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 adéles of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $\Phi :\mathbb{F}[t]\to {\mathrm{End}}_K({\mathbb{G}}_a)$ 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 $Φ(\F_p[t])$-submodule and an affine subvariety $X \subseteq \bG_a^g$ defined over $K$, we study the intersection of $X({\mathbf{A}}_K)$, the adèlic points of $X$, with $bar\Gamma$, the closure of $\Gamma$ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than $X(K)\cap \Gamma$.