A Hermite-Minkowski type theorem of varieties over finite fields

TL;DR

This paper extends the classical Hermite-Minkowski theorem to higher-dimensional varieties over finite fields by proving the finiteness of certain étale coverings with bounded ramification, based on Deligne's theorem.

Contribution

It introduces a higher-dimensional analogue of Hermite-Minkowski theorem using Deligne's finiteness results on l-adic sheaves.

Findings

Finiteness of étale coverings with bounded ramification over finite fields.

Extension of classical Hermite-Minkowski theorem to higher dimensions.

Application of Deligne's theorem to geometric finiteness problems.

Abstract

As an application of P. Delgine's theorem (Esnault and Kerz in Acta Math. Vietnam. 37:531-562, 2012) on a finiteness of $l$-adic sheaves on a variety over a finite field, we show the finiteness of \'etale coverings of such a variety with given degree whose ramification bounded along an effective Cartier divisor. This can be thought of a higher dimensional analogue of the classical Hermite-Minkowski theorem.