal.K/ ! GLn .Q` /.
Suppose that the families .` /`2L0 and .`0 /`2L0 both satisfy condition R.S=k/. Suppose that
one of the following two assertions is true:(a) `sss D .`0 /sss for all ` 2 L0 ,(b) ` is a direct summand of `0 for all ` 2 L0 .
Then the following hold: If the family .`0 /`2L0 satisfies S.S=k/, then so does .` /`2L0 .
Proof. The proof under hypothesis (a) is an immediate consequence of the simple factthat the kernel of ` .Gal.K// ! `ss .Gal.K// D `sss .Gal.K// is a pro-`-group.
Under hypothesis (b) the proof is trivial.
The following important result is taken from the Seminaire Bourbaki talk of Berthelot on
de Jong’s alteration technique (cf. [3, Theorem 6.3.2])
Theorem 7.3. Let X be a separated algebraic scheme over K. Then there exists a finite
extension k 0 =k, a finite separable extension K 0 =Kk 0 and a finite set of smooth projective varieties ¹Yi ºi D1;:::;r over K 0 such that for all ` 2 L0 the representation .`;X jGal.K 0 / /sss is a directLsummand of . i `;Yi /sss .
The following result is an immediate consequence of Theorem 7.3, Lemma 7.2 andCorollary 6.3.
Corollary 7.4. Let k be perfect of characteristic p > 0; recall that K D k.S /. Then forevery separated algebraic K-scheme X the family .`;X /`2L0 satisfies condition S.S=k/.5)
Proof of Theorem 7.3. For completeness we provide details of the proof in [3]. ForLq‹ 2 ¹c; ¿º we denote by `;X;‹ the representation of Gal.K/ on q0 .H‹ .XK; Q` //. It suffices to prove the theorem separately for the families .`;X;‹ /`2L0 . We also note that whenever
it is convenient, we are allowed (by passing from K to a finite extension) to assume that Xqq red ; Q` / for any
is geometrically reduced over K. This is so because H‹ .XK; Q` / Š H‹ .XK;
q 2 Z and ‹ 2 ¹¿; cº. We first consider the case of cohomology with compact supports. The
proof proceeds by induction on dim X .

Not assuming that k is perfect, one can show that after a finite field extension k 0 of k and a correspondingbase change S 0 of S, condition S.S 0 =k 0 / holds.
5)

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

22

Böckle, Gajda and Petersen, Independence of `-adic representations

The scheme XK is generically smooth. After passing from K to a finite extension, we
can find a dense open subscheme U  X that is smooth over K. By the long exact cohomology
sequence with supports (cf. [28, Remark III.1.30]) we have for any ` an exact sequence   ! Hci .UK; Q` / ! Hci .XK; Q` / ! Hci ..X X U /K; Q` / !    ;so that for all ` the representation .`;X;c /sss is a direct summand of .`;U;c /sss ˚ .`;X XU;c /sss .
By induction hypothesis, it thus suffices to treat the case that U is smooth over K. By induction
hypothesis, it is also sufficient to replace U by any smaller dense open subscheme, and it isclearly also sufficient to treat the case where U is in addition geometrically irreducible.
By de Jong’s theorem on alterations (cf. [6, Theorems 4.1, 4.2]), after passing from K toa finite extension, we can find a smooth projective scheme Y; an open subscheme U 0 of Y andan alteration W U 0 ! U . By replacing K yet another time by a finite extension, we can assume
that U 0 ! U is generically finite étale. And now we pass to an open subscheme V of U andto V 0 WD  1 .V /  U 0 such that V 0 ! V is finite étale. By the induction hypothesis applied
to Y X V 0 and again the long exact cohomology sequence for cohomology with supports, wefind that the assertion of the theorem holds true for the family .`;V 0 ;c /sss
. From now on 
`2L0
denotes the restriction to V 0 and Q`;X will be the constant sheaf Q` on any scheme X . Since is finite étale, say of degree d , there exists a trace morphism Trace W  Q`;V 0 ! Q`;V
whose composition with the canonical morphism Q`;V !  Q`;V 0 is multiplication by d(cf. [28, Lemma V.1.12]). In particular, we have that the constant sheaf Q`;V is a direct sum0 ; Q / Š H i .V  ;  Q /, we deduce that .
sss is a direct
mand of  Q`;V 0 . Since Hci .VK ```;V;c /c K
summand of .`;V 0 ;c /sss , and this completes the induction step.
Now we turn to the case ‹ D ¿. The case when X is smooth over K but not necessarily
projective is reduced, by Poincaré duality, to the case of compact supports. If X is connected,
2d q
one has H q .XK; Q` / Š Hc
.XK; Q` .d //_ for d D dim X (cf. [28, Corollary VI.11.12]),and one can reduce to the connected case by considering the connected components of X
separately.
Suppose now that X is an arbitrary separated algebraic scheme over K. By what we said
above, we may assume that X is geometrically reduced. Again we perform an induction on
dim X. The first step is a reduction to the case where X is irreducible, which may be thought ofas an induction by itself. Suppose X D X1 [ X2 where X1 is an irreducible component of X
and X2 is the closure of X X X1 . Consider the canonical morphism f W X1 t X2 ! X. It yields
a short exact sequence of sheaves(3)

0 ! Q`;X ! f Q`;X1 tX2 ! F ! 1

where F is a sheaf on X . Consider the inclusion i W X0 ,! X for X0 WD X1 \ X2 . We claimthat F Š i Q`;X0 . To see this observe first that if we compute the pullback of the sequence
along the open immersion j W X X X0 ,! X , then F vanishes and the morphism on the left
becomes an isomorphism. In particular, F is supported on X0 . To compute the pullback along
the closed immersion i , we may apply proper base change, since f is proper. But now the
restriction of f to X0 is simply the trivial double cover X0 tX0  X0 , so that i  F Š Q`;X0 .
This proves the claim because F Š i i  F , as F is supported on X0 . By an inductive application of the long exact cohomology sequences to sequences like (3), it suffices to prove thetheorem for schemes X that are geometrically integral and separated algebraic over K. In thiscase, the proof follows by resolving X by a smooth hypercovering X see [6, p. 51], [9, 6.2.5]

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

23

and the proof of [3, Theorem 6.3.2]. Since the hypercovering yields a spectral sequence thatcomputes the cohomology of X in terms of the cohomologies of the smooth Xi , for all `, andsince only those Xi with i  2 dim X , contribute to X, the induction step is complete because
we have reduced the case of arbitrary X to lower dimensions and to smooth Xi .
We now consider hypothesis (b) of Theorem 5.8. One has the following simple reduction
result.
Lemma 7.5. Let .` W Gal.K1 / ! G` /`2L0 be a family of Galois representations, andconsider the diagram of fields

/ K2
K1 
O

O

?

k1 



?
/ k2

2 of K2 , where ki is algebraically closed and Ki is finitely geninside an algebraic closure K
erated over ki for i D 1; 2. Then (1 ) implies (2 ) for the following assertion depending on i :(i ) There exists a constant c 2 N and a finite Galois extension Ki0 =Ki such that for all ` 2 L0
one has ` .Gal.Ki0 // 2 †` .c/.
Proof. Consider the following homomorphisms of Galois groups, given by restriction:
r3

r2

r1

1 /K10 / ! Gal.K10 /:Gal.K2 K10 / ! Gal.k2 K10 / ! Gal..k2 \ K
One easily verifies that r1 has closed normal image, r2 is surjective, and r3 has open image. Let
E=k2 K10 be a finite Galois extension such that r3 .Gal.K2 K10 // contains Gal.E/, and define K200
2 of r 1 .Gal.E//, i.e., as EK2 . Applying Corollary 3.4 to r1 , r2 and r3
as the fixed field in K
3shows that the conclusion of (2 ) holds with K200 in place of K20 . The lemma follows by anotherapplication of Corollary 3.4 for K20 the Galois closure of K200 over K2 .
Corollary 7.6. For k be algebraically closed, hypothesis (b) of Theorem 5.8 holds forthe family .`;X /`2L0 .
Proof. Let k0 be the prime field of K, let K0  K be an absolutely finitely generated
subfield field, and let X0 =K0 be a separated algebraic scheme with X D X0 Spec K0 Spec K.
Then we have .`;X /`2L0 D .`;X0 jGal.K/ /`2L0 . By [22, Section 3] there exists n 2 N suchthat dim.`;X0 /  n for all ` 2 L0 . For each ` we apply Theorem 3.6 to `;X0 , to obtain a short
exact sequence
1 ! M` ! `;X0 .Gal.K0 // ! H` ! 1
with H` 2 Jor` .J 0 .n// and M` 2 †` .2n 1 /. Because of Corollary 7.4 and Proposition 5.4,
there exists a smooth variety S0 over k0 with function field K0 D k0 .S0 / such that .`;X0 /`2L0
satisfies S.S0 =k0 / if p > 0, or satisfies R.S0 =k0 / if p D 0. Then Proposition 5.7 (a) yields
a finite Galois extension K00 =K0 such that `;X0 .Gal.kQ0 K00 // is a closed normal subgroupof M` for all ` 2 L0 . Corollary 3.4 together with Lemma 7.5 now provide a finite Galois extension K 0 of K such that for all ` 2 L0 one has ` .Gal.K 0 // 2 †` .2n 1 /, and so hypothesis (b)of Theorem 5.8 is satisfied.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

