Some new constructions of supercuspidal mod p representations of $\mathrm{GL}_2(F)$, for a $p$-adic field $F$

Abstract: Let $F/\mathbf{Q}_p$ be a finite extension. In contrast to the situation for complex representations, very little is known about the irreducible supercuspidal mod $p$ representations of $\GL_n(F)$, except in the case $\GL_2(\mathbf{Q}_p)$. 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 $\GL_2(F)$ with $\GL_2(\mathcal{O}_F)$-socle consistent with the Breuil-Mézard 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 shed light on the local Langlands correspondence for $r$. We then discuss works in progress with Ariel Weiss and with Reem Waxman that aim to give a "correct" generalization of the Breuil-Paskunas construction. A new feature is that we work with the category of mod $p$ representations of $\GL_2(R)$, 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)