Openness of the Galois Image for T-Modules of Dimension 1

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Journal of Number Theory 102 (2003) 306–338 http://www.elsevier.com/locate/jnt

Openness of the Galois image for t-modules of dimension 1

Francis Gardeyn1 Aprikosenstrasse 22, 8051 Zurich, Switzerland

Received 23 September 2002; revised 23 February 2003 Communicated by D. Goss

Abstract

Let C be a smooth projective absolutely irreducible curve over a finite field Fq; F its function field and A the subring of F of functions which are regular outside a fixed point N of C: For b every place c of A; we denote the completion of A at c by Ac: In [Pi2], Pink proved the Mumford–Tate conjecture for Drinfeld modules. Let f be a Drinfeld module of rank r defined over a finitely generated field K containing F: For every place c of A; we denote by Gc the image of the representation b rc : GK -Autb ðTcðfÞÞDGLrðAcÞ Ac

of the absolute Galois group GK of K on the Tate module TcðfÞ: The Mumford–Tate conjecture states that some subgroup of finite index of Gc is open inside a prescribed algebraic subgroup Hc of GL b : In fact, he proves this result for representations of GK on a finite r;Ac product of distinct Tate modules. A t-module over AK is a projective A#K-module of finite type endowed with a 1#j-semilinear injective homomorphism t; where j denotes the Frobenius morphism on K: Such a t-module is said to have dimension 1, if the K-vector space M=K Á tðMÞ has dimension 1. Drinfeld showed how to associate, in a functorial way, to every Drinfeld module over K a t- module MðfÞ over AK of dimension 1, called the t-motive of f: In this paper, we generalize Pink’s theorem to representations of Tate modules TcðMÞ of t-modules M of dimension 1 over AK : The key result can be formulated as follows: if we suppose that EndK ðMÞ¼A; then

E-mail address: [email protected]. 1 This work was part of an ETH-project supported by the Swiss Science Foundation. At the same time, the author was a Research Assistant of the Fund for Scientiﬁc Research Flanders.

0022-314X/03/$ - see front matter r 2003 Published by Elsevier Inc. doi:10.1016/S0022-314X(03)00106-9 ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 307 for every ﬁnite place c of F; the image Gc of the representation

rc : GK -Autb ðTcðMÞÞ Ac b is open in GLrðAcÞ; where r denotes the rank of M: As already demonstrated in the proof of the Tate conjecture for Drinfeld modules by Taguchi and Tamagawa, the relation between t-modules over AK and Galois representations b with coefficients in Ac is more natural and direct than that between Drinfeld modules (or, more generally, abelian t-modules) and their Tate modules. By this philosophy, the assumption that a t-module M is pure, or, equivalently, is the t-motive of a Drinfeld module f; should be and, indeed, is superfluous in proving a qualitative statement like the above Mumford–Tate conjecture. The main result of this paper is the corresponding statement for t- modules of dimension 1, i.e. whose maximal exterior power is the t-motive of a Drinfeld module. We stick to the basic outline of Pink’s proof: reducing ourselves to the case where the absolute endomorphism ring of M equals A; we first show that Gc is Zariski dense in GL b r;Ac and we use his results on compact Zariski dense subgroups of algebraic groups to conclude b that Gc if open in GLrðAcÞ: After a reduction to the case where K has transcendence degree 1 over Fq; the essential tools we will use are the Tate and semisimplicity theorem for simple t- modules, Serre’s Frobenius tori and the tori given by inertia at places of good reduction for M: r 2003 Published by Elsevier Inc.

MSC: 11G09

Keywords: t-Sheaves; t-Motives; Galois representation; Serre conjecture

1. Introduction

m For a finite field Fq of q ¼ p elements, we consider a smooth projective absolutely irreducible curve C with field of constants Fq; and denote its function field by F: Fixing a closed point N; we put C :¼ C \fNg and let

0 A :¼ H ðC; OCÞ be the ring of regular functions of C outside N: Let K be a ﬁeld containing F and denote the embedding F+K by i: On K; we have the Frobenius homomorphism

jAEndðKÞ : x/xq

(we denote the induced endomorphism in EndðSpec KÞ by j as well). Consider the Fq-scheme CK :¼ C#K; and its ring of global regular functions # # s AK :¼ A Fq K; both endowed with the endomorphism s :¼ id j: Let AK be the ARTICLE IN PRESS

308 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 ring AK ; viewed as an AK -algebra via s; and, for any AK -module M; put Ã s # s M :¼ AK AK M:

A t-module ðM; tÞ ( for short: M) over AK of rank r is a projective AK -module M of rank r endowed with an injective AK -linear homomorphism

t : sÃM-M:

Equivalently, M can be viewed as a t-sheaf, i.e. a locally free OCK -module of ﬁnite rank r; endowed with an injective morphism t : sÃM-M: A morphism of t-modules is an AK -linear morphism respecting the action of t: We denote the endomorphism ring of M over K by EndK ðMÞ: An isogeny between t-modules is an injective morphism of t-modules whose cokernel is a torsion AK -module. If all nontrivial sub-t-modules of a given t-module M are isogenous to M; then M is called simple. The tensor product M1#M2 of two t-modules over AK has M1#RM2 as the underlying R-module and a t-action deﬁned by

tðm1#m2Þ¼tm1#tm2 for miAMi: A t-module M is said to have (generic) characteristic i if the cokernel of t is supported on the point GðiÞ of CK deﬁned as the graph of i; and it has dimension 1if furthermore the restriction of coker t on GðiÞ has dimension 1. A t-module M on CK is called smooth (or, equivalently, of dimension 0) if t is an isomorphism.

1 Example 1.1. Put C ¼ A :¼ Spec Fq½t: A t-module M over AK DK½t can be viewed as a free K½t-module of ﬁnite rank endowed with a s-semi-linear injective morphism. Fixing a basis for M; we express t with respect to this basis by means of a matrix DM in MatrÂrðK½tÞ: Putting m :¼ðm1; y; mrÞ; this DM is determined by

tðmÞ¼m Á DM :

Let the matrix sU be obtained by applying s to the entries of U: If we replace m by 0 another basis m ¼ m Á U; with UAGLrðK½tÞ; then t is represented by

À1 s U Á DM Á U

0 with respect to the basis m : The determinant of a matrix DM representing t is independent of the choice of a basis m up to a unit in K: The t-module M is smooth if DM is invertible. Put y :¼ iðtÞ: The t-module M has characteristic i if and only if the annihilator of the K½t-module coker DM is contained in the ideal ðt À yÞCK½t; i.e. the determinant of DM equals

h Áðt À yÞd ; ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 309 where h is a unit in K: Its dimension is 1 if and only if

coker DM DK½t=ðt À yÞ; i.e. d ¼ 1: Let us recall how Drinfeld modules and t-motives are related to each other. We consider a Drinfeld A-module

- f : A EndFq ðGa;K Þ whose characteristic homomorphism is the homomorphism i: The K-vector space

MðfÞ :¼ EndFq ðGa;K Þ is endowed with the structure of an AK -module via

ða#cÞÁm :¼ cm3a; for all mAMðfÞ; cAK and aAA: Drinfeld proved that MðfÞ is a projective A#K- module, whose rank equals the rank of f as a Drinfeld module. The endomorphism tAEndðGa;K Þ acting on MðfÞ via t Á m :¼ t3m commutes with A and acts as Frobenius on K: We can now view MðfÞ as a projective AK -module, called the t-motive associated to f; and the injective morphism t : sÃMðfÞ-MðfÞ endows MðfÞ with the structure of a t-module over CK with characteristic i: As coker tDLieðGaÞ; the t-motive MðfÞ has dimension 1. The category of Drinfeld modules and that of the associated t-motives are antiequivalent.

