The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1326-1357
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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.
Cojocaru, Alina Carmen; Shulman, Andrew Michael. The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank. Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1326-1357. doi: 10.4153/CJM-2015-006-9
@article{10_4153_CJM_2015_006_9,
author = {Cojocaru, Alina Carmen and Shulman, Andrew Michael},
title = {The {Distribution} of the {First} {Elementary} {Divisor} of the {Reductions} of a {Generic} {Drinfeld} {Module} of {Arbitrary} {Rank}},
journal = {Canadian journal of mathematics},
pages = {1326--1357},
year = {2015},
volume = {67},
number = {6},
doi = {10.4153/CJM-2015-006-9},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-006-9/}
}
TY - JOUR AU - Cojocaru, Alina Carmen AU - Shulman, Andrew Michael TI - The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank JO - Canadian journal of mathematics PY - 2015 SP - 1326 EP - 1357 VL - 67 IS - 6 UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-006-9/ DO - 10.4153/CJM-2015-006-9 ID - 10_4153_CJM_2015_006_9 ER -
%0 Journal Article %A Cojocaru, Alina Carmen %A Shulman, Andrew Michael %T The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank %J Canadian journal of mathematics %D 2015 %P 1326-1357 %V 67 %N 6 %U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2015-006-9/ %R 10.4153/CJM-2015-006-9 %F 10_4153_CJM_2015_006_9
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