On the computation of Hermite-Humbert constants: the algorithm of Cohn revisited
Let \(K\) be a real quadratic number field of class number \(h_K=1\). \textit{H. Cohn} developed an algorithm for the numerical approximation of low-points in the fundamental domain for the Hilbert modular group of \(K\) [see Math. Comput. 19, 594--605 (1965; Zbl 0144.28501)]. These low-points correspond to extreme Humbert forms of real quadratic number fields. Moreover, he computed the Hermite-Humbert constants \(\gamma_{K,2}\) for the number fields \({\mathbb{Q}}(\sqrt{2})\), \({\mathbb{Q}}(\sqrt{3})\), and \({\mathbb{Q}}(\sqrt{5})\). In a previous paper, \textit{M. E. Pohst} and \textit{M. Wagner} used the theoretical background related to Humbert forms to develop an algorithm for computing extreme binary Humbert forms, i.e. those for which the Hermite-Humbert constant \(\gamma_{K,2}\) achieves a local maximum [see J. Théor. Nombres Bordx. 17, No. 3, 905--920 (2005; Zbl 1098.11063)]. Moreover, they implemented the algorithm in KANT/KASH to determine the extreme Humbert forms and compute the Hermite-Humbert constants for \({\mathbb{Q}}(\sqrt{13})\) and \({\mathbb{Q}}(\sqrt{17})\). In this paper, the authors modify the algorithm of Cohn to develop a new algorithm for the computation of Hermite-Humbert constants of real quadratic number fields \(K\) of class number one. This algorithm reduces the necessary amount of computations considerably. They use it to verify their early result for \({\mathbb{Q}}(\sqrt{13})\) and to compute the Hermite-Humbert constants of \({\mathbb{Q}}(\sqrt{6})\) and \({\mathbb{Q}}(\sqrt{21})\). Finally, they make conjectures for the Hermite-Humbert constants of \({\mathbb{Q}}(\sqrt{7})\) and \({\mathbb{Q}}(\sqrt{11})\).
scientific article; zbMATH DE number 5610513
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