##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Christian Klevdal

#### UCSD

## Strong independence of $\ell$ for Shimura varieties

##### Abstract:

(Joint with Stefan Patrikis.) In this talk, we discuss a strong form of independence of $\ell$ for canonical $\ell$-adic local systems on Shimura varieties, and sketch a proof of this for Shimura varieties arising from adjoint groups whose simple factors have real rank $\geq 2$. Notably, this includes all adjoint Shimura varieties which are not of abelian type. The key tools used are the existence of companions for $\ell$-adic local systems and the superrigidity theorem of Margulis for lattices in Lie groups of real rank $\geq 2$.

The independence of $\ell$ is motivated by a conjectural description of Shimura varieties as moduli spaces of motives. For certain Shimura varieties that arise as a moduli space of abelian varieties, the strong independence of $\ell$ is proven (at the level of Galois representations) by recent work of Kisin and Zhou, refining the independence of $\ell$ on the Tate module given by Deligne's work on the Weil conjectures.

### October 6, 2022

### 2:00 PM

APM 6402 and Zoom

See https://www.math.ucsd.edu/~nts

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