Cubic singular moduli, Ramanujan’s class invariants λ_nand the explicit Shimura Reciprocity Law

Schläfli modular equations for generalized Weber functions

“…Ramanujan's various claims on signature three invariants were established by Berndt, Chan, Kang, and Zhang [4]. Further values were established by Chan, Gee, and Tan [7], and by Chan, Liaw and Tan [8].…”

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

“…We assume that In [14], this empirical method will be put on a firm foundation, and it will be shown that λ n can be explicitly determined when n ≡ 1 (mod 4), 3 n and the class group of Q( √ −3n) is of the type Z 2 ⊕ Z 2 ⊕ · · · ⊕ Z 2 ⊕ Z 4 . This result is analogous to that associated with the Ramanujan-Weber class invariant G n proved in [13].…”

“…The class group of Q( √ −579) is Z 8 , and so it does not belong to the set of n's that we discuss here. However, we know that [14] t := λ 193 + λ A rigorous proof of (7.1) can be found in [14].…”

“…The purpose of this paper is to establish all the values of λ n in (1.2), including the ones that are not explicitly stated by Ramanujan, by using the modular j-invariant, modular equations, Kronecker's limit formula, and an empirical approach. Applications of values for λ n will appear in papers by Chan, W.-C. Liaw, and V. Tan [16], Chan, A. Gee, and Tan [14], and by Berndt and Chan [6].…”

“…Through Section 6, all values of λ n in (1.2) are calculated except for n = 73, 97, 193, 217, 241. In Section 7, we employ an empirical process, analogous to that employed by G. N. Watson [29], [30] in his calculations of class invariants, to determine λ n for these remaining values of n. This empirical method has been put on a firm foundation by Chan, A. Gee and V. Tan [14]. Their method works whenever 3 n, n is squarefree, and the class group of Q( √ −3n) takes the form Z 2 ⊕ Z 2 ⊕ · · · ⊕ Z 2k , with 1 ≤ k ≤ 4.…”