Endomorphisms of positive characteristic tori: entropy and zeta function

Abstract

Let $F$ be a finite field of order $q$ and characteristic $p$. Let ${\mathbb{Z}}_F=F[t]$, ${\mathbb{Q}}_F=F(t)$, ${\mathbb{R}}_F=F((1/t))$ equipped with the discrete valuation for which $1/t$ is a uniformizer, and let ${\mathbb{T}}_F={\mathbb{R}}_F/{\mathbb{Z}}_F$ which has the structure of a compact abelian group. Let $d$ be a positive integer and let $A$ be a $d\times d$-matrix with entries in ${\mathbb{Z}}_F$ and non-zero determinant. The multiplication-by-$A$ map is a surjective endomorphism on ${\mathbb{T}}_F^d$. First, we compute the entropy of this endomorphism; the result and arguments are analogous to those for the classical case ${\mathbb{T}}^d={\mathbb{R}}^d/{\mathbb{Z}}^d$. Second and most importantly, we resolve the algebraicity problem for the Artin-Mazur zeta function of all such endomorphisms. As a consequence of our main result, we provide a complete characterization and an explicit formula related to the entropy when the zeta function is algebraic.

Some minor changes in the introduction. This paper has been superseded by the paper arXiv:2206.00862