A Riemann-Hilbert Correspondence in Characteristic p
-- Jacob Lurie, October 11, 2019

<p>

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 k-vector spaces 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.