# $(\varphi, \Gamma)$-modules de de Rham et fonctions $L$ $p$-adiques

## TL;DR

This paper develops a method to construct $p$-adic $L$-functions from de Rham $p$-adic Galois representations, extending their analytic properties and establishing a functional equation in dimension two.

## Contribution

It introduces a new approach to constructing $p$-adic $L$-functions for de Rham representations, including cases with additive bad reduction and supercuspidal forms.

## Key findings

- Constructs $p$-adic $L$-functions for elliptic curves with bad reduction
- Provides $p$-adic $L$-functions for supercuspidal modular forms
- Proves a functional equation for $p$-adic $L$-functions in dimension two

## Abstract

We develop a variant of Coleman and Perrin Riou's methods giving, for a de Rham $p$-adic Galois representation, a construction of $p$-adic $L$ functions from a compatible system of global elements. As a result, we construct analytic functions on an open set of the $p$-adic weight space containing all locally algebraic characters of large enough conductor. Applied to Kato's Euler system, this gives $p$-adic $L$-functions for elliptic curves with additive bad reduction and, more generally, for modular forms which are supercuspidal at $p$. In the case of dimension $2$, we prove a functional equation for our $p$-adic $L$-functions.