A Galois criterion for good reduction of τ-sheaves Original Research Article

Let R be a complete discrete valuation image-algebra with fraction field K and perfect residue field k. For an irreducible smooth affine curve C, with field of constants image, let M denote a τ-sheaf over CK, endowed with a characteristic morphism ι :Spec K→C. Given a Tate module Tℓ(M) with trivial action of the inertia group IK, we construct a good model image for M over CR. This yields an analog for τ-sheaves of the classical Néron–Ogg–Shafareviimage 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.