The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank

Cojocaru, Alina Carmen ; Shulman, Andrew Michael

Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1326-1357

Résumé

Let $\psi$ be a generic Drinfeld module of rank $r\,\ge \,2$ . We study the first elementary divisor ${{d}_{1,\,\wp }}\,\left( \psi\right)$ of the reduction of $\psi$ modulo a prime $\wp $ , as $\wp $ varies. In particular, we prove the existence of the density of the primes $\wp $ for which ${{d}_{1,\,\wp }}\,\left( \psi\right)$ is fixed. For $r\,=\,2$ , we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp $ and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M. R. Murty.

Détail

DOI : 10.4153/CJM-2015-006-9

Mots-clés : 11R45, 11G09, 11R58, Drinfeld modules, density theorems

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