24

Böckle, Gajda and Petersen, Independence of `-adic representations

Using Proposition 5.4, Corollary 7.4 and Corollary 7.6, Theorem 5.8 yields our main
theorem:
Theorem 7.7. Let k be a field of characteristic p  0 and let K=k be a finitely generated extension. Let X=K be a separated algebraic scheme. Then there exists a finite extension
E=K and a constant c 2 N with the following properties:
Q(i) For every ` 2 L0 the group `;X .Gal.kE//
lies in †` .c/ and is generated by its `-Sylow
subgroups.
(ii) The family .`;X jGal.kE
Q / /`2L0 X¹2;3º is independent and the family .`;X jGal.kK/
Q /`2L0 is
almost independent.

References[1] E. Artin, The orders of the classical simple groups, Comm. Pure and Applied Math. 1 (1955), 455–472.
[2] M. Artin, A. Grothendieck and J.-L. Verdier, Séminaire de géométrie algébrique du Bois-Marie 1963–1964.
Théorie des topos et cohomologie étale des schémas (SGA 4), Lecture Notes in Math. 269, 270, 305, Springer,
Berlin 1972–1973.
[3] P. Berthelot, Altérations de variétés algébriques, in: Séminaire Bourbaki 1995/96. Exposés 805–819, Astérisque 241, Société Mathématique de France, Paris (1997), Exp. No. 815, 273–311.
[4] N. Bourbaki, Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 2 et 3 Springer, Berlin 1972.
[5] A. Cadoret and A. Tamagawa, On subgroups of GLr .F` / and representations of étale fundamental groups,
preprint 2013, http://www.math.polytechnique.fr/perso/cadoret.anna/Travaux.html.
[6] J. A. de Jong, Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci. 1 (1996), 51–93.
[7] P. Deligne, Les constantes des équations fonctionnelles des fonctions L, in: Modular functions of one variable. II (Antwerp 1972), Lecture Notes in Math. 349, Springer, Berlin (1973), 501–597.
[8] P. Deligne, La conjecture de Weil I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.
[9] P. Deligne, Théorie de Hodge III, Publ. Math. Inst. Hautes Études Sci. 44 (1975), 5–77.
[10] M. Demazure and A. Grothendieck, Schémas en groupes. II: Groupes de type multiplicatif, et structure desschémas en groupes généraux. Exposés VIII à XVIII. Séminaire de Géométrie Algébrique du Bois Marie
1962/64 (SGA 3), Lecture Notes in Math. 152, Springer, Berlin 1970.
[11] A. Devic and R. Pink, Adelic openness for Drinfeld modules in special characteristic, J. Number Theory 132(2012), 1583–1625.
[12] M. Dickinson, On the modularity of certain 2-adic Galois representations, Duke Math. J. 1 (2001), 319–383.
[13] W. Gajda aand S. Petersen, Independence of `-adic Galois representations over function fields, Compos. Math.
149 (2013), no. 7, 1091–1107.
[14] A. Grothendieck, Éléments de géométrie algébrique. I: Le langage des schémas, Publ. Math. Inst. Hautes
Études Sci. 4 (1960), 1–228.
[15] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes deschémas (Troisième partie), Publ. Math. Inst. Hautes Études Sci. 28 (1966), 1–255.
[16] A. Grothendieck, Séminaire de géométrie algébrique du Bois Marie 1960/61. Revêtements étales et groupe
fondamental (SGA 1), Lecture Notes in Math. 224, Springer, Berlin 1971.
[17] A. Grothendieck, Seminaire de géométrie algébrique Du Bois-Marie 1967–1969. Groupes de monodromie en
géométrie algébrique (SGA 7 I), Lecture Notes in Math. 288, Springer, Berlin 1972.
[18] C. Y. Hui and M. Larsen, Adelic openness without Mumford–Tate conjecture, preprint 2014, http://arxiv.
org/abs/1312.3812.
[19] J.-I. Igusa, Fibre systems of Jacobian varieties. III: Fibre systems of elliptic curves, Amer. J. Math. 81 (1959),
453–476.
[20] L. Illusie, Constructibilité générique et uniformité en `, preprint 2010, www.math.u-psud.fr/~illusie.
[21] N. M. Katz and S. Lang, Finiteness theorems in geometric classfield theory, Enseign. Math. (2) 27 (1981),
no. 3–4, 285–319.
[22] N. M. Katz and G. Laumon, Transformation de Fourier et majoration de sommes exponentielles, Publ. Math.
Inst. Hautes Études Sci. 62 (1986), 361–418; erratum, Publ. Math. Inst. Hautes Études Sci. 69 (1989), 244.

