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On the computation of Hermite-Humbert constants for real quadratic number fields. (English) Zbl 1098.11063

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, 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,…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 \(\gamma_{K}\).

MSC:

11Y40 Algebraic number theory computations
11R11 Quadratic extensions
11H55 Quadratic forms (reduction theory, extreme forms, etc.)

Citations:

Zbl 1139.11321

Software:

KANT/KASH
PDFBibTeX XMLCite
Full Text: DOI Numdam Numdam EuDML

References:

[1] R. Baeza, R. Coulangeon, M.I. Icaza, M. O’ Ryan, Hermite’s constant for quadratic number fields. Experimental Mathematics 10 (2001), 543-551. · Zbl 1042.11045
[2] R. Coulangeon, Voronoï theory over algebraic number fields. Monographies de l’Enseignement Mathématique 37 (2001), 147-162. · Zbl 1139.11321
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[4] H. Cohn, On the shape of the fundamental domain of the Hilbert modular group. Proc. Symp. Pure Math. 8 (1965), 190-202. · Zbl 0137.05702
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[6] M.I. Icaza, Hermite constant and extreme forms for algebraic number fields. J. London Math. Soc. 55 (1997), 11-22. · Zbl 0874.11047
[7] M.E. Pohst et al, The computer algebra system KASH/KANT, TU Berlin 2000,
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