On the Support Problem for Drinfeld Modules |
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Authors: | Anly Li |
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Institution: | 1. Department of Mathematics , Fu-Jen University , Taipei, Taiwan, P.R. China anlyli@math.fju.edu.tw |
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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 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 = End K (Φ) which is commutative and Λ = · y which is a cyclic -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 . |
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Keywords: | Drinfield modules Support problem |
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