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The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
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- Journal:
- Canadian Journal of Mathematics / Volume 67 / Issue 6 / 01 December 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1326-1357
- Print publication:
- 01 December 2015
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- Article
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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.
