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The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 2015
References
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