Cubic singular moduli, Ramanujan's class invariants \(\lambda_n\) and the explicit Shimura reciprocity law.
For a positive rational number \(n\), the cubic singular modulus is defined to be the unique positive number \(\alpha_n^*\) between \(0\) and \(1\) such that \[ \sqrt{n} = \frac{_2F_1(1/3,2/3;1;1-\alpha_n^*)}{_2F_1(1/3,2/3;1;\alpha_n^*)}, \] where \[ _2F_1(a,b;c;z) = \sum_{k = 0}^{\infty} \frac{(a)_k(b)_kz^k}{k!(c)_k}, \] with \(| z| < 1\) and \[ (a)_k = a(a-1)(a-2) \cdots (a+k-1). \] Computing the exact value of the cubic singular modulus has been useful in producing series representations for \(1/\pi\). In this paper the authors show that when the class group of \(\mathbb{Q}(\sqrt{-3n})\) takes the form \(\mathbb{Z}_2^t \oplus \mathbb{Z}_k\), with \(t \in \mathbb{N}\) and \(k = 4,6\), or \(8\), then \(\alpha_n^*\) can be determined explicitly. This is accomplished by exploiting the fact that \(1/\alpha_n^* = 1 + \mu_n^2\), where \[ \mu_n = \frac{\eta^6(\sqrt{-n/3})}{3\sqrt{3}\eta^6(\sqrt{-3n})}, \] and then using some algebraic number theory to determine \(\mu_n^2\). As an example the case \(n = 23\) is worked out in some detail. Their method also applies to the computation of a certain function \(\lambda_n\) considered by Ramanujan, namely \[ \lambda_n = \frac{\eta^6(\frac{1+\sqrt{-n/3}}{2})} {3\sqrt{3}\eta^6(\frac{1+\sqrt{-3n}}{2})}. \] The authors compute these \(\lambda_n\) explicitly for \(n = 73,97,193,217\), and \(241\).
scientific article; zbMATH DE number 2101038