A Galois criterion for good reduction of τ-sheaves

Abstract

Let R be a complete discrete valuation F q -algebra with fraction field K and perfect residue field k. For an irreducible smooth affine curve C, with field of constants F q , let M denote a τ-sheaf over C K , endowed with a characteristic morphism ι : Spec K→C . Given a Tate module T ℓ( M) with trivial action of the inertia group I K , we construct a good model M for M over C R . This yields an analog for τ-sheaves of the classical Néron–Ogg–Shafarevič theorem on good reduction of abelian varieties. We can actually extend this result to a criterion for nondegenerate and semistable reduction. As an application, we show how the local L-factor of a τ-sheaf at a place of bad reduction is related to the action of Frobenius on the associated Galois representations. Finally, we discuss the implications of these results to Drinfeld modules and their associated t-motives.