Sur la pr\'esentation des repr\'esentations supersinguli\`eres de $\mathrm{GL}_2(F)$

TL;DR

This paper proves that smooth irreducible supersingular representations with a central character of 2(F) are not of finite presentation, highlighting a fundamental property of these representations in the context of quadratic extensions.

Contribution

It establishes a new fundamental property of supersingular representations of 2(F), showing they lack finite presentation, which was previously unknown.

Findings

Supersingular representations are not of finite presentation.

The result applies specifically to 2(F) where F is a quadratic extension of 3.

This deepens understanding of the structure of supersingular representations.

Abstract

Let $F$ be a quadratic extension of $\mathbb{Q}_p$. We prove that smooth irreducible supersingular representations with central character of $\mathrm{GL}_2(F)$ are not of finite presentation.