Perfect unary forms over real quadratic fields
Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 759-775.

Soit F=(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 d200000, 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=(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 d200000. By relating perfect forms to continued fractions, we give bounds on h d and address some questions raised by Watanabe, Yano, and Hayashi.

DOI : 10.5802/jtnb.854
Classification : 11E12
Mots clés : quadratic forms, perfect forms, continued fractions, real quadratic fields
Yasaki, Dan 1

1 Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA
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 Yasaki, Dan. Perfect unary forms over real quadratic fields. Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 759-775. doi : 10.5802/jtnb.854. http://archive.numdam.org/articles/10.5802/jtnb.854/

[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 (6). 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 GL n over a totally real field. Preprint at http://www.math.sci.osaka-u.ac.jp/~twatanabe/voronoireduction.pdf. | MR

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