1 Example 1.2 (Drinfeld modules). Put C ¼ A :¼ Spec Fq½t: Let xAMðfÞ denote the iÀ1 identity morphism id : Ga;K -Ga;K : The elements mi :¼ t 3x for i ¼ 1; y; r yield a basis for the K½t-module MðfÞ (or, equivalently, for the coherent free sheaf MðfÞ 1 on AK ). If the Drinfeld module f is given by

Xr i f : t/ ait AEndðGa;K Þ i¼0

Â (where arAK and a0 ¼ y), then the action of t with respect to this basis is given by the matrix representation: 0 1 0 y 0 tÀy B ar C B 10Àa1 C B ar C t Áðm1; y; mrÞ¼ðm1; y; mrÞÁB C: ð1Þ @ & ^ A 01ÀarÀ1 ar ARTICLE IN PRESS

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1 Remark 1.3 (Purity). Put C ¼ A :¼ Spec Fq½t: Consider the Dieudonne´ module À1 # DNðMÞ :¼ Kððt ÞÞ K½tM of M: We recall that, following Anderson’s deﬁnition (cf. [An1, Section 1.9]), a t- motive M over K½t is pure of weight w; if there exists a non-zero positive integer z À1 2 and a K½½t -lattice MN in DNðMÞ such that

À1 z zw K½½t Á t ðMNÞ¼t MN: ð2Þ

If a pure t-motive has rank r; dimension d and weight w; then d w ¼ : r As Anderson shows, the t-motive of every Drinfeld module is pure of weight 1=r: Conversely, if a t-module M over K½t with characteristic i is pure (cf. [An1, Section 9]), then, upon replacing K by a ﬁnite inseparable extension, it is isomorphic to a t-motive and if it has dimension 1, then the associated abelian t-module is a Drinfeld module. This in part explains for the nomenclature of ‘dimension 1’. Another reason is that, as we will show in Section 4, if a t-module has dimension 1, then the connected part of the formal completion at a ﬁnite place c of F above which b M has good reduction corresponds to a one-dimensional formal Ac-module.

Remark 1.4 (Maximal exterior power). A t-module M over K½t with characteristic i has dimension 1 if and only if its maximal exterior power 4rM does. Every t-module over K½t of rank 1 and dimension 1 is pure, and therefore isomorphic to the t-motive 1 of a Drinfeld module of rank 1. Thus, if C ¼ A :¼ Spec Fq½t; we can reformulate the condition ‘M has dimension 1’ as follows: the maximal exterior power of M is the t-motive of a Drinfeld module.

1 Example 1.5 (t-Modules of dimension 1). Putting C ¼ A :¼ Spec Fq½t; we now give pure and non-pure examples of t-modules of dimension 1. Taking

K ¼ F ¼ FqðtÞ and putting y :¼ iðtÞAK; we deﬁne, for every f AFq½t; a rank 2 t-module Mf over K½t as follows: with respect to its K½t-basis ðm1; m2Þ; we put 0 t À y t Áðm1; m2Þ :¼ðm1; m2ÞÁ : À1 Àf

As the determinant of this representation matrix equals ðt À yÞ; the t-modules Mf have characteristic i and dimension 1 (cf. Example 1.1). One easily sees that the

2 z Note that, unlike Anderson, we do not assume K to be perfect, which implies that t ðMNÞ is not necessarily a K½½tÀ1-module. If K is pure, then the above deﬁnition coincides with Anderson’s. ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 311 maximal exterior power of Mf ; for any f ; is the t-motive associated to the Carlitz / module fC : t t þ y (cf. Remark 1.4). By reducing Mf modulo y; we obtain a t-module Mf over Fq½t (i.e. Mf has good reduction at y; cf. Section 2). Note that the action of s on Fq is trivial, so that t is a linear endomorphism of Mf ; its characteristic polynomial is

PðZÞ :¼ Z2 þ f Á Z þ t:

Let x be the place of K deﬁned by the ideal ðyÞ of Fq½y: As we will explain in Section 3, this characteristic polynomial coincides with the characteristic polynomial of the Frobenius substitution Frobx at the place x operating on the Galois modules VcðMf Þ associated with Mf ( for all ﬁnite places cax). If Mf is pure, it is the t-motive of a Drinfeld module ff over K: it is then well known (cf. [Go2, Theorem 3.2.3]) that the eigenvalues liAF of the endomorphism D Frobx of the Galois modules VcðMf Þ Vcðff Þ are pure, i.e. 1 deg l ¼ : i 2

From the Newton polygon associated to Pf ðZÞ; we see that this then implies that deg f ¼ 0: In conclusion, Mf is pure (and therefore is the t-motive of a Drinfeld module) if and only if f AFq:

For any closed point c of C; i.e. for any finite place of F; let kc denote the residue b b field of C at c; Ac the completion of A at c; and Fc its fraction field. For every finite set L of finite places of F; we also consider the ring Y b b FL :¼ Fc: cAL

We denote an algebraic, resp. separable closure of K by K; resp. KsepCK; and the sep absolute Galois group GalðK =KÞ by GK : We deﬁne the ring b b #c Ac;K :¼ Ac Fq K

b # as the completion of Ac Fq K with respect to its maximal ideal. We continuously b extend sAEndðAK Þ to an endomorphism of Ac;K : For a t-module M over AK ; we put b b # Mc :¼ Ac;K AK M: b Extending the action of t to Mc; the latter should be seen as a formal t-module over b Ac;K : We associate to it the module b b TcðMÞ¼ff AHomb ðMc; Ac;Ksep Þ; f 3t ¼ s3f g: ð3Þ Ac;K ARTICLE IN PRESS

312 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338

b This TcðMÞ; called the Tate module of M at c; is a free Ac-module of rank r; the rank of M; endowed with a continuous action of GK :

Remark 1.6. By Drinfeld (cf. [TW, Proposition 6.3]), the map b Mc/TcðMÞ b induces an antiequivalence between the categories of smooth t-modules Mc over b b b Ac;K and Ac½GK -modules Tc with free underlying Ac-module of ﬁnite type.

Remark 1.7. Let f be a Drinfeld A-module defined over K: For every finite place c of F; the Tate module TcðfÞ (defined as the projective limit of c-primary torsion b points) is isomorphic to TcðMðfÞÞ as an Ac½GK -module.

Example 1.8. We take up Example 1.1. Let c the ﬁnite place of F corresponding to b the ideal ðtÞ: The ring Ac is isomorphic to the power series ring Fq½½t and the Tate module TcðMÞ can be computed as follows: It is the Fq½½t-module consisting of sep "r vectors ðX1; y; XrÞAK ½½t satisfying

s s ð X1; y; XrÞ¼ðX1; y; XrÞÁD: ð4Þ

b b We consider the Fc½GK -module VcðMÞ :¼ Fc#TcðMÞ and its associated Galois representation b rc :¼ GK -Autb ðVcðMÞÞDGLrðFcÞ; Fc whose image we denote by Gc: Q c For every ﬁnite set L of ﬁnite places of F; we put VLðMÞ :¼ cAL VcðMÞ; and let GL denote the image of the representation

- D b rL :¼ GK Autb ðVLðMÞÞ GLrðFLÞ: FL

Let K be a ﬁnitely generated ﬁeld containing F and M a simple t-module M over AK of rank r; with characteristic i and dimension 1. Our main theorem is the following qualitative statement of the image of rL:

Theorem 1.9. Suppose that EndK ðMÞ¼A: For every ﬁnite set L of ﬁnite places of F; b the group GL is open in GLrðFLÞ:

The proof of this theorem will be given in Section 6.

There exists a natural embedding of EndK ðMÞ into Endb ðVLðMÞÞ: We consider FL the centralizer

C b HL :¼ Cent b ðEndK ðMÞÞ GLrðFLÞ: GLrðFLÞ ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 313

As we show in Proposition 2.3, Theorem 1.9 can then be generalized as follows:

Theorem 1.10 (Openness of the Galois image). For every ﬁnite set L of ﬁnite places of F; the group GL-HL is open in both GL and HL:

In the case that M is the t-motive of a Drinfeld module f deﬁned over K; then, by Remark 1.7, this theorem can be restated in terms of the Tate modules of f: The theorem in this Drinfeld module formulation was proved by Pink [Pi2, Theorem 0.2]. Although many of the techniques used in his proof will apply to our settings, its arguments make very strongly use of the fact that a t-motive associated to a Drinfeld module is pure, more precisely: that the eigenvalues of the Frobenius elements are pure (cf. 3.1). We stick to the basic outline of Pink’s proof: ﬁrst, in Section 5, we prove that Gc is Zariski dense in GL b : We then use Pink’s results from [Pi3] on compact Zariski r;Fc dense subgroups of algebraic groups, to conclude the proof of Theorem 1.9. Reducing ourselves to the case where K has transcendence degree 1 over Fq; the essential tools we will use are the Tate and semisimplicity theorem (cf. Section 2), Serre’s Frobenius tori (cf. Section 3) and the tori given by the image of inertia groups at a place x where M has a good model (cf. Section 4). At the same time, it is remarkable how a fairly low tech structure as a t-module of dimension 1, basically given by a square K½t-matrix with a prescribed determinant (cf. Example 1.1), allows such a nice qualitative statement as Theorem 1.10.

