Iwasawa theory of de Rham (\phi,\Gamma)-modules over the Robba rings

TL;DR

This paper develops a new framework for exponential maps in p-adic Hodge theory using (,)-modules over the Robba ring, generalizing existing constructions without relying on Fontaine's period rings.

Contribution

It generalizes Bloch-Kato and Perrin-Riou exponential maps to all de Rham (,)-modules over the Robba ring, avoiding Fontaine's rings and proving a determinant formula.

Findings

Generalized exponential maps for all de Rham (,)-modules

Interpolated Bloch-Kato and dual exponential maps

Proved a determinant formula extending Perrin-Riou's (V)-conjecture

Abstract

The aim of this article is to study Bloch-Kato's exponential map and Perrin-Riou's "big" exponential map purely in terms of (\phi,\Gamma)-modules over the Robba ring. We first generalize the definition of Bloch-Kato's exponential map for all the (\phi,\Gamma)-modules without using Fontaine's rings B_{cris}, B_{dR} of p-adic periods and then we generalize the construction of Perrin-Riou's "big" exponential map for all the de Rham (\phi,\Gamma)-modules and prove that this map interpolates our Bloch-Kato's exponential map and the dual exponential map. Finally, we prove a theorem concerning to the determinant of our "big" exponential map, which is a generalization of Perrin-Riou's \delta(V)-conjecture. The key ingredients for our study are Pottharst's theory of analytic Iwasawa cohomology and Berger's construction of p-adic differential equations associated to de Rham (\phi,\Gamma)-modules.