Completed cohomology and Kato's Euler system for modular forms

2018/12/08 by Yiwen Zhou, Zhou, Yiwen
Mathematics · Medicine · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Leprosy Research and Treatment #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1812.03272

openalex publication_date 2018/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we compare two different constructions of p-adic L-functions for modular forms and their relationship to Galois cohomology: one using Kato's Euler system and the other using Emerton's p-adically completed cohomology of modular curves. At a more technical level, we prove the equality of two elements of a local Iwasawa cohomology group, one arising from Kato's Euler system, and the other from the theory of modular symbols and p-adic local Langlands correspondence for GL₂(ℚ_p). We show that this equality holds even in the cases when the construction of p-adic L-functions is still unknown (i.e. when the modular form f is supercuspidal at p). Thus, we are able to give some representation-theoretic descriptions of Kato's Euler system.

Related

• Une factorisation de la cohomologie complétée et du système de Beilinson-Kato
• Relative completed cohomologies and modular symbols
• Etale cohomology, cofinite generation, and p-adic L-functions
• Iwasawa theory of overconvergent modular forms, I: Critical p-adic L-functions
• Refined applications of Kato's Euler systems for modular forms