Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$

TL;DR

This paper formulates conjectures on the structure of smooth mod p representations of GL_n over p-adic fields, supported by new finite length results for specific cases, advancing understanding of automorphic forms.

Contribution

It introduces conjectures on the structure of mod p automorphic representations of GL_n(K) and proves several cases for unramified quadratic extensions when n=2.

Findings

Conjectures on finite length and structure of mod p representations.

Proofs of several cases for n=2, unramified K.

New finite length results for specific cases.

Abstract

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular, when $K$ is unramified, we conjecture that they are of finite length and predict their internal structure (extensions, form of subquotients) from the structure of a certain algebraic representation of $\mathrm{GL}_n$. When $n=2$ and $K$ is unramified, we prove several cases of our conjectures, including new finite length results.