TY - JOUR

T1 - The growth of the discriminant of the endomorphism ring of the reduction of a rank 2 generic Drinfeld module

AU - Cojocaru, Alina Carmen

AU - Papikian, Mihran

N1 - Funding Information:
A.C.C. was partially supported by a Collaboration Grant for Mathematicians from the Simons Foundation under Award No. 318454.M.P. was partially supported by a Collaboration Grant for Mathematicians from the Simons Foundation under Award No. 637364.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2022/8

Y1 - 2022/8

N2 - For q an odd prime power, A=Fq[T], and F=Fq(T), let ψ:A→F{τ} 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 ψ, with residue field Fp. We study the growth of the absolute value |Δp| of the discriminant of the Fp-endomorphism ring of the reduction of ψ modulo p and prove that, for all p, |Δp| grows with |p|. Moreover, we prove that, for a density 1 of primes p, |Δp| is as close as possible to its upper bound |ap2−4μpp|, where X2+apX+μpp∈A[X] is the characteristic polynomial of τdegp.

AB - For q an odd prime power, A=Fq[T], and F=Fq(T), let ψ:A→F{τ} 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 ψ, with residue field Fp. We study the growth of the absolute value |Δp| of the discriminant of the Fp-endomorphism ring of the reduction of ψ modulo p and prove that, for all p, |Δp| grows with |p|. Moreover, we prove that, for a density 1 of primes p, |Δp| is as close as possible to its upper bound |ap2−4μpp|, where X2+apX+μpp∈A[X] is the characteristic polynomial of τdegp.

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U2 - 10.1016/j.jnt.2021.03.026

DO - 10.1016/j.jnt.2021.03.026

M3 - Article

AN - SCOPUS:85111037467

VL - 237

SP - 15

EP - 39

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -