Perfect unary forms over real quadratic fields

Dan  Yasaki

doi:10.5802/jtnb.854

Dan Yasaki ¹

¹Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA

Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 3, pp. 759-775

Abstract (Original language)
Résumé

Let $F = ℚ (\sqrt{d})$ be a real quadratic field with ring of integers $𝒪$ . In this paper we analyze the number $h_{d}$ of ${GL}_{1} (𝒪)$ -orbits of homothety classes of perfect unary forms over $F$ as a function of $d$ . We compute $h_{d}$ exactly for square-free $d \leq 200000$ . By relating perfect forms to continued fractions, we give bounds on $h_{d}$ and address some questions raised by Watanabe, Yano, and Hayashi.

Soit $F = ℚ (\sqrt{d})$ un corps quadratique réel avec anneau d’entiers $𝒪$ . Dans cet article, nous analysons le nombre $h_{d}$ de ${GL}_{1} (𝒪)$ -orbites de classes d’homothétie des formes parfaites unaires sur $F$ en fonction de $d$ . Nous calculons $h_{d}$ exactement pour $d \leq 200000$ , sans carré. En reliant les formes parfaites aux fractions continues, nous donnons des bornes sur $h_{d}$ et répondons à certaines questions de Watanabe, Yano et Hayashi.

Article information
Cite this article

Received: 2012-07-09
Revised: 2013-03-03
Published online: 2014-02-17

Zbl MR

DOI: 10.5802/jtnb.854

Classification: 11E12
Keywords: quadratic forms, perfect forms, continued fractions, real quadratic fields

Author's affiliations:

Dan Yasaki ¹

¹ Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA

Dan  Yasaki. Perfect unary forms over real quadratic fields. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 3, pp. 759-775. doi: 10.5802/jtnb.854

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