Number Theory Seminar
A.C. Cojocaru
University of Illinois at Chicago
The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank
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.
Tuesday April 9, 2013 at 1:00 PM in SEO 636