Question 1.11. What is a good formulation for a conjecture on the image of the Galois representations associated to t-modules with characteristic i and dimension dX2? The tensor product of simple t-modules of dimension 1 of rank bigger than 1 b yields a t-module M with EndK ðMÞ¼A for which Gc-SLrðFcÞ is not open in b SLrðFcÞ!

Question 1.12. Does a statement like Theorem 1.10 hold if the characteristic i of M is special, i.e. if i : A-K is not injective?

Remark 1.13 (Hodge structures). Suppose that M is the t-motive of a Drinfeld module f: In [Pi1], Pink associates a Hodge structure to f: In [Pi1, Theorem 0.6],he establishes that the Hodge group H of this structure, an algebraic group over F; equals

CentGLr;F ðEndK ðfÞÞ: b The group HL can therefore be interpreted as HðFLÞ: This yields a precise analog of Serre’s conjecture in the classical theory of motives over number ﬁelds, cf. [Se6, Section 9–13, Conjecture 11.4]. Up to now, there exists no theory of Hodge structures for general t-modules. ARTICLE IN PRESS

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Remark 1.14. A generalization of Theorem 1.10 for infinite sets L of finite places c has been conjectured. Let Fad denote the finite adeles of F: The subset Y VadðMÞC VcðMÞ caN consisting of ðvcÞc such that vcATcðMÞ is a free Fad-module endowed with a continuous GK -action. We denote by Gad the image of the representation - rad : GK EndFad ðVadðMÞÞ ð5Þ and put C Had :¼ CentGLrðFadÞðEndK ðMÞÞ GLrðFadÞ:

Conjecture 1.15 (Adelic openness of the Galois image). The group Gad-Had is open in both Gad and Had:

Remark 1.16. For the t-motive of a Drinfeld A-module fM of rank 1, or equivalently, for a t-module over AK of rank 1 and dimension 1, this conjecture, and a fortiori Theorem 1.10, is a well known consequence of the abelian class ﬁeld theory of F (cf. Theorem [Go1, of Section 7.7]). In [Ga1, Section 3.III], we explain how Theorem 1.10 implies Conjecture 1.15 in the case where the rank r of M equals 2. The proof of this implication is based on an analog of Serre’s theory of abelian l-adic representations (cf. [Se1]) and his famous theorem from [Se3] on the image of Galois on the torsion of elliptic curves over a number ﬁeld.

2. The endomorphism ring of a s-module

In this section, we study the endomorphism ring EndK ðMÞ and recall the Tate and semistability conjectures. We explain how one can reduce the proof of Theorem 1.9 to the case where the field has transcendence degree 1, and we show how Theorem 1.9 implies Theorem 1.10. Let K be a finitely generated field containing F: We fix a model X for K; i.e. an Fq-scheme X of finite type over Spec Fq with function field K: For a point x of K; we denote the residue field by kx and the local ring of functions at x by Rx: We consider the Frobenius homomorphism A / q j EndFq ðRxÞ : x x ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 315

# # and the endomorphism s :¼ id j on the ring Ax :¼ A Fq Rx: A t-module M over 3 Ax is a ﬁnitely generated torsion free Ax-module endowed with an injective Ax- linear homomorphism t : sÃM-M: Let M be a t-module over AK : A model Mx over Ax of a t-module M on AK is a t- module over Ax such that # M :¼ AK Ax Mx; in other words, viewed as a t-sheaf on CK ; M is the generic ﬁbre of Mx: Consider the reduction # Mx :¼ Akx ARx Mx

of Mx to Akx ; a projective Akx -module endowed with the induced homomorphism

Ã t : s Mx-Mx:

The t-module Mx is called good at the point x if Mx is a t-module (i.e. t is injective). If Mx is good then it is a maximal model, i.e. it contains every other model of M over good Ax: There exists an open subscheme X of X such that there exists a good model Mx for M at x: Let M be a simple t-module over AK of rank r; with characteristic i: Taguchi (in the case of Drinfeld modules; cf. [Ta1,Ta2]) and Tamagawa ( for general t-modules) [Tam]) proved the so-called Tate and semisimplicity conjectures:

Theorem 2.1 (Taguchi, Tamagawa). (i) (Tate conjecture.) The homomorphism b Fc#AEndK ðMÞ-Endb ðVcðMÞÞ Fc½GK is an isomorphism. b (ii) (Semisimplicity conjecture.) The Fc½GK -module VcðMÞ is semi-simple.

We now have the following result on the endomorphism ring EndK ðMÞ; as well as on the absolute endomorphism ring EndK ðMÞ:

Proposition 2.2 (Endomorphism ring). The ring EndK ðMÞ (resp. EndK ðMÞÞ is a 2 ﬁnitely generated A-algebra of rank at most r : If M has dimension 1, then EndK ðMÞ (resp. EndK ðMÞÞ is commutative and has A-rank at most r:

Proof. Let us ﬁrst prove the result for the ring E :¼ EndK ðMÞ: This E has the structure of an A-algebra via the inclusion A+E: We put

# E :¼ EndK ðMÞ :¼ F AE:

3 Note that we do not assume a model to be locally free, unlike in our more detailed study of models of t- sheaves in [Ga2]. ARTICLE IN PRESS

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We claim that, for every aAE; there exists an ba such that ba Á aAA: If M has rank 1, then clearly EndK ðMÞ¼A: In general, if the endomorphism a is (locally) represented by a matrix B w.r.t. some basis of M; then its determinant is an element of A: The endomorphism ba represented by the adjoint matrix Bad then satisﬁes the condition. This proves that E is a torsion free A-module and that E is a division F-algebra. good We choose a model M for M over CX : For every point xAX ; Mx is maximal and therefore EndK ðMÞ-invariant. By reduction, we obtain a ring homomorphism - j : E Endkx ðMxÞ:

Without loss of generality, we may reduce ourselves to the case that ADFq½t (cf. 1.1), and, upon replacing t by some power, that kx ¼ Fq; as we only risk to increase the endomorphism ring. Then Endkx ðMÞ is just the full matrix ring of rank r Â r over A; so for sure it is finitely generated as an A-module. We now claim that - # E F AEndkx ðMÞ is injective. The ring E being a division F-algebra, it suffices to show that it is not a zero morphism. That is obvious because it is non-zero on the subring ACE of multiplication-by-a endomorphisms. Thus j is injective, and hence EndK ðMÞ is finitely generated as an A-module. It follows from the Tate conjecture (Theorem 2.1) that, for every finite place c of b b F; the F-algebra Fc#AE embeds into Endb ðVcðMÞÞ; which is an Fc-algebra of rank Fc r: This shows that E has F-dimension at most r2; if E is commutative then its dimension is at most r:

To prove the result for EndK ðMÞ; we choose an infinite tower of finitely generated subfield fields KiCK such that ,Ki ¼ K: For every Ki; the 2 ring EndKi ðMÞ is finitely generated over A; of rank pr : It follows from b b the Tate conjecture that the Ac-algebra Ac#AE is saturated in Endb ðTcðMÞÞ; i.e. Ac b the quotient of Endb ðTcðMÞÞ by Ac#AE is torsion free. Therefore, if, for KiCKj; Ac we have a EndKi ðMÞ EndKj ðMÞ; then this implies that

rankA EndKi ðMÞorankA EndKj ðMÞ:

2 As these ranks are bounded by r ; we obtain a Kj such that

EndKj ðMÞ¼EndK ðMÞ; which, by the above, proves the result. ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 317

Finally, suppose that M has dimension 1. Endomorphisms, by deﬁnition, commute with t and so we have a natural homomorphism:

E-End ðcoker tÞ: OCK

As coker t is supported on the closed point GðiÞ of CK and has rank 1 on the point GðiÞ; we can identify the latter with K ¼ EndK ðKÞ: We claim that

j : E-K is a injection. For any non-zero aAACE; we know, because M has characteristic i; that the element jðaÞ acts on coker t as iðaÞAKÂ: Therefore we can extend j to a non- zero ring homomorphism E-K; which, the ring E being a division F-algebra, is injective. Hence E is a (commutative) ﬁeld extension of F; of rank at most r; and E is & a commutative projective A-algebra of rank at most r; idem for EndK ðMÞ:

Suppose now that M has dimension 1. Using the Tate conjecture, we can reduce the proof of Theorem 1.10 to the case where K has transcendence degree 1 and

EndK ðMÞ¼A:

Proposition 2.3. Theorem 1.9 implies Theorem 1.10.

