On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field

TL;DR

This paper proves that the pro-semisimple completion of the fundamental group of a smooth variety over a finite field is independent of the chosen non-Archimedean place, extending to a crystalline setting for curves, based on Langlands conjectures.

Contribution

It establishes the independence of the pro-semisimple completion from the place $\lambda$, including a crystalline generalization for curves, using Langlands correspondences and a reconstruction theorem.

Findings

Pro-semisimple completion is independent of $\lambda$.

Crystalline generalization for curves.

Formulation of reciprocity conjectures involving all $l$-adic cohomologies.

Abstract

Let $\Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $\lambda$ of the field of algebraic numbers which is prime to p, consider the $\lambda$-adic pro-semisimple completion of $\Pi$ as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on $\lambda$. If dim X=1 we also prove a crystalline generalization of this fact. We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its crystalline analog (proved by T. Abe) using a reconstruction theorem in the spirit of Kazhdan-Larsen-Varshavsky. We also formulate two related conjectures, each of which is a "reciprocity law" involving a sum over all $l$-adic cohomology theories (including the crystalline theory for $l=p$).