On the computation of Hermite-Humbert constants for real quadratic number fields
Let \(K/{\mathbb{Q}}\) be a totally real quadratic number field, with ring of integers \({\mathcal O}_K\) and discriminant \(d_K\). Let \(S=(S_1, S_2)\) be a positive real binary Humbert form over \(K\), where \(S_1\), \(S_2\) are positive definite \(2\times 2\) real matrices. In this paper, when the class number \(h_K = 1\), the authors use 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 \(\gamma_K\) achieves a local maximum. Extreme forms have two properties: perfection and eutacticity. In 2001, \textit{R. Coulangeon} [see Euclidean lattices, spherical designs and modular forms. On the works of Boris Venkov. Genéve: L'Enseignement Mathématique. Monogr. Enseign. Math. 37, 147--162 (2001; Zbl 1139.11321)] proved that the number of extreme forms is finite up to unimodular equivalence. The main algorithm is divided into several subalgorithms that help to compute extreme Humbert forms, minimal vectors, all 3-sets, test equivalence of extreme Humbert forms,\dots The authors implement these algorithms in KANT/KASH and use them to determine the extreme Humbert forms for \({\mathbb{Q}}(\sqrt{13})\) and \({\mathbb{Q}}(\sqrt{17})\). In fact, for \({\mathbb{Q}}(\sqrt{13})\), they obtain 4000 triples, 770 Humbert forms with more than 4 minimal vectors, 3 Humbert forms with more than 4 minimal vectors up to unimodular equivalence. They compute the Hermite-Humbert constant \[ \gamma_{K, 2} = \sqrt{\frac{1476+144\sqrt{91}}{175}}=4.0353243\dots \] In the case of \({\mathbb{Q}}(\sqrt{17})\), the authors are not able to show that each extreme Humbert form has a unimodular pair of minimal vectors. But they compute extreme forms with pairs of unimodular minimal vectors. So they obtain 80000 triples, solve polynomial equations to determine Humbert forms, compute minimal vectors. They find two extreme Humbert forms giving \[ \gamma_{K}(S) = \sqrt{\frac{784+128\sqrt{34}}{225}}=2.607989300\dots \] and \[ \gamma_{K}(S) = \sqrt{\frac{1408+128\sqrt{85}}{405}}=2.527919014... \] as local maxima of \(\gamma_{K}\).
scientific article; zbMATH DE number 5016593