Local–global problem for Drinfeld modules

Abstract

Let K be a function field with an A-algebra structure. The ring A arises in the definition of the Drinfeld module φ over K. By E( K) we denote K together with the A-module structure induced on it by φ. For any principal prime ideal ( a)⊂ A, we study the question whether an element x∈ E( K) which is an a-fold in E( K ν ) for every place ν of K, is an a-fold in E( K). In particular, we study the group S(a,K)≔ ker E(K)/aE(K)→ ∏ ν E(K ν)/aE(K ν) for Drinfeld modules of rank 2. We show that this finite group is trivial in many cases, but can become arbitrarily large.