The structure of analytic τ-sheaves

Abstract

Let R be a complete discrete valuation F q -algebra whose residue field is algebraic over F q , and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve C ̃ K≔C× Spec K , seen as a rigid analytic space. One motivation is the Tate uniformization theorem for t-motives of Drinfeld modules, which we want to extend to general τ-sheaves. On the other hand, we are interested in the action of inertia on a generic Tate module T ℓ( M) of M. For a given τ-sheaf M on C ̃ K , we prove the existence of a maximal model M max for M on C ̃ R , an R-model of C ̃ K , and, over a finite separable extension R′ of R, of nondegenerate models M for M. We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′⊂ C, and a filtration 0⊂ M ̃ 1⊂⋯⊂ M ̃ | C ̃ K′′ of sub- τ-sheaves of M ̃ | C ̃ ′ K′ on C K′ , such that its subquotients are τ-sheaves with a good model. As a consequence, the action of the inertia group I K on the Tate modules T ℓ( M ̃ ) associated to M ̃ is potentially unipotent for almost all closed points ℓ of C.