The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank

Alina Carmen Cojocaru; Andrew Michael Shulman

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arXiv · 1304.2100

The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank

Alina Carmen Cojocaru·Andrew Michael Shulman

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Abstract

Let $ψ$ be a generic Drinfeld module of rank $r \geq 2$. We study the first elementary divisor $d_{1, \wp}(ψ)$ of the reduction of $ψ$ modulo a prime $\wp$, as $\wp$ varies. In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (ψ)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $ψ$ modulo $\wp$ and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. Serre 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.

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Alina Carmen Cojocaru, Andrew Michael Shulman. 2013-11-12. The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank. https://arxiv.org/abs/1304.2100

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