Ramanujan’s class invariants, Kronecker’s limit formula, and modular equations

Berndt, Bruce; Chan, Heng Huat; Zhang, Liang-Cheng

doi:10.1090/S0002-9947-97-01738-8

Ramanujan’s class invariants, Kronecker’s limit formula, and modular equations
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by Bruce C. Berndt, Heng Huat Chan and Liang-Cheng Zhang PDF

Trans. Amer. Math. Soc. 349 (1997), 2125-2173 Request permission

Abstract:

In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan’s class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker’s limit formula, the second employs modular equations, and the third uses class field theory to make Watson’s “empirical method”rigorous.

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