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Ramanujan and the Modular j-Invariant
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- Journal:
- Canadian Mathematical Bulletin / Volume 42 / Issue 4 / 01 December 1999
- Published online by Cambridge University Press:
- 20 November 2018, pp. 427-440
- Print publication:
- 01 December 1999
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- Article
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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 
$\mathbb{Q}\left( \sqrt{-n} \right)$. This shows that ${{t}_{n}}$ is a new class invariant according to H. Weber’s definition of class invariants.
