Automatic de Rhamness of arithmetic local systems

Abstract: It turns out that a geometrically irreducible p-adic etale local system on a smooth variety over a p-adic field can be made de Rham by twisting by a character of the Galois group of the base field. This implies that, assuming the relative Fontaine-Mazur conjecture, any "arithmetic" irreducible local system on a smooth variety over complex numbers comes from geometry.

The proof uses the p-adic Riemann-Hilbert correspondence of Liu and Zhu. Time permitting, I'll also discuss a generalization of this result to not necessarily irreducible local systems that was observed by Beilinson. In particular, it implies that the action of the Galois group on the pro-algebraic completion of the fundamental group is de Rham in the appropriate sense.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)