Modular equations for congruence subgroups of genus zero (II)

Bumkyu Cho

doi:10.1016/j.jnt.2020.10.016

Modular equations for congruence subgroups of genus zero (II)

Bumkyu Cho

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup Γ_H(N,t) of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about Γ₁(m)⋂Γ₀(mN). Furthermore we show that the similar result holds for a certain congruence subgroup Γ of genus zero with [Γ:Γ_H(N,t)]=2. Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.

Original language	English
Pages (from-to)	48-79
Number of pages	32
Journal	Journal of Number Theory
Volume	231
DOIs	https://doi.org/10.1016/j.jnt.2020.10.016
State	Published - Feb 2022

Keywords

Kronecker's congruence relation
Modular equations
Modular polynomials

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10.1016/j.jnt.2020.10.016

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title = "Modular equations for congruence subgroups of genus zero (II)",

abstract = "We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup ΓH(N,t) of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about Γ1(m)⋂Γ0(mN). Furthermore we show that the similar result holds for a certain congruence subgroup Γ of genus zero with [Γ:ΓH(N,t)]=2. Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.",

keywords = "Kronecker's congruence relation, Modular equations, Modular polynomials",

author = "Bumkyu Cho",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2022",

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doi = "10.1016/j.jnt.2020.10.016",

language = "English",

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T1 - Modular equations for congruence subgroups of genus zero (II)

AU - Cho, Bumkyu

PY - 2022/2

Y1 - 2022/2

N2 - We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup ΓH(N,t) of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about Γ1(m)⋂Γ0(mN). Furthermore we show that the similar result holds for a certain congruence subgroup Γ of genus zero with [Γ:ΓH(N,t)]=2. Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.

AB - We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup ΓH(N,t) of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about Γ1(m)⋂Γ0(mN). Furthermore we show that the similar result holds for a certain congruence subgroup Γ of genus zero with [Γ:ΓH(N,t)]=2. Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.

KW - Kronecker's congruence relation

KW - Modular equations

KW - Modular polynomials

UR - https://www.scopus.com/pages/publications/85098650338

U2 - 10.1016/j.jnt.2020.10.016

DO - 10.1016/j.jnt.2020.10.016

M3 - Article

AN - SCOPUS:85098650338

SN - 0022-314X

VL - 231

SP - 48

EP - 79

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

Modular equations for congruence subgroups of genus zero (II)

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Keywords

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