Transactions of the American Mathematical Society

Author(s): Bruce C. Berndt; Heng Huat Chan; Liang-Cheng Zhang
Journal: Trans. Amer. Math. Soc. 349 (1997), 2125-2173.
MSC (1991): Primary 11R29; Secondary 11R04, 11R37, 11R42, 11F27, 33D10
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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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Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11R29, 11R04, 11R37, 11R42, 11F27, 33D10

Bruce C. Berndt
Affiliation: Department of Mathematics, 1409 West Green Street, University of Illinois, Urbana, Illinois 61801
Email: berndt@math.uiuc.edu

Heng Huat Chan
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, National Chung Cheng University, Minhsiung, Chiyai 621, Taiwan, R.O.C.
Email: hhchan@mthmp.math.ccu.edu.tw

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