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.