Some fundamental groups in arithmetic geometry

TL;DR

This paper discusses fundamental groups in arithmetic geometry, focusing on Deligne's finiteness theorem for b5-adic representations, crystalline versions, and the role of the geometric e9tale fundamental group in controlling crystals over various sites.

Contribution

It provides an overview of recent results on the finiteness properties of b5-adic representations and the relationship between fundamental groups and crystals in positive characteristic.

Findings

Deligne's finiteness theorem for b5-adic representations

Crystalline version of the finiteness theorem

The geometric e9tale fundamental group controls crystals

Abstract

Those are the notes for the 2015 Summer Research Institute on Algebraic Geometry. We report on Deligne's finiteness theorem for $\ell$-adic representations on smooth varieties defined over a finite field, on its crystalline version, and on how the geometric \'etale fundamental group of a smooth projective variety defined over a characteristic $p>0$ field controls crystals on the infinitesimal site and should control those on the crystalline site. v2: last results added to the report, and some typos corrected.