scholarly journals Modular equations of a continued fraction of order six

2019 ◽  
Vol 17 (1) ◽  
pp. 202-219
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Positive Integer ◽  
Continued Fraction ◽  
Quadratic Field ◽  
Function Theory ◽  
Algebraic Integer ◽  
Modular Function ◽  
Modular Equations ◽  
Modular Equation ◽  
Explicit Procedure ◽  
Ray Class Field

Abstract We study a continued fraction X(τ) of order six by using the modular function theory. We first prove the modularity of X(τ), and then we obtain the modular equation of X(τ) of level n for any positive integer n; this includes the result of Vasuki et al. for n = 2, 3, 5, 7 and 11. As examples, we present the explicit modular equation of level p for all primes p less than 19. We also prove that the ray class field modulo 6 over an imaginary quadratic field K can be obtained by the value X2 (τ). Furthermore, we show that the value 1/X(τ) is an algebraic integer, and we present an explicit procedure for evaluating the values of X(τ) for infinitely many τ’s in K.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

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.

scholarly journals On a problem of Hasse and Ramachandra

2019 ◽  
Vol 17 (1) ◽  
pp. 131-140
Author(s):  
Ja Kyung Koo ◽  
Dong Hwa Shin ◽  
Dong Sung Yoon
Keyword(s):  
Positive Integer ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Integral Ideal ◽  
Weber Function ◽  
Class Field ◽  
Ray Class Field

Abstract Let K be an imaginary quadratic field, and let 𝔣 be a nontrivial integral ideal of K. Hasse and Ramachandra asked whether the ray class field of K modulo 𝔣 can be generated by a single value of the Weber function. We completely resolve this question when 𝔣 = (N) for any positive integer N excluding 2, 3, 4 and 6.

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 Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Elliptic Curves ◽  
Cusp Form ◽  
Mellin Transform ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Function Fields ◽  
Complex Multiplication ◽  
Modular Function ◽  
Imaginary Quadratic Field

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

scholarly journals Solvability of the diophantine equation x2 − Dy2 = ± 2 and new invariants for real quadratic fields

1994 ◽  
Vol 134 ◽  
pp. 137-149 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
Real Quadratic

In our recent papers [3, 4, 5], we defined some new D-invariants for any square-free positive integer D and considered their properties and interrelations among them. Especially, as an application of it, we discussed in [5] the characterization of real quadratic field Q() of so-called Richaud-Degert type in terms of these new D-invariants.

scholarly journals On Ramanujan's cubic continued fraction as a modular function

2010 ◽  
Vol 62 (4) ◽  
pp. 579-603 ◽  
Author(s):  
Bumkyu Cho ◽  
Ja Kyung Koo ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Modular Function

scholarly journals A NEW CLASS OF MODULAR EQUATION FOR WEBER FUNCTIONS

2007 ◽  
Vol 03 (01) ◽  
pp. 141-157 ◽  
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.

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