Facteurs epsilon <i>p</i> -adiques

Abstract

Abstract We develop and study the epsilon factor of a ‘local system’ of p -adic coefficients over the spectrum of a complete discrete valuation field K with finite residue field of characteristic p &gt;0. In the equal characteristic case, we define the epsilon factor of an overconvergent F -isocrystal over Spec ( K ), using the p -adic monodromy theorem. We conjecture a global formula, the p -adic product formula, analogous to Deligne’s formula for étale ℓ -adic sheaves proved by Laumon, which explains the importance of this local invariant. Namely, for an overconvergent F -isocrystal over an open subset of a projective smooth curve X , the constant of the functional equation of the L -series is expressed as a product of the local epsilon factors at the points of X . We prove the conjecture for rank-one overconvergent F -isocrystals and for finite unit-root overconvergent F -isocrystals. In the mixed characteristic case, we study the behavior of the epsilon factor by deformation to the field of norms.