[cancelled] Supercuspidal mod $p$ representations of $\mathrm{GL}_2(F)$, beyond the generic unramified case

Abstract: Let $F / \mathbf{Q}_p$ be a $p$-adic field. In contrast to the situation for complex representations, no classification of the irreducible supercuspidal mod $p$ representations of $\mathrm{GL}_n(F)$ is known, except in the case $\mathrm{GL}_2(\mathbf{Q}_p)$. If $F / \mathbf{Q}_p$ is unramified and $r$ is a generic irreducible two-dimensional mod $p$ representation of the absolute Galois group of $F$, then nearly 15 years ago Breuil and Paskunas gave a beautiful construction of an infinite family of diagrams giving rise to supercuspidal mod $p$ representations of $\mathrm{GL}_2(F)$ with $\mathrm{GL}_2(\mathcal{O}_F)$-socle determined by Serre’s modularity conjecture for $r$. While their construction is not exhaustive, various local-global compatibility results obtained by a number of mathematicians in the intervening years indicate that it is sufficiently general to capture the mod p local Langlands correspondence for generic Galois representations.

In this talk we will review the ideas mentioned above and discuss how to move beyond them to consider ramified $p$-adic fields $F$, or non-generic representations $r$ for unramified $F$. We will describe a simple construction of supercuspidal representations for certain ramified $F$ and generic $r$; while this is the first such example for ramified $F$, it involves a breakage of symmetry that makes it unlikely to figure in the local Langlands correspondence for $r$. We then discuss works in progress with Ariel Weiss and with Reem Waxman that shed new light on Breuil-Paskunas and aim to give a “correct” generalization of their construction. A new feature is that we work with the category of mod $p$ representations of $\mathrm{GL}_2(R)$, where $R$ is a quotient ring of $\mathcal{O}_F$ that is larger than the residue field.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)