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

JOURNAL OF NUMBER THEORY

卷:237

Article

Cojocaru, Alina Carmen^1,2  Papikian, Mihran³ 

[1] Univ Illinois, Dept Math Stat & Comp Sci, 851 S Morgan St,322 SEO, Chicago, IL 60607 USA

[2] Romanian Acad, Inst Math Sim Stoilow, Sect 1, 21 Calea Grivitei St, Bucharest 010702, Romania

[3] Penn State Univ, Dept Math, University Pk, PA 16802 USA

关键词: Drinfeld modules;    Orders;    Endomorphism rings;   

DOI  :  10.1016/j.jnt.2021.03.026

来源: Elsevier

【 摘 要 】

For q an odd prime power, A = F-q[T], and F = F-q(T), let psi : A -> F{tau} be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let p = pA be a prime of A of good reduction for psi, with residue field F-p. We study the growth of the absolute value |Delta(p)| of the discriminant of the F-p-endomorphism ring of the reduction of psi modulo p and prove that, for all p, |Delta(p)| grows with |p|. Moreover, we prove that, for a density 1 of primes p, |Delta(p)| is as close as possible to its upper bound |a(p)(2) - 4 mu(p)p|, where X-2 + a(p)X + mu(p)p is an element of A[X] is the characteristic polynomial of tau(deg p). (c) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free