Geodesic
Ramanujan and the Modular j-Invariant
Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 427-440

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1999-050-1
Mots-clés : 33C05, 33E05, 11R20, 11R29, modular functions, the Borweins’ cubic theta-functions, Hilbert class fields 
Berndt, Bruce C.; Chan, Heng Huat. Ramanujan and the Modular j-Invariant. Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 427-440. doi: 10.4153/CMB-1999-050-1
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