The structure of analytic τ-sheaves

AbstractLet R be a complete discrete valuation Fq-algebra whose residue field is algebraic over Fq, and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve C̃K≔C×SpecK, 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 C̃K, we prove the existence of a maximal model Mmax for M on C̃R, an R-model of C̃K, 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 filtration0⊂M̃1⊂⋯⊂M̃|C̃K′′of sub-τ-sheaves of M̃|C̃′K′ 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 Tℓ(M̃) associated to M̃ is potentially unipotent for almost all closed points ℓ of C