Conjectures and results on modular representations of GL_n(K) for a p-adic field K

2021/02/11 by Christophe Breuil, Breuil, Christophe, Florian Herzig +7
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT

paper · pdf · doi:10.48550/arxiv.2102.06188

published as Mem. Amer. Math. Soc. 315 (2025), no. 1598, v+163pp · Minor updates after the referee's report

arxiv created 2023/10/02 · arxiv updated 2026/08/03

Abstract

Let p be a prime number and K a finite extension of ℚ_p. We state conjectures on the smooth representations of 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 GL_n. When n=2 and K is unramified, we prove several cases of our conjectures, including new finite length results.

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