Survey on some arithmetic properties of rigid local systems

Abstract: A central conjecture of Simpson predicts that complex rigid local systems on a smooth complex variety come from geometry. In the last couple of years, we proved some arithmetic consequences of it: integrality (using the arithmetic Langlands program), F-isocrystal properties, crystallinity of the underlying p-adic representation (using the Cartier operator over the Witt vectors and the Higgs-de Rham flow) (for Shimura varieties of real rank at least 2, this is the corner piece of Pila-Shankar-Tsimerman's proof of the André-Oort conjecture), weak integrality of the character variety (using de Jong's conjecture proved with the geometric Langlands program) (yielding a new obstruction for a finitely presented group to be the topological fundamental group of a smooth complex variety).

We'll survey some aspects of this (please ask if there is something on which you would like me to focus on). The talk is based mostly on joint work with Michael Groechenig, also, even if less, with Johan de Jong.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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