Effective isotrivial Mordell-Lang in positive characteristic

TL;DR

This paper makes the isotrivial Mordell-Lang theorem effective for a broader class of algebraic groups over finite fields, using automata theory to describe intersection sets and solve key diophantine problems.

Contribution

It extends the effective description of $X igcap \Gamma$ to arbitrary commutative algebraic groups and finitely generated modules, employing automata for concrete characterization.

Findings

Automata provide a concrete description of $X igcap \Gamma$.

A dichotomy theorem for point growth in $X(K)$ is established.

Decision procedures for nonemptiness, infiniteness, and coset presence are developed.

Abstract

The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\cap\Gamma$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $\Gamma$ is a finitely generated subgroup of $G$ that is invariant under the $q$-power Frobenius endomorphism $F$. That description is here made effective, and extended to arbitrary commutative algebraic groups $G$ and arbitrary finitely generated $\mathbb{Z}[F]$-submodules $\Gamma$. The approach is to use finite automata to give a concrete description of $X\cap \Gamma$. These methods and results have new applications even when specialised to the case when $G$ is an abelian variety over a finite field, $X\subseteq G$ a subvariety defined over a function field $K$, and $\Gamma=G(K)$. As an application of the automata-theoretic approach, a dichotomy theorem is established for the growth of the number of points in $X(K)$ of bounded height. As an application of the effective description of $X\cap\Gamma$, decision procedures are given for the following three diophantine problems: Is $X(K)$ nonempty? Is it infinite? Does it contain an infinite coset?