Bun_G minicourse: Lefschetz formula for diamonds

Abstract: This talk is the third part of a six-part series "$\mathrm{Bun}_G$, Shtukas, and the Local Langlands Program", held Tuesdays and Thursdays between 5 and 21 October, 2021.

Series abstract: The recent manuscript of Fargues-Scholze aims to "geometrize" the Langlands program for a p-adic group $G$, by relating the players in that story to the stack $\mathrm{Bun}_G$. Following a strategy of V. Lafforgue, the main result of [FS] is the construction of an L-parameter attached to a smooth irreducible representation of $G$.

The goal of this series is to review the main ideas of this work, and to discuss two related results: progress on the Kottwitz conjecture for local shtuka spaces by Hansen-Kaletha-Weinstein, and the construction of eigensheaves on $\mathrm{Bun}_G$ when $G=\mathrm{GL}_n$ by Anschütz-le Bras.

Talk abstract: In this talk we will discuss a very general form of the Lefschetz-Verdier trace formula which applies to stacks (both of schemes and of diamonds). As an application, we will show that if a locally pro-$p$ group $G$ acts on a proper diamond $X$, and if $A$ is a $G$-equivariant $\ell$-adic sheaf on $X$ which is "dualizable" (= universally locally acyclic), then the cohomology $R\Gamma(X,A)$ is an admissible representation of $G$, whose Harish-Chandra distribution can be computed in terms of local terms living on the fixed-point locus of $G$ on $X$.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

Series comments: The Zoom Meeting ID is: 995 3670 1681. The password for the series is: *The first three-digit prime*. Please visit the external homepage for notes and videos from past talks.