$(\phi ,\mathrm{\Gamma })$-modules de de Rham et fonctions $L$ $p$-adiques

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.