Algebraic equations on the adèlic closure of a Drinfeld module

摘要

Let k be a field of positive characteristic and K = k(V) a function field of a variety V over k and let A K be the ring of adèles 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 K g and A K g as (Phi left( {mathbb{F}_p [t]} right))-modules under the diagonal action induced by Φ. For Γ ⊆ K g 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(A K ), the adèlic points of X, with (bar Gamma), the closure of Γ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than X(K) ∩ Γ.