Local ε-conjecture and p-adic differential equations

Laurent Berger attached a p-adic differential equation N_​ (M) with a Frobenius structure to an arbitrary de Rham (φ, Γ) -module M over a Robba ring. In this article, we compare the local epsilon conjecture for the cyclotomic deformation of M with that of N_​ (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_​ (M). The result can be regarded as a refined interpolation formula of the big exponential map.