Algebraic equations on the adèlic 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 AK be the ring of adeles of K with respect to the places on K corresponding to the divisors on V. Given a Drinfeld module \(\Phi :\mathbb{F}[t] \to End_K (\mathbb{G}_a )\) over K and a positive integer g we regard both Kg and AKg as \(\Phi \left( {\mathbb{F}_p [t]} \right)\)-modules under the diagonal action induced by Φ. For Γ ⊆ Kg a finitely generated \(\Phi \left( {\mathbb{F}_p [t]} \right)\)-submodule and an affine subvariety \(X \subseteq \mathbb{G}_a^g\) defined over K, we study the intersection of X(AK), the adelic points of X, with \(\bar \Gamma\), the closure of Γ with respect to the adelic topology, showing under various hypotheses that this intersection is no more than X(K) ∩ Γ.