Proof. The ideas of this proof originate from [Pi2, Theorem 0.2] (cf. p. 408). Without loss of generality, we may replace K by a finite extension such that 0 0 # E :¼ EndK ðMÞ¼EndK ðMÞ: If we put A :¼ E and F :¼ F AE; then by Proposi- tion 2.2, F 0 is a finite extension of F: We put C0 :¼ Spec A0 and consider the finite morphism f : C0-C: 0 0 0 The t-sheaf M on CK ; endowed with an action of A ; induces a t-sheaf M on C K 0 0 such that fÃM ¼ M; denoting the induced morphism C K -CK again by f : However, C0 is not necessarily smooth, so we consider the normalization C˜0 of C0 and the morphism f˜: C˜0-C0: Consider the t-sheaf

M% :¼ f˜ÃM0

˜0 ˜ ˜Ã ˜Ã ˜ on C K : We get adjunction morphisms fÃf -id and id-f fÃ; which are isomorph- 0 ˜ % isms outside the finite set S of singularities of C : Thus we see that the t-sheaf fÃM 0 0 on C K is isogenous to M : As now Tate modules are determined by t-sheaves up to isogeny (by the Tate conjecture, Theorem 2.1), we can reduce ourselves to the case that C˜0 ¼ C0: By the Tate conjecture, GLCHL; for every finite set L of finite places of F: On the other hand, by Theorem 1.9 and the above, this implies, for any finite set L of finite ARTICLE IN PRESS

318 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 places of F; that the image of the representation Y 0 0 GK -VL0 ðM Þ :¼ AutðVc0 ðM ÞÞ c0AL0 is open, where L0 is the set of places of F 0 above L: Finally, the isomorphism D" 0 VcðMÞ c0jc Vc0 ðM Þ of Tate modules induces an isomorphism

D 0 0 HL AutF L0 ðVL ðMÞÞ:

This concludes the proof. &

Proposition 2.4. If Theorem 1.9 holds for all finite extensions of F; then it also holds for every finitely generated field K containing F:

Proof. We use ideas from [Pi2, Theorem 1.4]. Let K be finitely generated field containing F and M a t-module on AK with characteristic i and dimension 1. By the b Tate conjecture (Theorem 2.1) the sub-Fc-algebra b FcGc of Endb ðVcðMÞÞ generated by Gc is equal to Endb ðVcðMÞÞ; and this for any fixed Fc Fc finite place c of F: By Pink [Pi2, Lemma 1.5], there is an open normal subgroup 0 0 G1CGc such that for any subgroup O CGc for which O G1 ¼ Gc; we have

b 0 b FcO ¼ FcGc

b b (denoting by FcO0 the sub-Fc-algebra of Endb ðVcðMÞÞ generated by O0). Taking the Fc extension K˜ of K fixed by G1; one denotes by X˜ be the normalization of X in K˜ and by p the morphism X˜-X: By Pink [Pi2, Lemma 1.6], there exists a point x in the open subscheme X good of X À1 such that kx; the residue field of x; is a finite extension of F; and such that p ðxÞ is 0 irreducible. Letting O x be the image of Gkx on VcðMÞ; seen as a subgroup of Gc; we then have

0 O xG1 ¼ Gc:

Hence

b 0 FcO x ¼ EndðVcðMÞÞ; and therefore the reduction Mx of the good model Mx of M at x satisﬁes

EndK ðMxÞ¼A; ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 319 by the Tate conjecture (Theorem 2.1). As, by assumption, Theorem 1.9 holds for the b finite extension of kx of F; we get that the image of Gkx is open in GLrðFLÞ; for every b finite set L of finite places of F: A fortiori, GL is open in GLrðFLÞ: &

3. Frobenius tori

Let K be a finite field extension of F (so K is a field of transcendence degree 1 over Fq). We fix a projective smooth Fq-curve X whose function field is isomorphic to K to serve as a model for K: b For a place x of K; i.e. a closed point of X; let cx be the place of F below x: Let Kx b be the completion of K at x and Rx its valuation ring. Further, we denote the b absolute Galois group of Kx by Gx; its inertia group by Ix; and the Frobenius generator of Gx=Ix by jx: good Let M be a t-module over AK of rank r; with characteristic i: For a place xAX ; we put dx :¼½kx : Fq and we define

dx A PxðM; ZÞ :¼ det ðt À Z j MxÞ Akx ½Z: ð6Þ A#kx

The identity

dx PxðM; U Þ¼detðt À U j MxÞAA½U A

proves that PxðM; ZÞ actually has coefﬁcients in A: We consider an element txAGLrðFÞ with characteristic polynomial PxðM; ZÞ: Some power of tx is semisimple and lies in a unique conjugacy class. The Zariski closure Tx of the connected component of the group generated by tx is therefore a well-deﬁned diagonalisable subgroup of GLr;F : Following Serre (cf. [Se4]), it is called the Frobenius torus at x: It contains essential information on the action of the cac Frobenius jx on all VcðMÞ; for x; as is shown in Proposition 3.1. sep b Note that the choice of an embedding of K into a separable closure of Kx corresponds to the choice of a conjugate of the group GxCGK : Any lift of jx in GK inside GK is called a Frobenius substitution Frobx:

good Proposition 3.1 (Strictly compatible system). (i) For every xAX ; the GK -module VcðMÞ is unramified for all finite places cacx (up to conjugacy, the action of a Frobenius substitution Frobx on VcðMÞ is then well defined ). (ii) The characteristic polynomial

b P ðVcðMÞ; ZÞ :¼ detðFrob À Z j VcðMÞÞAAc½Tð7Þ x b x Fc ARTICLE IN PRESS

320 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338

(which is independent of the choice of Frobx), satisﬁes

PxðVcðMÞ; ZÞ¼PxðM; ZÞ; for all cacx:

Remark 3.2. Following Serre [Se1], the system of Galois modules VcðMÞ is thus a b strictly compatible system of integral Fc-representations over the set of finite places c good of F; with exceptional set X\X : i.e. PxðVcðMÞ; ZÞ has coefficients in A and is good independent of c; for all xAX and all finite places cacx of F: It is also worth mentioning that, by the proposition, the Frobenius torus Tx is a b subtorus of GLr;F ; which, upon specialization to Fc; is conjugate to the torus spanned by the image of Frobx under the representation rc:

b b c b We set Ac;x :¼ Ac#Rx (the completion of Ac#Rx) with respect to its maximal b ideal) and we continuously extend sAEndðAxÞ to an endomorphism of Ac;x:

good Proof of Proposition 3.1. (i) For every finite place xAX ; the cokernel of t on Mx good is supported on the graph GðiÞ of i : A-Rx in Ax: If x is an infinite place in X ; then Mx is smooth, i.e. t operates as an isomorphism on it. good This shows that, for every place x of X and for every finite place cacx of F; the completion

c b # Mc;x :¼ Ac;x Ax Mx ð8Þ is smooth. By the equivalence between smooth formal t-modules and Galois representations mentioned in (cf. 1.6), this implies that it is unramified at xAX good for cacx as it is endowed with a continuous action of Gx: t b (ii) The action of the transpose jx of jx on the unramified Ac½Gx-module TcðMÞ dx c corresponds, by the definition of TcðMÞ; to the action of t on Mc;x; which is induced by that on the reduction Mx: Therefore

