Algebraic equations on the adelic closure of a Drinfeld module

TL;DR

This paper investigates the intersection properties of algebraic equations on the adelic closure of Drinfeld modules over function fields, establishing conditions under which the adelic points intersect finitely with finitely generated modules.

Contribution

It provides new results on the intersection of adelic points with finitely generated modules in the context of Drinfeld modules, extending the understanding of algebraic equations in positive characteristic.

Findings

Under certain hypotheses, the intersection of adelic points with the closure of finitely generated modules is contained within the rational points.

The study generalizes classical results to the setting of adelic points and Drinfeld modules over function fields.

Conditions are identified where the adelic closure does not enlarge the intersection beyond rational points.

Abstract

Let $k$ be a field of positive characteristic and $K = k(V)$ a function field of a variety $V$ over $k$ and let ${\mathbf A}_K$ be a ring of ad\'{e}les of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $\Phi:{\mathbb F}[t] \to \operatorname{End}_K({\mathbb G}_a)$ over $K$ and a positive integer $g$ we regard both $K^g$ and ${\mathbf A}_K^g$ as $\Phi({\mathbb F}_p[t])$-modules under the diagonal action induced by $\Phi$. For $\Gamma \subseteq K^g$ a finitely generated $\Phi(\F_p[t])$-submodule and an affine subvariety $X \subseteq \bG_a^g$ defined over $K$, we study the intersection of $X({\mathbf A}_K)$, the ad\`{e}lic points of $X$, with $bar{\Gamma}$, the closure of $\Gamma$ with respect to the ad\`{e}lic topology, showing under various hypotheses that this intersection is no more than $X(K) \cap \Gamma$.