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

[摘要] Let C be a smooth projective absolutely irreducible curve over a finite field F(q), F its function field and A the subring of F of functions which are regular outside a fixed point infinity of C. For every place l of A, we denote the completion of A at l by (A) over cap (/). In[Pi2], Pink proved the Mumford-Tate conjecture for Drinfeld modules. Let phi be a Drinfeld module of rank r defined over a finitely generated field K containing F. For every place l of A, we denote by Gamma(l) the image of the representation r(l) : Gamma(K) --> Aut((A) over cap/) (T(/) (phi)) congruent to GL(r) ((A) over cap (/)) of the absolute Galois group Gamma(K) of K on the Tate module T(l)(phi). The Mumford-Tate conjecture states that some subgroup of finite index of Gamma(l) is open inside a prescribed algebraic subgroup H(l) of GL(r,(A) over capl). In fact, he proves this result for representations Of Gamma(K) on a finite product of distinct Tate modules. A tau-module over A(K) is a projective Acircle timesK-module of finite type endowed with a 1circle timesphi-semilinear injective homomorphism tau, where phi denotes the Frobenius morphism on K. Such a tau-module is said to have dimension l, if the K-vector space M/K.tau(M) has dimension l. Drinfeld showed how to associate, in a functorial way, to every Drinfeld module over K a tau-module M(phi) over A(K) of dimension l, called the t-motive of phi. In this paper, we generalize Pink's theorem to representations of Tate modules T(l)(M) Of T-Modules M of dimension l over A(K). The key result can be formulated as follows: if we suppose that End(K)(M) = A, then for every finite place l of F, the image Gamma(l) of the representation rho(l): Gamma(K) --> Aut((A) over capl)(T(l)(M)) is open in GL(r)(($) over capl), 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 A(K) and Galois representations with coefficients in (A) over cap (t) 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 tau-module M is pure, or, equivalently, is the t-motive of a Drinfeld module phi, 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 tau-modules of dimension l, 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 Gamma(l) is Zariski dense in GL(r,(A) over capl) and we use his results on compact Zariski dense subgroups of algebraic groups to conclude that Gamma(l) if open in After a reduction to the case where K has transcendence degree 1 over F(q), the essential tools we will use are the Tate and sernisimplicity theorem for simple tau-modules, Serre's Frobenius tori and the tori given by inertia at places of good reduction for M. (C) 2003 Published by Elsevier Inc.

[发布日期] 2003-10-01

[关键词] tau-sheaves;t-motives;Galois representation;Serre conjecture