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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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##### References:
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