The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank

Abstract

AbstractLet ψ be a generic Drinfeld module of rank r ≥ 2. We study the first elementary divisor d1,℘ (ψ) of the reduction of ψ modulo a prime ℘, as ℘ varies. In particular, we prove the existence of the density of the primes ℘ for which d1,℘ (ψ) is fixed. For r = 2, we also study the second elementary divisor (the exponent) of the reduction of ψ modulo ℘ and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study 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.