2021/04/19 by Pierre Colmez, Shanwen Wang, Colmez, Pierre +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2104.09200
openalex publication_date 2021/04/19 · openalex created_date 2021/04/26 · openalex updated_date 2026/07/28
We show that the modular symbol (0,∞), 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,∞) (an algebraic analog of Rankin's method). The proof uses the p-adic local Langlands correspondence for \mathbf GL2(\mathbf Qp) 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.