Perfect unary forms over real quadratic fields

Yasaki, Dan

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

Soit $F=\mathbb{Q}\left(\sqrt{d}\right)$ un corps quadratique réel avec anneau d’entiers $𝒪$. Dans cet article, nous analysons le nombre $h_d$ de ${GL}_1\left(𝒪\right)$-orbites de classes d’homothétie des formes parfaites unaires sur $F$ en fonction de $d$. Nous calculons $h_d$ exactement pour $d\le 200000$, 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.

Let $F=\mathbb{Q}\left(\sqrt{d}\right)$ be a real quadratic field with ring of integers $𝒪$. In this paper we analyze the number $h_d$ of ${GL}_1\left(𝒪\right)$-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.

[1] Eva Bayer-Fluckiger and Gabriele Nebe, On the Euclidean minimum of some real number fields. J. Théor. Nombres Bordeaux, 17(2) (2005), 437–454. | Numdam | MR | Zbl

[2] Paul E. Gunnells and Dan Yasaki, Hecke operators and Hilbert modular forms. In Algorithmic number theory, volume 5011 of Lecture Notes in Comput. Sci., pages 387–401. Springer, Berlin, 2008. | MR | Zbl

[3] A. Hurwitz, Ueber die Reduction der binären quadratischen Formen. Math. Ann., 45(1) (1894), 85–117. | MR

[4] Max Koecher, Beiträge zu einer Reduktionstheorie in Positivitätsbereichen. I. Math. Ann., 141 (1960), 384–432. | MR | Zbl

[5] Alar Leibak, The complete enumeration of binary perfect forms over the algebraic number field $\mathbb{Q}\left(\sqrt{6}\right)$. Proc. Estonian Acad. Sci. Phys. Math., 54(4) (2005), 212–234. | MR | Zbl

[6] Trygve Nagel, Zur arithmetik der polynome. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1 (1922), 178–193. 10.1007/BF02940590. | MR

[7] Heidrun E. Ong, Perfect quadratic forms over real-quadratic number fields. Geom. Dedicata, 20(1) (1986), 51–77. | MR | Zbl

[8] Kenji Okuda and Syouji Yano, A generalization of Voronoï’s theorem to algebraic lattices. J. Théor. Nombres Bordeaux, 22(3) (2010), 727–740. | Numdam | MR | Zbl

[9] Kenneth H. Rosen, Elementary number theory and its applications. Pearson, Reading, MA, sixth edition, 2010. | Zbl

[10] Achill Schürmann, Enumerating perfect forms. In Quadratic forms—algebra, arithmetic, and geometry, volume 493 of Contemp. Math., pages 359–377. Amer. Math. Soc., Providence, RI, 2009. | MR | Zbl

[11] François Sigrist, Cyclotomic quadratic forms. J. Théor. Nombres Bordeaux, 12(2) (2000), 519–530. Colloque International de Théorie des Nombres (Talence, 1999). | Numdam | MR | Zbl

[12] G. Voronoǐ, Sur quelques propriétés des formes quadratiques positives parfaites. J. Reine Angew. Math., 133 (1908), 97–178.

[13] Takao Watanabe, Syouji Yano, and Takuma Hayashi, Voronoï’s reduction theory of ${\mathrm{GL}}_n$ over a totally real field. Preprint at http://www.math.sci.osaka-u.ac.jp/~twatanabe/voronoireduction.pdf. | MR