P ðM; ZÞ¼detðFrob À Z j TcðMÞÞ ¼ P ðVcðMÞ; ZÞ: & x b x x Ac

4. Image of inertia

Let K be a finite field extension of F with smooth projective model X: We fix a good place xAX and we adopt the notations from the previous section. Let c :¼ cx be b½s b ½s the place of F below x and Ac the unramified extension of degree s of Ac: Let T b½s b denote the torus of GL b obtained by restriction of scalars from Ac to Ac; so s;Ac

½s b b½s b T ðAcÞ¼Ac CGLsðAcÞ: ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 321

Let M be a t-module over AK of rank r; with characteristic i and dimension 1. The following theorem gives a lower bound on the image of the representation of the inertia group on VcðMÞ:

b b Theorem 4.1. Suppose that the extension Kx=Fc is unramiﬁed. Then there exists an exact sequence

et 0-Vc -VcðMÞ-Vc -0 ð9Þ

b et of Fc½Gx-modules with s :¼ dimVc X1 such that the module Vc is unramiﬁed and such that the image of the inertia group Ix under the representation

b rc : Gx-AutðVc ÞDGLsðFcÞ

s b contains a subgroup conjugate to T ½ ðAcÞ:

The proof of this theorem will take up the remainder of this section. First, we prove (Lemma 4.2) that (9) corresponds to a ‘connected-e´ tale’ short exact sequence

cet c c 0-Mc -Mc-Mc-0

b c of t-modules over Ac;x; where (via Anderson’s equivalence Proposition 4.3) Mc is b associated to a formal 1-dimensional Ac-module. Finally, we prove in Theorem 4.5 the required result on the image of inertia the Tate module of such a formal module. b c A locally free Ac;x-module Mc;x endowed with an injective morphism

Ã c c t : s Mc;x-Mc;x

b is called a t-module over Ac;x: We consider its reduction Mc;x modulo the maximal b ideal of Ac;x: c b We call a t-module Mc;x over Ac;x connected if its reduction Mc;x is nilpotent, i.e. n Ã n t : ðs Þ Mc;x-Mc;x is the zero morphism for some n40:

c b Lemma 4.2. For every t-module Mc over Ac;x; there exists an exact sequence

cet c c 0-Mc -Mc-Mc-0 ð10Þ

c b cet c of t-modules Mc over Ac;x; where Mc is smooth and Mc is connected.

Proof. The reduction Mc;x; which is a free kc#kx-module endowed with a s- semilinear endomorphism t; contains a t-invariant direct summand M1 on which t acts injectively such that t acts nilpotently on the quotient Mnil :¼ Mc;x=M1: We ARTICLE IN PRESS

322 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 choose a basis

0 y m :¼ðm1; ; mr1 Þ for M1; which we then extend to a basis m for Mc;x: Any lift m ¼ðmiÞ of m yields a b 0 Ac;x-basis of the latter; let m denote the ordered subset of m which reduces to y ðm1; ; mr1 Þ: Let

A # D1 Matr1Âr1 ðkc kxÞ

A b 0 (resp. D MatrÂrðAc;kx Þ) be the matrix representation of t with respect to the basis m 0 0 b b (resp. m): i.e. tðm Þ¼m Á D1: Consider any lift D1 of D1 with coefﬁcients in Ac;x: cet c We construct a basis n for a sub-t-module Mc of rank r1 of Mc;x such that t b operates as D1 with respect to n: Writing n as m Á Z; for some matrix Z; we obtain, upon comparing the action of t with respect to n and m; the equation

b s Z Á D1 ¼ D Á Z: ð11Þ

s For the reduction Z Á D1 ¼ D Á Z of this equation, a solution is given by the matrix expressing m0 in terms of m: b b As D1 is invertible and Ac;x is complete with respect to its maximal ideal, we can determine a solution Z for (11) by iteration. The basis n generates an smooth t- cet c c c cet invariant direct summand Mc of Mc of rank r1: The quotient Mc :¼ Mc=Mc is connected because its reduction is isomorphic to Mnil: &

Via the antiequivalence between formal t-modules and Galois modules (cf. Remark 1.6), sequence (10) yields a short exact sequence

et 0-Tc -TcðMÞ-Tc -0 of free Ac-modules endowed with a continuous action of Gx: Tensoring with Fc cet yields a sequence as in (9). As Mc is smooth, the Tate module TcðMÞ is unramiﬁed. What remains to be shown is the statement on the image of inertia on Vc : b b b Let us ﬁx an isomorphism AcDkc½½l: Extending i to an embedding Ac-Rx; we c b put y :¼ iðlÞ: A t-module Mc;x over Ac;x has dimension 1if b coker tb DAc;x=ðl À yÞ: Mc;x

For our given t-module M over AK with characteristic i and dimension 1 and a c b # good model Mx over Rx; the completion Mc;x :¼ Ac;x Ax Mx; as well as its c c connected quotient M have dimension 1. We will now explain how M is related to b a formal 1-dimensional Ac-module. ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 323

b b Let G b denote the formal additive group over Rx: Its endomorphism ring is a;Rx b isomorphic to the skew power series ring Rx½½j generated by the endomorphism jAEndðRxÞ and deﬁned by the relation

j Á f ¼j f Á j: b Following Anderson (cf. [An2, Section 3.4]), a formal 1-dimensional Ac-module E b over Rx of height s is a continuous homomorphism b b E : Ac-Rx½½j such that b b EðlÞy mod Rx Á jðRx½½jÞ and

sÁdeg c b EðlÞj mod p Á Rx½½j: ð12Þ

deg c Let us denote the order of the residue ﬁeld kc by qc ¼ q : For every i; the kernel

E½li :¼ ker EðliÞ

b i is is a finite flat scheme of Ac=l -modules of order qc : One defines the Tate module b TcðEÞ associated to E; a free Ac-module with continuous action of GK as

i sep Tcð Þ¼lim ½l ðK Þ: E ’ E

We now quote:

Proposition 4.3 (Anderson, [An2, 3.4]). There is an antiequivalence of categories M b between the categories of formal one-dimensional Ac-modules of height s and the b category of connected one-dimensional t-modules over Ac;x of rank s: Moreover,

TcðEÞDTðM ðEÞÞ:

b Let EM denote the formal one-dimensional Ac-module of height s associated to Mc under this equivalence. As TcðEM ÞDTc ; the proof of Theorem 4.1 will be concluded by Proposition 4.5, dealing with the image of inertia on the Tate module b of a formal one-dimensional Ac-module.

b Remark 4.4. The formal one-dimensional Ac-module Ef associated to the t-motive of a Drinfeld A-module f can also be retrieved by a different method. Fixing a place x of K of good reduction, we set c :¼ cx: Taguchi proved in [Ta1] that the system ARTICLE IN PRESS

324 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338

n ff½l gnAN forms a l-divisible group. For such l-divisible groups there exists a connected-e´ tale exact sequence [Ta1, Remark, p. 296] and an equivalence between connected l-divisible groups and formal A-modules [Ta1, Proposition 1.4].

b Let E be a formal one-dimensional Ac-module of height s: Following ideas of Fontaine (cf. [Abr,Fon]) on the division points of formal groups over unramiﬁed extensions of Zp; we study the image of the inertia group Ix under the representation E b rc : Gx-AutðTcðEÞÞDGLsðAcÞ: b We set G :¼ GLsðAcÞ and, for every iX1; we consider the subgroup

i i b G :¼ 1 þ l Á MatsÂsðAcÞ

b ½i i iþ1 of GLsðAcÞ: We further set G :¼ G =G ; which is isomorphic to MatsÂsðkcÞ via the identiﬁcation

½i i ui :MatsÂsðkcÞ-* G : y/1 þ l Á y:

i E i We set, for iX0; I :¼ rc ðIxÞ-G and

I ½i :¼ I i=I iþ1CG½i: ð13Þ

½s b½s b We denote by kc the extension of kc of degree s inside Ac CMatsÂsðAcÞ: We ½s Â b consider the action by conjugation of the ﬁnite group J :¼ðkc Þ on MatsÂsðAcÞ: For any integer iAZ=sZ; the isotypical component

i b j 1Àqc UðiÞ :¼fuAMatsÂs ðAcÞ; 8jAJ: u ¼ j Á ug; ð14Þ

b½s is endowed with the structure of an Ac -module (cf. [Abr, Section 1; Fon]) and we obtain the following decomposition: b M U :¼ MatsÂsðAcÞ¼ UðiÞ: iAZ=sZ

