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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

Alina Carmen Cojocaru  and
Andrew Michael Shulman
Alina Carmen Cojocaru
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, 60607, IL, USA. email: cojocaru@math.uic.edu Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucharest, 010702, Sector 1, Romania. email: ashulm2@uic.edu
Andrew Michael Shulman
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, 60607, IL, USA. email: cojocaru@math.uic.edu
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Abstract

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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.

Keywords

11R4511G0911R58Drinfeld modulesdensity theorems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1326 - 1357
Copyright
Copyright © Canadian Mathematical Society 2015

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