The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank

TL;DR

This paper investigates the distribution of elementary divisors of reductions of generic Drinfeld modules, establishing density results and average size properties, extending analogies with elliptic curves and primitive root conjectures.

Contribution

It proves the existence of natural densities for primes with fixed elementary divisors and analyzes the average size of the second divisor for rank 2 modules, advancing understanding of Drinfeld modules.

Findings

Density of primes with fixed first elementary divisor established.

Average size of the second elementary divisor shown to be large for rank 2.

Results extend analogies with elliptic curves and primitive root conjectures.

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 prove the existence of 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, moreover, by refinements to Serre's study developed by the first author and M. R. Murty.