On the category of finitely presented mod $p$ representations of $GL_2(F)$

TL;DR

This paper proves that the category of finitely presented smooth mod p representations of GL_2(F) over a finite extension of _p is an abelian subcategory, using amalgamated products of completed group rings.

Contribution

It establishes the abelian property of finitely presented smooth mod p representations of GL_2(F), a significant structural result in the representation theory of p-adic groups.

Findings

The category of finitely presented smooth mod p representations is abelian.

The proof employs amalgamated products of completed group rings.

This result clarifies the structure of representations over finite extensions of _p.

Abstract

Let $F$ be a finite extension of $\mathbb{Q}_p$. We prove that the category of finitely presented smooth $Z$-finite representations of $GL_2(F)$ over a finite extension of $\mathbb{F}_p$ is an abelian subcategory of the category of all smooth representations. The proof uses amalgamated products of completed group rings.