Algebraic equations on the adelic closure of a Drinfeld module

2010/12/08 by Dragos Ghioca, Thomas Scanlon, Ghioca, Dragos +1
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Meromorphic and Entire Functions #math.AG #math.LO #math.NT #msc:03C98 #msc:11G09 #msc:14G17

paper · pdf · doi:10.48550/arxiv.1012.1825

arxiv created 2010/12/08 · arxiv updated 2010/12/09

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 Φ:\mathbb F[t] → End_K(\mathbb G_a) over K and a positive integer g we regard both K^g and \mathbf A_K^g as Φ(\mathbb F_p[t])-modules under the diagonal action induced by Φ. For Γ⊆ K^g a finitely generated Φ(\F_p[t])-submodule and an affine subvariety X ⊆ \bG_a^g defined over K, we study the intersection of X(\mathbf A_K), the adèlic points of X, with barΓ, the closure of Γ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than X(K) ∩ Γ.

Related

• Adelic Openness for Drinfeld Modules in Special Characteristic
• Monodromy Groups associated to Non-Isotrivial Drinfeld Modules in Generic Characteristic
• Finding Endomorphisms of Drinfeld modules
• Surjectivity of the adelic Galois Representation associated to a Drinfeld Module of prime rank
• Characterization of subfields of adelic algebras by a product formula