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On Russell-Type Modular Equations

Open Access article
 Printed: Feb 2000
  • Heng Huat Chan
  • Wen-Chin Liaw
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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, 33C05, 11F11 show english descriptions unknown classification 33D10
Classical hypergeometric functions, ${}_2F_1$
Holomorphic modular forms of integral weight
33D10 - unknown classification 33D10
33C05 - Classical hypergeometric functions, ${}_2F_1$
11F11 - Holomorphic modular forms of integral weight

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