Smallness of fundamental groups for arithmetic schemes

TL;DR

This paper proves the smallness of fundamental groups for arithmetic schemes, extending classical theorems to higher dimensions and deriving finiteness results for their representations over algebraically closed fields.

Contribution

It establishes the smallness property for fundamental groups of arithmetic schemes, generalizing the Hermite-Minkowski theorem to higher dimensions.

Findings

Proved smallness of fundamental groups for arithmetic schemes.

Extended Hermite-Minkowski theorem to higher dimensions.

Established finiteness of certain representations of fundamental groups.

Abstract

The smallness is proved of fundamental groups for arithmetic schemes. This is a higher dimensional analogue of the Hermite-Minkowski theorem. We also refer to the case of varieties over finite fields. As an application, we prove certain finiteness results of representations of the fundamental groups over algebraically closed fields.