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On Russell-Type Modular Equations |
Canad. J. Math. 52(2000), 31-46
Published:2000-02-01
Printed: Feb 2000
Printed: Feb 2000
- Heng Huat Chan
- Wen-Chin Liaw
Abstract
In this paper, we revisit Russell-type modular equations, a
collection of modular equations first studied systematically by
R.~Russell in 1887. We give a proof of Russell's main theorem and
indicate the relations between such equations and the constructions
of Hilbert class fields of imaginary quadratic fields. Motivated by
Russell's theorem, we state and prove its cubic analogue which
allows us to construct Russell-type modular equations in the theory
of signature~$3$.
| MSC Classifications: |
33D10 - unknown classification 33D10 33C05 - Classical hypergeometric functions, ${}_2F_1$ 11F11 - Holomorphic modular forms of integral weight |