Product formula for p-adic epsilon factors

TL;DR

This paper establishes a product formula for p-adic epsilon factors of arithmetic D-modules on smooth proper curves over finite fields, extending known formulas in l-adic cohomology to the p-adic setting.

Contribution

It proves a p-adic product formula for epsilon factors, including for overconvergent F-isocrystals, using microlocal techniques and regular stationary phase theorems.

Findings

Proved a p-adic epsilon factor product formula for arithmetic D-modules.

Extended the formula to overconvergent F-isocrystals.

Developed microlocal methods for regular stationary phase in p-adic context.

Abstract

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-adic \'etale cohomology (for a prime l different from 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.