Ramanujan and the modular j-invariant

Bruce C. Berndt, Heng Huat Chan

Research output: Contribution to journalArticlepeer-review


A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about tn 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, tn generates the Hilbert class field of ℚ(√-n). This shows that tn is a new class invariant according to H. Weber's definition of class invariants.

Original languageEnglish (US)
Pages (from-to)427-440
Number of pages14
JournalCanadian Mathematical Bulletin
Issue number4
StatePublished - Dec 1999
Externally publishedYes


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

ASJC Scopus subject areas

  • General Mathematics


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