Perfect unary forms over real quadratic fields

Dan Yasaki[1]

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

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 3, page 759-775
  • ISSN: 1246-7405

Abstract

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Let F = ℚ ( 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 ≤ 200000 . By relating perfect forms to continued fractions, we give bounds on h d and address some questions raised by Watanabe, Yano, and Hayashi.

How to cite

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Yasaki, Dan. "Perfect unary forms over real quadratic fields." Journal de Théorie des Nombres de Bordeaux 25.3 (2013): 759-775. <http://eudml.org/doc/275681>.

@article{Yasaki2013,
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.},
affiliation = {Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA},
author = {Yasaki, Dan},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {quadratic forms; perfect forms; continued fractions; real quadratic fields; continued fraction},
language = {eng},
month = {11},
number = {3},
pages = {759-775},
publisher = {Société Arithmétique de Bordeaux},
title = {Perfect unary forms over real quadratic fields},
url = {http://eudml.org/doc/275681},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Yasaki, Dan
TI - Perfect unary forms over real quadratic fields
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/11//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 3
SP - 759
EP - 775
AB - 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.
LA - eng
KW - quadratic forms; perfect forms; continued fractions; real quadratic fields; continued fraction
UR - http://eudml.org/doc/275681
ER -

References

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  9. Kenneth H. Rosen, Elementary number theory and its applications. Pearson, Reading, MA, sixth edition, 2010. Zbl0964.11002
  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. Zbl1209.11063MR2537111
  11. François Sigrist, Cyclotomic quadratic forms. J. Théor. Nombres Bordeaux, 12(2) (2000), 519–530. Colloque International de Théorie des Nombres (Talence, 1999). Zbl0977.11029MR1823201
  12. G. Voronoǐ, Sur quelques propriétés des formes quadratiques positives parfaites. J. Reine Angew. Math., 133 (1908), 97–178. Zbl38.0261.01
  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. Zbl1298.11067MR3074816

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