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# On Russell-Type Modular Equations

Published:2000-02-01
Printed: Feb 2000
• Heng Huat Chan
• Wen-Chin Liaw
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## 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 33D1033C05 - Classical hypergeometric functions, ${}_2F_1$ 11F11 - Holomorphic modular forms of integral weight