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

Christophe Breuil; Florian Herzig; Yongquan Hu; Stefano Morra; Benjamin Schraen

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

Number Theory 2023-10-03 v4 Representation Theory

Authors: Christophe Breuil , Florian Herzig , Yongquan Hu , Stefano Morra , Benjamin Schraen

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.

Keywords

modular representations gl_n p-adic field automorphic forms finite length unramified extension

Cite

@article{arxiv.2102.06188,
  title  = {Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$},
  author = {Christophe Breuil and Florian Herzig and Yongquan Hu and Stefano Morra and Benjamin Schraen},
  journal= {arXiv preprint arXiv:2102.06188},
  year   = {2023}
}

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

Abstract

Keywords

Cite

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