Ramanujan and the modular j-invariant

Bruce C. Berndt; Heng Huat Chan

doi:10.4153/CMB-1999-050-1

Ramanujan and the modular j-invariant

Bruce C. Berndt, Heng Huat Chan

Research output: Contribution to journal › Article › peer-review

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 ℚ(√-n). This shows that t_n is a new class invariant according to H. Weber's definition of class invariants.

Original language	English (US)
Pages (from-to)	427-440
Number of pages	14
Journal	Canadian Mathematical Bulletin
Volume	42
Issue number	4
DOIs	https://doi.org/10.4153/CMB-1999-050-1
State	Published - Dec 1999
Externally published	Yes

Keywords

Hilbert class fields
Modular functions
The Borweins' cubic theta-functions

ASJC Scopus subject areas

Mathematics(all)

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10.4153/CMB-1999-050-1

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Ramanujan and the modular j-invariant

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