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Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study thefirst elementary divisor $d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo aprime $\wp$, as $\wp$ varies. In particular, we prove the existence of thedensity 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 reductionof $\psi$ modulo $\wp$ and prove that, on average, it has a large norm. Ourwork is motivated by the study of J.-P. Serre of an elliptic curve analogue ofArtin's Primitive Root Conjecture, and, moreover, by refinements to Serre'sstudy developed by the first author and M. R. Murty.

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