The structure of analytic τ-sheaves Original Research Article

Let R be a complete discrete valuation image-algebra whose residue field is algebraic over image, and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve image, 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 Tℓ(M) of M. For a given τ-sheaf M on image, we prove the existence of a maximal model image for M on image, an R-model of image, and, over a finite separable extension R′ of R, of nondegenerate models image for M. We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′subset ofC, and a filtrationimageof sub-τ-sheaves of image on CK′, such that its subquotients are τ-sheaves with a good model. As a consequence, the action of the inertia group IK on the Tate modules image associated to image is potentially unipotent for almost all closed points ℓ of C.