Bun_G minicourse: The Kottwitz conjecture

Abstract: This talk is the fourth 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.

Recordings and slides will appear here: sites.google.com/view/rampageseminar/home

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 lecture, we will give a detailed sketch of the proof of the main theorem of [HKW], building on the material in the first three lectures. The idea that the Kottwitz conjecture should follow from some form of the Lefschetz trace formula goes back to Harris in the '90s. We will try to emphasize the new ingredients which allow us to implement this idea in full generality.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

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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.

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