Algebraic versus topological entropy for surfaces over finite fields

TL;DR

This paper compares algebraic and topological entropy for surface automorphisms over finite fields, showing that entropy values are determined by the span of the Néron-Severi group within -adic cohomology, paralleling complex de Rham cohomology.

Contribution

It establishes a link between entropy and the Néron-Severi group in -adic cohomology for surfaces over finite fields, extending known complex results.

Findings

Entropy is determined by the span of the Néron-Severi group in -adic cohomology.

The result parallels the complex case of de Rham cohomology.

The paper clarifies the relationship between algebraic and topological entropy in this setting.

Abstract

We show that, as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of the surface over a finite field $\F_q $ is taken on the span of the N\'eron-Severi group inside of $\ell$-adic cohomology. v2: (some) typos removed, exposition (partly) improved.