##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Christian Klevdal

#### University of Utah

## Integrality of G-local systems

##### Abstract:

Simpson conjectured that for a reductive group $G$, rigid $G$-local systems on a smooth projective complex variety are integral. I will discuss a proof of integrality for cohomologically rigid $G$-local systems. This generalizes and is inspired by work of Esnault and Groechenig for $GL_n$. Surprisingly, the main tools used in the proof (for general $G$ and $GL_n$) are the work of L. Lafforgue on the Langlands program for curves over function fields, and work of Drinfeld on companions of $\ell$-adic sheaves. The major differences between general $G$ and $GL_n$ are first to make sense of companions for $G$-local systems, and second to show that the monodromy group of a rigid G-local system is semisimple. \\ \\ All work is joint with Stefan Patrikis.

Host: Kiran Kedlaya

### April 29, 2021

### 2:00 PM

### Location: See https://www.math.ucsd.edu/\~{}nts/

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