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

2013/04/08 by Alina Carmen Cojocaru, Cojocaru, Alina Carmen, Andrew Michael Shulman +1
Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Meromorphic and Entire Functions #African history and culture studies

paper · pdf · doi:10.48550/arxiv.1304.2100

Abstract

Let \ψ be a generic Drinfeld module of rank r \≥ 2. We study the\nfirst elementary divisor d_{1, wp}(\ψ) of the reduction of \ψ modulo a\nprime wp, as wp varies. In particular, we prove the existence of the\ndensity of the primes wp for which d_{1, wp} (\ψ) is fixed. For r =\n2, we also study the second elementary divisor (the exponent) of the reduction\nof \ψ modulo wp and prove that, on average, it has a large norm. Our\nwork is motivated by the study of J.-P. Serre of an elliptic curve analogue of\nArtin's Primitive Root Conjecture, and, moreover, by refinements to Serre's\nstudy developed by the first author and M. R. Murty.\n

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