## Elementary divisors of reductions of generic Drinfeld modules

##### Abstract

Let q be a power of an odd prime, $A := F_q[T]$, and $k := F_q(T)$. Let $\psi$ be a Drinfeld $A$-
module over a fi nite extension $K$ of $k$ of rank $r \geq 2$. Let $\wp$ be a prime of $K$, of good reduction for $\psi$, $\F_{\wp}$ the residue fi eld at $\wp$, and consider the reduced Drinfeld $A$-module $\psi \otimes \wp$ over $\F_{\wp}$. The $A$-module action of $\psi \otimes \wp$ on $\F_{\wp}$, denote $\F_{\wp}$, makes $\F_{\wp}$ isomorphic, as an $A$-module, to $A / d_{1,\wp}(\psi)A \times \ldots \times A / d_{r,\wp}(\psi) A$ for uniquely determined monic polynomials
$d_{1,\wp}(\psi),\ldots,d_{r,\wp}(\psi)$, depending on $\psi$ and $\wp$, such that $d_{1,\wp}(\psi) | \ldots | d_{r,\wp}(\psi)$. The elements $d_{1,\wp}(\psi),\ldots,d_{r,\wp}(\psi)$ are called the elementary divisors of $\psi$ modulo $\wp$. In this thesis, we study the growth of the largest elementary divisor, $d_{r,\wp}(\psi)$, as the prime $\wp$ varies, in analogy
with a result by W. Duke pertaining to elliptic curves. We also consider the distribution of
the smallest elementary divisor, $d_{1,\wp}(\psi)$, again as $\wp$ varies, in analogy with work started by J.-P. Serre related to Lang and Trotter's elliptic curve formulation of Artin's primitive root conjecture. One of our main results is that for a density 1 of primes $\wp$ of $K$, the infinity norm of $d_{r,\wp}(\psi)$ is as large as possible. More precisely, we show that for any function $f$ de fined on the primes of $K$ with values in $A$ such that $f$ grows very slowly as the degree of the primes in $K$ increases to infinity, then almost all primes $\wp$ of $K$ satisfy $|d_{r,\wp}(\psi)|_{\ifnty} > \frac{|\wp|_{\infty}}{|f(\wp)|_{\infty}$.

##### Subject

Drinfeld moduleselliptic curves

reduction

elementary divisors