# Some fundamental groups in arithmetic geometry

Research paper by Hélène Esnault

Indexed on: 30 Dec '15Published on: 30 Dec '15Published in: Mathematics - Algebraic Geometry

#### 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.