scholarly journals Class invariants from a new kind of Weber-like modular equation

2015 ◽  
Vol 40 (2) ◽  
pp. 413-446 ◽  
Author(s):  
William B. Hart
Keyword(s):  
Modular Equation ◽  
Class Invariants

scholarly journals The modular equation and modular forms of weight one

1985 ◽  
Vol 100 ◽  
pp. 145-162 ◽  
Author(s):  
Toyokazu Hiramatsu ◽  
Yoshio Mimura
Keyword(s):  
Modular Forms ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Algebraic Integer ◽  
Complex Multiplication ◽  
Cusp Forms ◽  
Imaginary Quadratic Fields ◽  
Modular Equation ◽  
Class Invariants ◽  
Weight One

This is a continuation of the previous paper [8] concerning the relation between the arithmetic of imaginary quadratic fields and cusp forms of weight one on a certain congruence subgroup. Let K be an imaginary quadratic field, say K = with a prime number q ≡ − 1 mod 8, and let h be the class number of K. By the classical theory of complex multiplication, the Hubert class field L of K can be generated by any one of the class invariants over K, which is necessarily an algebraic integer, and a defining equation of which is denoted byΦ(x) = 0.

The Field Generated by the Discriminant of the Class Invariants of an Imaginary Quadratic Field

1983 ◽  
Vol 26 (3) ◽  
pp. 280-282 ◽  
Author(s):  
D. S. Dummit ◽  
R. Gold ◽  
H. Kisilevsky
Keyword(s):  
Quadratic Field ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Square Root ◽  
Modular Equation ◽  
Class Invariants ◽  
Special Value

AbstractThis note determines the quadratic field generated by the square root of the discriminant of the modular equation satisfied by the special value j(α) of the modular function α for a an integer in an imaginary quadratic field.

scholarly journals On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Congruence Subgroups ◽  
Maass Form ◽  
Weakly Holomorphic Modular Forms ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Higher Genus ◽  
Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

Inequalities and infinite product formula for Ramanujan generalized modular equation function

2017 ◽  
Vol 46 (1) ◽  
pp. 189-200 ◽  
Author(s):  
Miao-Kun Wang ◽  
Yong-Min Li ◽  
Yu-Ming Chu
Keyword(s):  
Infinite Product ◽  
Product Formula ◽  
Modular Equation

Computing Polynomials of the Ramanujan tn Class Invariants

2009 ◽  
Vol 52 (4) ◽  
pp. 583-597 ◽  
Author(s):  
Elisavet Konstantinou ◽  
Aristides Kontogeorgis
Keyword(s):  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Class Field ◽  
Minimal Polynomials ◽  
Hilbert Polynomials ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Hilbert Class Field

AbstractWe compute the minimal polynomials of the Ramanujan values tn, where n ≡ 11 mod 24, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field and have much smaller coefficients than the Hilbert polynomials.

scholarly journals Introductory remarks on complex multiplication

1982 ◽  
Vol 5 (4) ◽  
pp. 675-690 ◽  
Author(s):  
Harvey Cohn
Keyword(s):  
Number Theory ◽  
Complex Multiplication ◽  
Elliptic Integrals ◽  
Modular Equation ◽  
Advanced Form ◽  
Classical Mathematics ◽  
Active Interest

Complex multiplication in its simplest form is a geometric tiling property. In its advanced form it is a unifying motivation of classical mathematics from elliptic integrals to number theory; and it is still of active interest. This interrelation is explored in an introductory expository fashion with emphasis on a central historical problem, the modular equation betweenj(z)andj(2z).

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