Abstract :
Let C be a smooth projective absolutely irreducible curve over a finite field , F its function field and A the subring of F of functions which are regular outside a fixed point ∞ of C. For every place ℓ of A, we denote the completion of A at ℓ by .
In [Pi2], Pink proved the Mumford–Tate conjecture for Drinfeld modules. Let φ 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 Γℓ the image of the representation of the absolute Galois group ΓK of K on the Tate module Tℓ(φ). The Mumford–Tate conjecture states that some subgroup of finite index of Γℓ is open inside a prescribed algebraic subgroup Hℓ of . In fact, he proves this result for representations of ΓK on a finite product of distinct Tate modules.
A τ-module over AK is a projective A K-module of finite type endowed with a 1 -semilinear injective homomorphism τ, where denotes the Frobenius morphism on K. Such a τ-module is said to have dimension 1, if the K-vector space M/K•τ(M) has dimension 1.
Drinfeld showed how to associate, in a functorial way, to every Drinfeld module over K a τ-module M(φ) over AK of dimension 1, called the t-motive of φ. In this paper, we generalize Pinkʹs theorem to representations of Tate modules Tℓ(M) of τ-modules M of dimension 1 over AK. The key result can be formulated as follows: if we suppose that EndK(M)=A, then for every finite place ℓ of F, the image Γℓ of the representation is open in , 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 τ-modules over AK and Galois representations with coefficients in 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 τ-module M is pure, or, equivalently, is the t-motive of a Drinfeld module φ, 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 τ-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 Pinkʹs proof: reducing ourselves to the case where the absolute endomorphism ring of M equals A, we first show that Γℓ is Zariski dense in and we use his results on compact Zariski dense subgroups of algebraic groups to conclude that Γℓ if open in . After a reduction to the case where K has transcendence degree 1 over , the essential tools we will use are the Tate and semisimplicity theorem for simple τ-modules, Serreʹs Frobenius tori and the tori given by inertia at places of good reduction for M.
