Une factorisation de la cohomologie complétée et du système de Beilinson-Kato

Abstract

We show that the modular symbol $(0,\infty )$, considered as an element of the dual of Emerton's completed cohomology, interpolates Kato's Euler system at classical points, and we deduce from this a factorisation of Beilinson-Kato's system as a product of two symbols $(0,\infty )$ (an algebraic analog of Rankin's method). The proof uses the $p$-adic local Langlands correspondence for ${\mathbf{G}L}_2({\mathbf{Q}}_p)$ and Emerton's factorization of the completed cohomology of the tower of modular curves for which we provide a new proof resting upon the construction of a Kirillov model for the completed cohomology, and which we refine by imposing conditions at classical points; the existence of such a refinement is a manifestation of an analyticity property for $p$-adic periods of modular forms.

187pages, in French. This new version includes a proof of Emerton's factorization of completed cohomology (with some local conditions removed) which makes the factorization of Beilinson-Kato system more streamlined. The paper is dedicated to the memory of Jan Nekovář and Joël Bella\"ıche