On the factorization of the discriminant of a classical modular equation

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.

Download Full-text

Introductory remarks on complex multiplication

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171282000623 ◽

1982 ◽

Vol 5 (4) ◽

pp. 675-690 ◽

Cited By ~ 4

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).

Download Full-text

III. Addition to memoir on the transformation of elliptic functions

Proceedings of the Royal Society of London ◽

10.1098/rspl.1878.0034 ◽

1878 ◽

Vol 27 (185-189) ◽

pp. 177-179

Keyword(s):

Elliptic Functions ◽

Modular Equation

I have recently succeeded in completing a theory considered in my “Memoir on the Transformation of Elliptic Functions,” Phil. Trans., t. 164 (1874), pp. 397—456, that of the septic transformation, n = 7 . We have here 1 - y /1 + y = 1- x /1 + x ( α - βx + γx 2 - δx 3 / α + βx 2 + γx 2 + δx 3 ), a solution of M dy /√1- y 2 . 1- v 2 y 2 = dx /√1 - x 2 . 1 - u 8 x 2 , where 1/M = 1 + 23/ a ; and the rations α: β: γ: δ , and the uv -modular equation are determined by the equations.

Download Full-text

A NEW CLASS OF MODULAR EQUATION FOR WEBER FUNCTIONS

International Journal of Number Theory ◽

10.1142/s1793042107000845 ◽

2007 ◽

Vol 03 (01) ◽

pp. 141-157 ◽

Cited By ~ 1

Author(s):

WILLIAM B. HART

Keyword(s):

Modular Equations ◽

Modular Equation ◽

New Class ◽

New Type

We describe the construction of a new type of modular equation for Weber functions. These bear some relationship to Weber's modular equations of the irrational kind. Numerous examples of these equations are explicitly computed. We also obtain some modular equations of the irrational kind which are not present in Weber's work.

Download Full-text