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

Abstract

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 ∞ of C. For every place ℓ of A, we denote the completion of A at ℓ by A ℓ . 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 Γ ℓ the image of the representation ρ ℓ : Γ K→ Aut A ℓ (T ℓ(φ))≅ GL r( A ℓ) of the absolute Galois group Γ K of K on the Tate module T ℓ( φ). The Mumford–Tate conjecture states that some subgroup of finite index of Γ ℓ is open inside a prescribed algebraic subgroup H ℓ of GL r, A ℓ . In fact, he proves this result for representations of Γ K on a finite product of distinct Tate modules. A τ-module over A K is a projective A⊗ K-module of finite type endowed with a 1⊗ ϕ-semilinear injective homomorphism τ, where ϕ 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 A K of dimension 1, called the t-motive of φ. In this paper, we generalize Pink's theorem to representations of Tate modules T ℓ( M) of τ-modules M of dimension 1 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 Γ ℓ of the representation ρ ℓ : Γ K→ Aut A ℓ (T ℓ(M)) is open in GL r( A ℓ) , 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 A K and Galois representations with coefficients in A ℓ 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 Pink's proof: reducing ourselves to the case where the absolute endomorphism ring of M equals A, we first show that Γ ℓ is Zariski dense in GL r, A ℓ and we use his results on compact Zariski dense subgroups of algebraic groups to conclude that Γ ℓ if open in GL r( A ℓ) . 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 semisimplicity theorem for simple τ-modules, Serre's Frobenius tori and the tori given by inertia at places of good reduction for M.