The growth of the discriminant of the endomorphism ring of the reduction of a rank 2 generic Drinfeld module

2020/02/22 by Cojocaru, Alina Carmen, Papikian, Mihran
#11G09 #11R29 #11R44 (Secondary) #11R58 (Primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2002.09582

Abstract

Let ψ: A → F\τ\ be a Drinfeld A-module over F of rank 2 and without complex multiplication, where A = \mathbbF_q[T], F = \mathbbF_q(T), and q is an odd prime power. For a prime \mathfrakp = p A of A of good reduction for ψ and with residue field \mathbbF_\mathfrakp, we study the growth of the absolute value |Δ_\mathfrakp| of the discriminant of the \mathbbF_\mathfrakp-endomorphism ring of the reduction of ψ modulo \mathfrakp. We prove that for all \mathfrakp, |Δ_\mathfrakp| grows with |p|. Moreover, we prove that for a density 1 of primes \mathfrakp, |Δ_\mathfrakp| is as close as possible to its upper bound |a_\mathfrakp² - 4 μ_\mathfrakpp|, where X²+a_\mathfrakpX+μ_\mathfrakp p ∈ A[X] is the characteristic polynomial of τ^{deg p}.

Related

• Drinfeld modules, Frobenius endomorphisms, and CM-liftings
• Endomorphism Rings and Isogenies Classes for Drinfeld Modules of Rank 2 Over Finite Fields
• The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank
• Endomorphism rings of reductions of Drinfeld modules
• The normal order of of the divisor-counting function for invariants of rank 2 Drinfeld modules