Likewise, we have a decomposition M H :¼ MatsÂsðkcÞ¼ HðiÞ; iAZ=sZ

½s where each HðiÞ is a one-dimensional kc -vector space on which J operates through i the character j/j1Àqc :

b b Proposition 4.5. Suppose that the extension Kx=Fc is unramiﬁed. Then there exists a function

n : Z=sZ-N,fþNg ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 325 satisfying nð0Þ¼1 and nði þ j ÞpnðiÞþnð j Þ such that 0 1 X E ½s Â @ nðiÞ A rc ðIxÞ¼ðkc Þ r 1 þ l Á UðiÞ : ð15Þ iAZ=sZ

In particular,

½s Â ðAc Þ CrcðIxÞ:

Let us, before starting the proof of the proposition, briefly recall the definition of a ðq sÞ fundamental character. Fixing a solution psCF c of X c À lX ¼ 0; we define a character of the inertia group

½s Â Â zc;s : Ix-ðkc Þ Ckc : s/sðpsÞ=ps

b b which factors through the tame inertia group. If the extension Kx=Fc is unramiﬁed, ½s this character, together with its Galðkc =kcÞ-conjugate

iÀ1 ðqc Þ zc;s for i ¼ 1; y; s is called a fundamental character of level s for kc (cf. [Se3, Section 1]) (this set of fundamental characters is independent of the choice for ps). b s Consider an Fq-linear polynomial PðZÞAKx of degree qc satisfying

PðZÞ¼Z Á EðZÞ; where EðZÞ is an Eisenstein polynomial. Following Serre (cf. [Se3, Section 1]), the Fq-vector space W of roots of P is endowed with the structure of a one-dimensional ½s kc -vector space such that the action of Ix on W is given by a fundamental character.

Proof of Proposition 4.5. The inertia group Ix is isomorphic to the semidirect t r p t p product Ix Ix ; where Ix is the quotient group of tame inertia and Ix is the subgroup t - of wild inertia (higher ramiﬁcation subgroup). We ﬁx a section Ix Ix: b Tame inertia: By assumption, y :¼ iðlÞ is a uniformizer of Kx: By the theory of Newton polygons, the l-torsion points z1 in E½l are roots of a polynomial PðZÞ¼ s Z Á EðZÞ of degree qc ; where EðZÞ is an Eisenstein polynomial. Hence the action of Ix factors through tame inertia, operating via a fundamental character of level s: The image I ½0 of

t - Ix GLsðkcÞCH

½s can be identiﬁed, as a group, with the multiplicative group J of kc : ARTICLE IN PRESS

326 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338

# E t + + Moreover, the restriction of the representation F c b rc to Ix Ix Gx on Ac

c# c Aut c ðF b T ðEÞÞ F Ac is then isomorphic to the direct sum:

Ms iÀ1 Ms ðqc Þ t - ÂC zc;s : Ix kc GLsðF cÞ: ð16Þ i¼1 i¼1

bur b Wild inertia: Let L0 :¼ Kx denote the maximal unramiﬁed extension of Kx; and, for every iX1; set

i bsep Li :¼ L0ðE½l ÞCKx S and LN :¼ iX1 Li: Without any risk of confusion, we will denote the tame inertia bur subgroup of GalðLN=Kx Þ by J: ½i ½i The group GalðLi=LiÀ1Þ is isomorphic to the sub-Z½J-module I of G : Note that via the identiﬁcation G½iDH; for iX1; we have a decomposition M G½i ¼ Hð j Þ: jAZ=sZ

For ja0; the Z½J-modules Hð j Þ are simple. For each iAN; deﬁne ZðiÞ as the subset of Z=sZ such that M I ½iD Hð j Þ; ð17Þ jAZðiÞ

What we need to show is the following:

(i) for every iX1; 0AZðiÞ and (ii) for every i; i0X1; if jAZðiÞ and j0AZði0Þ then j þ j0AZði þ i0Þ:

Indeed, (i) and (ii) imply that if jAZðiÞ; then

i E 1 þ l Uð j ÞCrc ðIxÞ:

If we then put

nð j Þ :¼ inffiAN; jAZðiÞgAN,fNg; then n clearly is the required function. i (i) For every iX1; we recursively ﬁx an element ziAE½l such that z1a0 and

EðlÞðziþ1Þ¼zi ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 327 and we put

L˜i :¼ L0ðziÞCLi:

The extensions L˜i=L˜iÀ1 are Galois: any conjugate sðziÞ of zi is contained in zi þ E½l; and E½lCL˜1 ¼ L1: Since the group GalðLN=L˜iÞ is a subgroup of the higher ramification subgroup of ˜ | ˜ GalðLN=L0Þ; we have J-GalðLN=LiÞ¼ : Let us consider the fixed field Li of the semidirect subgroup JrGalðLN=L˜iÞ of GalðLN=L0Þ:

˜ ˜ Note that the extension Li=LiÀ1 is a Galois extension, with Galois group isomorphic to JrGalðL˜i=L˜iÀ1Þ: The extension L˜i=L˜0 has degree

s sÁðiÀ1Þ ðqc À 1ÞÁqc ; as one sees from the slopes of the Newton polygon. For any sAJ; we have, with respect to the normalized valuation v of L˜i; that

sÁðiÀ1Þ vðsðziÞÀziÞ¼qc :

As ziþ1 is a uniformizer of L˜i; the subgroup

C ˜ GalðLi=LiÀ1Þ GalðLi=LiÀ1Þ ARTICLE IN PRESS

328 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 is, by Serre [Se2, IV, Section 1], contained in the mðiÞth higher ramiﬁcation group GmðiÞ of GalðLi=KÞ; where we put

sÁðiÀ1Þ mðiÞ :¼ qc À 1:

Therefore J acts trivially by conjugation on GalðL˜i=L˜iÀ1Þ; by Serre [Se2, IV, Section 2, Propostion 9, p. 77]). Let us prove by induction that, for every iX1;

(a) the extensions Li=L˜i and L˜iþ1=L˜i are linearly disjoint and (b)

M GalðLiþ1=LiL˜iþ1ÞD Hð j Þ: ð18Þ jAZðiÞ\f0g

For i ¼ 1; both statements are trivial. Suppose now that, for a given i41; the claim 0 holds for all i oi: From (18), we see that no element of GalðLi=L˜iÞ is invariant under the action of J: As GalðL˜iþ1=L˜iÞ is invariant under this action, part (a) follows. Hence

½i GalðLiL˜iþ1=LiÞDHð0ÞCI ; which by (17) proves part (b). This concludes the induction. ½i In conclusion, the quotient GalðLiL˜iþ1=LiÞ of I ; invariant under conjugation by s ½i J; has order qc : As a Z½J-module, it must therefore coincide with Hð0ÞCI ; i.e. 0AZðiÞ: (ii) Note that, for the Lie bracket ½Á; Á; we have

½Hð j Þ; Hð j0Þ ¼ Hð j þ j0Þ; if ð j; j Það0; 0Þ (cf. [Fon, Section 7]). By the commutative diagram

½i ½i0 - ½iþi0 / À1 À1 G Â G G : ðg1; g2Þ g1g2g1 g2 mm m ð19Þ

H Â H - H : ðh1; h2Þ / ½h1; h2 this implies that if jAZðiÞ and j0AZði0Þ; then

j þ j0AZði þ i0Þ for ð j; j Það0; 0Þ; and so, by (ii), if jAZðiÞ; then jAZði0Þ for every i04i: & ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 329

5. Zariski density of Cc

Let K be a ﬁnite ﬁeld extension of F and M a t-module over AK of rank r; with characteristic i and dimension 1. In this section, we will prove the following theorem:

c Theorem 5.1. Suppose that EndK ðMÞ¼A: Then for every ﬁnite place of F; the group Gc is Zariski dense in GL b : r;Fc

For a ﬁnite place c of F; let Gc denote the Zariski closure of Gc inside GL b : The r;Fc following criterion for the connectedness of Gc was proved by Serre, independently of the characteristic of the ﬁeld K:

Theorem 5.2 (Serre [Se4, App.]). The group Gc is connected if and only if, for every function f AZ½c1; y; cr; the set of places x of K with

f ðrcðFrobxÞÞ ¼ 0 has density 0 or 1.

As f ðrcðFrobxÞÞAA is independent of c ( for all but a finite number of places of K), this shows that Gc is connected for one place c; if and only if it is connected for all c: Without loss of generality, we can replace K by a finite separable extension so that GcCGc: Hence, we may assume that Gc is connected for all c: Without loss of generality, we may replace K by a finite separable extension, so that we can suppose that EndK ðMÞ¼A: By the Tate and semisimplicity conjecture, we know that GK acts absolutely irreducibly on VcðMÞ; for every finite place c of F: The tautological representation of Gc is absolutely irreducible as well, and it follows from this that Gc is a reductive connected group (cf. [Pi2, Fact A.1]). We want to apply the following result:

Proposition 5.3 (Pink [Pi2, Proposition A.3]). Let L be an algebraically closed ﬁeld and GCGLr;L a reductive connected linear algebraic group acting irreducibly on an r- dimensional L-vector space V: Suppose that G possesses a cocharacter which has weight 1 with multiplicity 1 and weight 0 with multiplicity r À 1 on V: Then G ¼ GLr;L:

b Let F c denote a fixed algebraic closure of Fc: The rest of this section is devoted to finding a cocharacter for F c Âb Gc in its representation on F c#VcðMÞ as in Fc Proposition 5.3. We recall that, in his proof of Theorem 1.9 for t-motives of Drinfeld modules, Pink first proves that the set of finite places x of K at which f has ordinary reduction has positive density, by an argument involving the purity of the eigenvalues of the Frobenius elements. The Frobenius torus at such a place then yields the desired cocharacter. ARTICLE IN PRESS

330 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338

In our argument below, we will first prove that the maximal tori for Gc look the same for all c and then use the image of tame inertia inside these tori to prove the existence of such a cocharacter. We define the functions c1; c2; y; cr :GLr-Ga as the coefficients of the characteristic polynomial of elements in GLr; i.e. for every gAGLr; the characteristic polynomial PgðXÞ of g is equal to

r rÀ1 X þ c1ðgÞX þ ? þ crðgÞ:

We also consider the morphism

- rÀ1 c ¼ðc1; y; crÞ :GLr Ga Â Gm:

For every diagonalisable group T in GLr; the image cðTÞ is a closed subvariety of rÀ1 Ga Â Gm (cf. [Se5, Section 5]). For every ﬁnite place c of F; we ﬁx a maximal torus Tc of Gc and an isomorphism

r F cDF c#VcðMÞ such that

r F c Âb TcCG CGL : Fc m;F c r;F c

r cC C Also, we choose a torus Y Gm;F GLr;F such that

F c Â YcDF c Âb Tc F Fc

# r r and such that the representations of these tori on F c F F ¼ F c coincide. Note the action of the symmetric group

r r SrDNorm ðG Þ=Cent ðG Þ; GLr;F m;F GLr;F m;F

A sending the torus Yc to all of its conjugates inside GLr;F : For every s Sr; we denote the resulting copy of Yc by Yc;s:

C c Proposition 5.4. The tori Yc GLr;F are conjugate for all :

Proof. This proof was inspired by the arguments of [L-P, Section 8]. Let c and c0 be ﬁnite places of F: For any place xAX good not lying above c or c0; we ﬁx an element Tx of the conjugacy class of Frobenius tori for x such that C cC C c0 C Tx Y GLr;F : Let sx be the element in Sr such that Tx Y ;sx GLr;F : Let us now assume that

r cg c0 C Y Y ;s Gm;F ; ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 331 for all sASr: Then the image under c of the union of subgroups [ U :¼ ðYc-Yc0;sÞ sASr is a proper closed subvariety of cðYcÞ: On the other hand, the set

0 good 0 U ¼fcðTxÞ; xAX not above c or c g is a Zariski dense subset of cðYcÞðFÞ; the set images rcðFrobxÞ of Frobenius substitutions being dense in Gc by the Chebotarev density theorem. Therefore, there exists a place xAX good such that

cðTxÞecðUÞðFÞ:

g g 0 This implies that Tx U; and, in particular, that Tx Yc ;sx ; which gives a contradiction. 0 In conclusion, for any places c and c of F; we have YcCYc0;s; for at least one 0 sASr: By symmetry, this shows that the maximal tori Yc and Yc are conjugate. &

r C c c Let us ﬁx a subtorus T Gm;F of GLr;F which is conjugate to Y for every :

r Proposition 5.5. The torus T possesses, in its representation on F ; a cocharacter of weight 1 with multiplicity 1 and weight 0 with multiplicity r À 1:

Proof. Let us fix, for every finite place c of F; a place xc of K lying above c: We c b b denote by L the set of finite places of F which do not ramify in Kxc =Fc and such that M has good reduction at xc: By Theorem 4.1, the image

t C b Dc :¼ rcðIxc Þ GLrðFcÞ of the tame inertia group Txc on VcðMÞ is isomorphic to

½sc Â rÀsc + b ðkc Þ Âf1gCGLsc ðkcÞÂf1g GcðFcÞ;

where sc denotes height of Mc;xc : By (16), the action of inertia on Tc is given by the iÀ1 direct sum of the qc th powers of a fundamental character zc;s of level s; for i ¼ 1; y; s: There exists an spr and an inﬁnite subset L1 of L such that, for all cAL1; sc ¼ s: As Gc is connected, the cyclic group Dc is contained in a maximal torus Tc of Gc: Under the ﬁxed isomorphism of the character groups XðTÞDXðTcÞ; we can now r c consider the action of the standard basis characters wi of Gm;F on D : For every c AL1; there exists a renumbering sc of the standard basis characters wi such that

iÀ1 s * qc ½ Â t wiðaÞ¼a for all aAðkc Þ ¼ zc;sðIxÞ and i ¼ 1; y; s and ARTICLE IN PRESS

332 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338

* wi ¼ 1 for all i ¼ s þ 1; y; r:

There exists an inﬁnite subset L2 of L1; such that the above holds with the same c renumbering sc of the characters wi for all AL2: Let us put ˜ s rÀs T :¼ T-ðGm Âfeg Þ:

Suppose that T˜ satisﬁes a relation

Xs w :¼ ai Á wi; i¼1

½s Â with aiAZ: Considering the action of w on the elements of ðkc Þ ; we obtain, for all c in L2 that

Xs iÀ1 s ai Á qc 0 mod qc À 1: i¼1

As L2 is an inﬁnite set, the qc are unbounded, which yields w ¼ 0: Hence T˜ ¼ s rÀs * ˜ Gm Âfeg : In particular, we see that T T possesses a cocharacter of weight 1 with multiplicity 1 and weight 0 with multiplicity r À 1: &

As the representations

F c Â YcDF c Âb Tc F Fc

r concide on F c ; this theorem now implies that the maximal torus F c Âb Tc of F c Âb Fc Fc Gc has a cocharacter of weight 1 with multiplicity 1 and weight 0 with multiplicity r r À 1 in its representation on F c : By Proposition 5.3, it follows that

F c Âb Gc ¼ GL ; Fc r;F c which concludes the proof of Theorem 5.1.

