A.C. Cojocaru : The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank

Posted by Alina Carmen Cojocaru , part of the Number Theory Seminar.

At

April 9, 2013, 1 p.m.

In

SEO 636

Abstract

Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study the first elementary divisor $d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies. In particular, we obtain the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ 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 the study of J.-P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, in particular, by refinements to Serre's study developed by A.C. Cojocaru and M. R. Murty. This is joint work with Drew Shulman.