Tohoku Mathematical Journal

On Ramanujan's cubic continued fraction as a modular function

Bumkyu Cho, Ja Kyung Koo, and Yoon Kyung Park

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Abstract

We first extend the results of Chan and Baruah on the modular equations of Ramanujan's cubic continued fraction $C(\tau)$ to all primes $p$ by finding the affine models of modular curves and then derive Kronecker's congruence relations for these modular equations. We further show that by its singular values we can generate ray class fields modulo 6 over imaginary quadratic fields and find their class polynomials after proving that $1/C(\tau)$ is an algebraic integer.

Article information

Source
Tohoku Math. J. (2), Volume 62, Number 4 (2010), 579-603.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1294170348

Digital Object Identifier
doi:10.2748/tmj/1294170348

Mathematical Reviews number (MathSciNet)
MR2768761

Zentralblatt MATH identifier
1248.11110

Subjects
Primary: 11Y65: Continued fraction calculations
Secondary: 11F11: Holomorphic modular forms of integral weight 11R37: Class field theory 11R04: Algebraic numbers; rings of algebraic integers 14H55: Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx]

Keywords
Ramanujan cubic continued fraction modular form class field theory

Citation

Cho, Bumkyu; Koo, Ja Kyung; Park, Yoon Kyung. On Ramanujan's cubic continued fraction as a modular function. Tohoku Math. J. (2) 62 (2010), no. 4, 579--603. doi:10.2748/tmj/1294170348. https://projecteuclid.org/euclid.tmj/1294170348


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