Facteurs epsilon p-adiques

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 > 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.

Product formula forp-adic epsilon factors

“…If Λ ′ /K is totally ramified, we may put σ Λ ′ := id Λ ′ ; then the category F -Isoc † (U, X/K) Λ ′ identifies to F -Isoc † (U, X/Λ ′ ), cf. Example 7.3.3, and so we finish by [Mr,Theorem 4.3.15]. Let us treat the general case.…”

“…where the first equality follows from the localization triangle (3.1.9.1) and 3.1.5.1, and the second (resp. third) from [Mr,(3.4 7.2.7 Remark. -(i) Note that in loc.…”

“…In the following, we will always assume µ(A/̟A) = 1; to lighten the notation, we put ε(M, ω) := ε π (M, ω, µ). Moreover, for a free differential R-module M with Frobenius structure, we define ε rig 0 (M, ω) := ε 0 (M, ψ π (ω), µ) and ε rig (M, ω) := ε(M, ψ π (ω), µ) using the notation of [Mr,3.4.4]. For a complex C of F -D an S ,Q -modules with bounded holonomic cohomology, we put 7.2.…”

“…Let us denote by Déco( K ur ⊗ K V ηx ) the sub-K ur -vector space of K ur ⊗ K V ηx , spanned by the finite orbits under the action of π 1 (η x ,η x ). By [Mr,3.3.6], we have Ψ(−1)(M ηx ) = Déco( K ur ⊗ K V ηx ). Since M has finite monodromy, we get Ψ(−1)(M ηx ) = K ur ⊗ K V ηx endowed with the diagonal action of π 1 (η x ,η x ) (it acts on K ur via the residual action).…”

Combinatorial degenerations of surfaces and Calabi–Yau threefolds

“…Then (ϕ, ∇)-modules over R K should be considered as p-adic analogues of Galois representations, for example, they satisfy a local monodromy theorem (see [Ked04]) and hence have a canonical monodromy operator N attached to them (see [Mar08]). More specifically, the connection ∇ should be viewed as an analogue of the action of the inertia subgroup I F and the Frobenius ϕ the action of some Frobenius lift in G F .…”