Une factorisation de la cohomologie compl\'et\'ee et du syst\`eme de Beilinson-Kato

arXiv2104.09200MaRDI QIDQ6504457

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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 ${mathbfGL}_2({mathbfQ}_p)$ and Emerton's description of the completed cohomology 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.

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