Local $\epsilon$-conjecture and $p$-adic differential equations

Abstract

Laurent Berger attached a $p$-adic differential equation ${\mathbf{N}}_{\mathrm{rig}}(M)$ with a Frobenius structure to an arbitrary de Rham $(\phi ,\Gamma )$-module $M$ over a Robba ring. In this article, we compare the local epsilon conjecture for the cyclotomic deformation of $M$ with that of ${\mathbf{N}}_{\mathrm{rig}}(M)$. 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 ${\mathbf{N}}_{\mathrm{rig}}(M)$. The result can be regarded as a refined interpolation formula of the big exponential map.