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM

Böckle, Gajda and Petersen, Independence of `-adic representations

25[23] M. Kerz and A. Schmidt, On different notions of tameness in arithmetic geometry, Math. Ann. 346 (2010),
no. 3, 641–668.
[24] W. Kimmerle, R. Lyons, E. Sandling and D. Teague, Composition factors from the group ring and Artin’s
theorem on orders of simple groups, Proc. Lond. Math. Soc. (3) 60 (1990), no. 1, 89–122.
[25] M. Larsen and R. Pink, Finite subgroups of algebraic groups, J. Amer. Math. Soc. 24 (2011), no. 4, 1105–1158.
[26] W. Lütkebohmert, On compactification of schemes, Manuscripta Math. 80 (1993), 95–111.
[27] G. Malle and D. Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Stud. Adv.
Math. 133, Cambridge Mathematical Society, Cambridge 2011.
[28] J. Milne, Étale cohomology, Princeton University Press, Princeton 1980.
[29] M. Nori, On subgroups of GLn .Fp /, Invent. Math. 88 (1987), 257–275.
[30] F. Orgogozo, Sur les propriétiés d’uniformité des images directes en cohomologie étale, preprint 2012,
http://fabrice.orgogozo.perso.math.cnrs.fr/articles/uniformite.pdf.
[31] R. Ramakrishna, Infinitely ramified representations, Ann. of Math. (2) 151 (2000), 793–815.
[32] J.-P. Serre, Represéntations `-adiques, in: Œuvres. Collected papers. Vol. III: 1972-–1984, Springer, Berlin(1986), 384–400.
[33] J.-P. Serre, Propriétés conjecturales des groupes de Galois motiviques et des représentations `-adiques, in:
Motives. Proceedings of the summer research conference on motives (Seattle 1991), Proc. Sympos. Pure
Math. 55, American Mathematical Society, Providence (1994), 377–400.
[34] J.-P. Serre, Abelian l-adic representations and elliptic curves, Res. Notes Math. 7, A K Peters, Wellesley 1998.
[35] J.-P. Serre, Une critère d’indépendance pour une famille de représentations `-adiques, Comment. Math. Helv.
88 (2013), 543–576.
[36] J. Tate, Number theoretic background, in: Automorphic forms, representations and L-functions (Corvallis
1977), Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence (1979), 3–26.
[37] M. Vaquié, Valuations, in: Resolution of singularities (Obergurgl 1997), Progr. Math. 181, Birkhäuser, Basel(2000), 539–590.

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, Polande-mail: gajda@amu.edu.pl
Sebastian Petersen, Fachbereich 10, Universität Kassel,
Wilhelmshöher Allee 73, 34121 Kassel, Germany
e-mail: petersen@mathematik.uni-kassel.de
Eingegangen 15. Oktober 2013, in revidierter Fassung 16. Februar 2015

Authenticated | gajda@amu.edu.pl author's copy
Download Date | 7/24/15 7:12 AM