Local epsilon conjecture and p-adic differential equations

TL;DR

This paper links the local epsilon conjecture for de Rham (phi, Gamma)-modules to their associated p-adic differential equations, simplifying the conjecture's verification via an isomorphism of fundamental lines.

Contribution

It establishes an isomorphism between fundamental lines of cyclotomic deformations, reducing the local epsilon conjecture for modules to that of their p-adic differential equations.

Findings

Defined an isomorphism between fundamental lines of deformations

Reduced the local epsilon conjecture to the case of p-adic differential equations

Provided a new approach to verify the epsilon conjecture in p-adic Hodge theory

Abstract

Laurent Berger attached a p-adic differential equation N_rig(M) with a Frobenius structure to an arbitrary de Rham (phi, Gamma)-module over a Robba ring. In this article, we compare the local epsilon conjecture for the cyclotomic deformation of M with that of N_rig(M). We first define an isomorphism between the fundamental lines of their cyclotomic deformations using the second author's results on the big exponential map. As a main result of the article, we show that this isomorphism enables us to reduce the local epsilon conjecture for the cyclotomic deformation of M to that of N_rig(M).