On the Support Problem for Drinfeld Modules

Abstract

Let Φ be a Drinfeld A-module over an A-field K of generic characteristic. We will prove the following two results which are analogous to ones in number fields. Case 1. Φ is of rank one. Suppose that P and Q are two nontorsion points in Φ(K). If for any element a ∈ A and almost all prime ideals 𝒫 in 𝔸 one has that Φ a (P) ≡ 0 (mod 𝒫) ⇒ Φ a (Q) ≡ 0 (mod 𝒫), then Q = Φ m (P) for some m ∈ A. Case 2. Φ is of general rank ≥ 1. Let x, y ∈ Φ(K) be two K-rational points. Denote 𝒪 = End K (Φ) which is commutative and Λ = 𝒪 · y which is a cyclic 𝒪-module. Let red v :Φ(K) → Φ(k v ) be the reduction map at a place v of K with residue field k v . If red v (x) ∈ red v (Λ) for almost all places v of K. Then f(x) = g(y), for some nonzero elements f and g in 𝒪.