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 > 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 forétale -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.

Let X be a smooth proper curve over a finite field of characteristic p. We prove a product formula for p-adic epsilon factors of arithmetic D-modules on X. In particular we deduce the analogous formula for overconvergent F -isocrystals, which was conjectured previously. The p-adic product formula is the equivalent in rigid cohomology of the Deligne-Laumon formula for epsilon factors in ℓ-adicétale cohomology (for ℓ = p). One of the main tools in the proof of this p-adic formula is a theorem of regular stationary phase for arithmetic D-modules that we prove by microlocal techniques.

On the continuity of the finite Bloch-Kato cohomology ADRIAN IOVITA (*) -ADRIANO MARMORA (**) ABSTRACT -Let K 0 be an unramified, complete discrete valuation field of mixed characteristics (0; p) with perfect residue field. We consider two finite, free Z p -representations of G K0 , T 1 and T 2 , such that T i Zp Q p , for i 1; 2, are crystalline representations with Hodge-Tate weights between 0 and r p À 2: Let K be a totally ramified extension of degree e of K 0 . Supposing that p ! 3 and e(r À 1) p À 1, we prove that for every integer n ! 1 and i 1; 2, the inclusion H 1 f (K; T i )=p n H 1 f (K; T i ) ,3 H 1 (K; T i =p n T i ) of the finite Bloch-Kato cohomology into the Galois cohomology is functorial with respect to morphisms as Z=p n Z[G K0 ]-modules from T 1 =p n T 1 to T 2 =p n T 2 . In the appendix we give a related result for p 2.

Given a Galois cover of curves over F p , we relate the padic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a 'weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of p-adic valuations of epsilon constants.