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# Ramanujan and the Modular $j$-Invariant

Published:1999-12-01
Printed: Dec 1999
• Bruce C. Berndt
• Heng Huat Chan
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## Abstract

A new infinite product $t_n$ was introduced by S.~Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about $t_n$ by establishing new connections between the modular $j$-invariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, $t_n$ generates the Hilbert class field of $\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class invariant according to H.~Weber's definition of class invariants.
 Keywords: modular functions, the Borweins' cubic theta-functions, Hilbert class fields
 MSC Classifications: 33C05 - Classical hypergeometric functions, ${}_2F_1$ 33E05 - Elliptic functions and integrals 11R20 - Other abelian and metabelian extensions 11R29 - Class numbers, class groups, discriminants