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Dan Yasaki
Perfect unary forms over real quadratic fields
Journal de théorie des nombres de Bordeaux, 25 no. 3 (2013), p. 759-775, doi: 10.5802/jtnb.854
Article PDF | Reviews MR 3179682 | Zbl 06291373
Class. Math.: 11E12
Keywords: quadratic forms, perfect forms, continued fractions, real quadratic fields

Résumé - Abstract

Let $F = \mathbb{Q}(\sqrt{d})$ be a real quadratic field with ring of integers $\mathcal{O}$. In this paper we analyze the number $h_d$ of $\operatorname{GL}_1(\mathcal{O})$-orbits of homothety classes of perfect unary forms over $F$ as a function of $d$. We compute $h_d$ exactly for square-free $d \le 200000$. By relating perfect forms to continued fractions, we give bounds on $h_d$ and address some questions raised by Watanabe, Yano, and Hayashi.

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