6. Openness of CK

Let K be a finite field extension of F and M a t-module over AK of rank r; with characteristic i and dimension 1. In this section we conclude the proof of the openness of the L-adic representations, Theorem 1.9. Let L be a finite set of finite places of F: The maximal exterior power 4rM is t- module over AK of rank 1 and dimension 1, and hence Theorem 1.9 is known for r r r 4 M; by Remark 1.16. As VLð4 MÞD4 VLðMÞ; it follows from this that the image ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 333 of the composite morphism

r GK - Aut ðVLðMÞÞ - Aut ð4 VLðMÞÞ Fb Fb L DDL

b det bÂ GLrðFLÞ ! FL

b is open. What remains to be shown is that GL-SLrðFLÞ is open in b SLrðFLÞ: Let c be a ﬁnite place of F: We consider the absolutely simple connected adjoint group PGL b : Without loss of generality, we may replace K by a ﬁnite separable r;Fc extension, so that we can suppose that EndK ðMÞ¼EndK ðMÞ: By Theorem 5.1, the image of GK under the composite map

rc b P b GK !GLrðFcÞ !PGLrðFcÞ is Zariski dense for every ﬁnite place c: We use Pink’s results [Pi3] on compact Zariski dense (abstract) subgroups of algebraic groups, which we can formulate as follows:

Theorem 6.1 (Pink [Pi3, Main Theorem 0.2, Proposition 0.4(a)]). For every ﬁnite set L of ﬁnite places, there exist

* b b a subring ELCFL; * b an algebraic group H over EL such that b FL Âb HDPGL b ; EL r;FL

* an inner automorphism j of PGL b r;FL such that * b b the image of GL in PGLrðFLÞ is contained in jðHðELÞÞ and * 0 b the commutator subgroup GL of GL is open in H˜ðELÞ; where H˜ is the maximal b covering of H; a model of SL b over EL: r;FL

An algebraic function f : PGLr-Ga is called central if f 3a ¼ f for any inner automorphism a of PGLr: Denoting as before the functions that give the coefﬁcients of the characteristic polynomial of a matrix gAGLr by c1; c2; y; cr; the functions

r À1 À1 f1 :¼ c1 Á cr and f2 :¼ c1 Á crÀ1 Á cr are examples of such central functions on PGLr: ARTICLE IN PRESS

334 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338

b Let L be a ﬁnite set of ﬁnite places of F and let BL denote the total b ring of quotients of the closure of the subring of FL generated by 1 and the elements f ðgÞ; where f is a central function on PGLr and gAGL: b b Clearly, BLCEL:

Proposition 6.2 (Pink [Pi3, Proposition 0.4(a+c)]). For a ﬁnite place c of F; the ﬁeld b b bp Bfcg either equals Efcg or Efcg:

good b b Lemma 6.3. For a place xAX ; we put c :¼ cx: If the field extension Kx=Fc is unramified, then (i) b b bp we have Bfcg ¼ EfcggFc and (ii) for every finite place c0ac of F;

b b a Bfc;c0g-ðFc Âf0gÞ f0g: b Proof. Let us ﬁx an isomorphism FcDkcððlÞÞ: Our proof uses in an essential way the information on the image of inertia from Theorem 4.1. (i) We will prove that there exists an element gAGc such that, for some central bp b bp b function f on PGLr; we have f ðgÞeFc : Then BfcggFc ; which implies that Bfcg ¼ b Efcg; by Proposition 6.2. First, suppose that s41: Choose a polynomial

XsÀ1 s i PðZÞ¼Z þ hiZ Akc½Z i¼0

½s which deﬁnes the ﬁeld extension kc (cf. the notations of Section 4). For the embedding

b½s Â b ðAc Þ +GLsðAcÞ;

0 b½s we can ﬁnd an element g AAc whose characteristic polynomial equals

XsÀ2 s p sÀ1 i b Pg0 ðZÞ :¼ Z þðhsÀ1 þ l ÞZ þ hiZ þðh0 þ lÞAFc½Z; i¼1 by Hensel’s lemma. 0 By Theorem 4.1, there exists an element gAIx; such that g ¼ rcðgÞ acts as g on the submodule Tc of TcðMÞ and trivially on the quotient. Thus the characteristic polynomial of g on VcðMÞ equals

rÀs PgðZÞÁðZ À 1Þ : ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 335

In particular,

p c1ðgÞ¼ÀTrðgÞ¼l þ hsÀ1 Àðr À sÞa0 and

r rÀs crðgÞ¼ðÀ1Þ detðgÞ¼ðÀ1Þ ðh0 þ lÞa0:

bp Â bp bp But then c1ðgÞAðFc Þ and crðgÞeFc ; which shows that f1ðgÞeFc : If s ¼ 1; we need a different trick. There exists, by Theorem 4.1, an element gAIx; such that g ¼ rcðgÞ acts as h :¼ 1 þ l on the submodule TcðEÞ of TcðMÞ and trivially on the quotient. We ﬁx a basis such that g is in diagonal form:

g ¼ diagðh; 1; y; 1Þ:

As GK is Zariski dense in GLb ; there exists an element a ¼ðaijÞAGcCGLb ; with Fc Fc inverse b ¼ðbijÞ; such that

a1 :¼ a11a0;

Xr a2 :¼ ajja0; j¼2 a b1 :¼ b11 0;

Xr a b2 :¼ bjj 0: j¼2

Then n n n Àc1ða Á g Þ¼Trða Á g Þ¼a1 Á h þ a2;

crÀ1 n n À1 Àn for all nX0: Similarly, ða Á g Þ¼ÀTrðða Á g Þ Þ¼b1 Á h þ b2: Thus we obtain cr that n n Àn f2ða Á g Þ¼w1 Á h þ w2 Á h þ w3 ð20Þ b for elements wiAFc; with w1; w2 non-zero. n bp We suppose that jn :¼ f2ða Á g ÞAFc ; for all nX0; and aim to ﬁnd some b bp contradiction. The elements wiAFc are in Fc as they are the unique solutions of bp the following system of linear equations with coefﬁcients in Fc : 8 <> w1 þ w2 þ w3 ¼ j0; p Àp w1 Á h þ w2 Á h þ w3 ¼ jp; ð21Þ :> 2p À2p w1 Á h þ w2 Á h þ w3 ¼ j2p: ARTICLE IN PRESS

336 F. Gardeyn / Journal of Number Theory 102 (2003) 306–338

2m If p ¼ 2; then we choose odd m such that 0aw1 Á h þ w2: As

2m Àm jm ¼ðw1 Á h þ w2ÞÁh þ w3;

2m b2 m b2 where jm; w3 and w1 Á h þ w2 are elements of Fc ; we obtain that h AFc ; a contradiction. P À1A bp N pi If p42; let us put o :¼ w1 Á w2 Fc : Expanding o as ibÀN opil ; we obtain X À1 p i 2 w2 Á j1 À w3 ¼ opiðl Þ Áð1 þ lÞþð1 À l þ l þ yÞ: ð22Þ

bp In this equation, the left-hand side is an element of Fc ; whereas the right-hand side clearly is not (cf. the coefﬁcient of l2), which gives a contradiction. good 0 (ii) As xAX ; the action of Ix on Vc0 ðMÞ is trivial for c ac: For s41; we immediately see that, with gAIx as above,

a b b 0 f1ðrLðgÞÞ À f1ðrLð1ÞÞABfc;c0g-Fc Âf0g:

For s ¼ 1; using gAIx and aAGL as above, we note that the function

/ n n f2ða Á rLðg ÞÞ À f2ða Á rLð1ÞÞ b b is a non-constant function with values in Bfc;c0g-ðFc Âf0gÞ: &

good For every xAX not lying above c; the eigenvalues of rcðFrobxÞ lie in F and are independent of cacx; by Proposition 3.1. Consider the subﬁeld B of F generated by 1 and by the set of elements f ðrcðFrobxÞÞ; where f is a central function on PGLr and good xAX : The set of Frobenius substitutions being dense in GK ; the closure of B in b b FL coincides with BL:

Proposition 6.4. B ¼ F:

b bp c Proof. By Lemma 6.3(i), B is not a finite field, as BfcggFqCFc ; for any : As F has transcendence degree 1 over Fq; this implies that F=B is a finite extension. Furthermore Lemma 6.3(i) implies that F=B is separable. Consider the set LB of finite places b of B which do not ramify in K=B and such that all places x of K above b are in X good (this excludes only a finite number of b places of B). For bALB; let Bb denote the completion of B at b; and, for a place x above b; let c denote the place of F below x: If there exists another place c0 of F above b and L ¼fc; c0g; then the closure of B b b in Ffc;c0g equals Bb; diagonally embedded; hence b b b f0g¼Bfc;c0g-ðFc Âf0gÞCFfc;c0g; ARTICLE IN PRESS

F. Gardeyn / Journal of Number Theory 102 (2003) 306–338 337 which yields a contradiction with Lemma 6.3(ii). In conclusion, only one place of F can lie above each bALB; which shows that B ¼ F: &

b b b b b As F is dense in FL; this implies that BL ¼ FL and therefore EL ¼ FL: By 0 b Theorem 6.1, it follows that G L is open in SLrðFLÞ; which concludes the proof of Theorem 1.9. &

Acknowledgments

It is a great pleasure to thank Richard Pink for his detailed explanations of several of his results, for plenty of helpful remarks and for his enthusiastic and motivating support.

References

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