# Completed cohomology and Kato's Euler system for modular forms

## TL;DR

This paper establishes a deep connection between two constructions of $p$-adic $L$-functions for modular forms, proving their equivalence in a broad setting and providing new representation-theoretic insights into Kato's Euler system.

## Contribution

It proves the equality of elements from Kato's Euler system and modular symbols in local Iwasawa cohomology, even for supercuspidal forms, linking different approaches to $p$-adic $L$-functions.

## Key findings

- Proves equality of two cohomology elements from different constructions.
- Extends the understanding of $p$-adic $L$-functions to supercuspidal cases.
- Provides representation-theoretic descriptions of Kato's Euler system.

## 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 $GL_2(\mathbb{Q}_p)$. 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.