Orders in quadratic fields, II

Mollin, R. A.; Zhang, L.-C.

doi:10.3792/pjaa.69.368

Cited by 5 publications

(2 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is known [21] that each ideal class in the class group C K contains primitive ideals which are Z-modules of the form A = [a, b + Ω], where a and b are rational integers, a > 0, a|N (b + Ω), |b| ≤ a/2, a is the smallest positive integer in A and N (A) = a. Hence, Siegel's Theorem [27, p. 72], obtained from Kronecker's limit formula, can be stated as follows.…”

Section: −1mentioning

confidence: 99%

A certain quotient of eta-functions found in Ramanujan’s lost notebook

Berndt¹,

Chan

Kang

et al. 2002

Pacific J. Math.

View full text Add to dashboard Cite

In his lost notebook, Ramanujan defined a parameter λ n by a certain quotient of Dedekind eta-functions at the argument q = exp(−π n/3). He then recorded a table of several values of λ n . To prove these values (and others), we develop several methods, which include modular equations, the modular j-invariant, Kronecker's limit formula, Ramanujan's "cubic theory" of elliptic functions, and an empirical process.

show abstract

Section: −1mentioning

confidence: 99%

A certain quotient of eta-functions found in Ramanujan’s lost notebook

Berndt¹,

Chan

Kang

et al. 2002

Pacific J. Math.

View full text Add to dashboard Cite

show abstract

“…It is known [15] that each ideal class contains primitive ideals which are Z-modules of the form B = [a, b+Ω], where a and b are rational integers, a > 0, a|N(b+Ω), |b| ≤ a/2, a is the smallest positive integer in B, and N (B) = a.…”

Section: For a Fixed Non-zero Integral Idealmentioning

confidence: 99%

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

Berndt

Chan

Zhang

1997

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract