##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Jacob Lurie

#### Harvard University and UCSD

## A Riemann-Hilbert Correspondence in Characteristic p

##### Abstract:

Let k be a perfect field of characteristic p, and let Gal(k) denote the absolute Galois group of k. By a classical result of Katz, the category of finite-dimensional $F_p-vector spaces$ with an action of Gal(k) is equivalent to the category of finite-dimensional vector spaces over k with a Frobenius-semilinear automorphism. In this talk, I'll discuss some joint work with Bhargav Bhatt which generalizes Katz's result, replacing the field k by an arbitrary $F_p-scheme X$. In this case, there is a correspondence relating p-torsion etale sheaves on X to quasi-coherent sheaves on X equipped with a Frobenius-semilinear automorphism, which can be viewed as a ``mod p'' version of the Riemann-Hilbert correspondence for complex algebraic varieties.

Cristian Popescu

### December 6, 2018

### 1:00 PM

### AP&M 7321

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