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On the reduction of a non-torsion point of a Drinfeld module. (English) Zbl 1144.11046
Fix a number field \(K\) and an elliptic curve \(E\) over \(K\). Let \(Q \in E(K)\) be a point of infinite order. Denote by \(Q\bmod\pi\) the reduction of \(Q\) modulo a prime \(\pi\) of \(K\) (this is well-defined for all but finitely many primes \(\pi\)). J. H. Silverman [J. Number Theory 30, No. 2, 226–237 (1988; Zbl 0654.10019)] (for \(K=\mathbb Q\)) and J. Cheon and S. Hahn [Acta Arith. 88, No. 3, 219–222 (1999; Zbl 0933.11029)] (for general \(K\)) have shown that: for all but finitely many positive integers \(n\) there exists a prime \(\pi\) such that the exact order of \(Q\bmod\pi\) is \(n\).
The present paper proves the analogous result for a non-torsion point on a Drinfeld \(F_q[T]\)-module of arbitrary rank over a global function field (not necessarily of generic characteristic). Independently, the same result has also been obtained by D. Ghioca and T. J. Tucker [Trans. Am. Math. Soc. 360, No. 9, 4863–4887 (2008; Zbl 1178.11046)].

MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
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