Algebraic equations on the adelic closure of a Drinfeld module

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 $X(K) \cap \Gamma$.