Cyclicity of finite Drinfeld modules

• Autores: Wentang Kuo, Yu-Ru Liu
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 80, Nº 3, 2009, págs. 567-584
• Idioma: inglés
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• Resumen

  • Let A = Fq[T] be the polynomial ring over the finite field Fq, let k = Fq(T) be the rational function field, and let K be a finite extension of k. For a prime P of K, we denote by OP the valuation ring of P, byMP the maximal ideal of OP, and by FP the residue field OP/MP. Let �Ó be a Drinfeld A-module over K of rank r. If �Ó has good reduction at P, let �Ó . FP denote the reduction of �Ó at P and let �Ó(FP) denote the A-module (�Ó . FP)(FP). If �Ó is of rank 2 with End �PK (�Ó) = A, then we obtain an asymptotic formula for the number of primes P of K of degree x for which �Ó(FP) is cyclic. This result can be viewed as a Drinfeld module analogue of Serre�fs cyclicity result on elliptic curves. We also show that when �Ó is of rank r  3 a similar result follows.