CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Ramanujan and the Modular $j$-Invariant

  Published:1999-12-01
 Printed: Dec 1999
  • Bruce C. Berndt
  • Heng Huat Chan
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

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 modular functions, the Borweins' cubic theta-functions, Hilbert class fields
MSC Classifications: 33C05, 33E05, 11R20, 11R29 show english descriptions Classical hypergeometric functions, ${}_2F_1$
Elliptic functions and integrals
Other abelian and metabelian extensions
Class numbers, class groups, discriminants
33C05 - Classical hypergeometric functions, ${}_2F_1$
33E05 - Elliptic functions and integrals
11R20 - Other abelian and metabelian extensions
11R29 - Class numbers, class groups, discriminants
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/