Ramanujan and the Modular j-Invariant
Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 427-440
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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.
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
@article{10_4153_CMB_1999_050_1,
author = {Berndt, Bruce C. and Chan, Heng Huat},
title = {Ramanujan and the {Modular} {j-Invariant}},
journal = {Canadian mathematical bulletin},
pages = {427--440},
year = {1999},
volume = {42},
number = {4},
doi = {10.4153/CMB-1999-050-1},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-050-1/}
}
TY - JOUR AU - Berndt, Bruce C. AU - Chan, Heng Huat TI - Ramanujan and the Modular j-Invariant JO - Canadian mathematical bulletin PY - 1999 SP - 427 EP - 440 VL - 42 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-050-1/ DO - 10.4153/CMB-1999-050-1 ID - 10_4153_CMB_1999_050_1 ER -
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