A generalization of Voronoï’s Theorem to algebraic lattices

Kenji Okuda; Syouji Yano[1]

  • [1] Department of Mathematics, Graduate School of Science, Osaka-University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 3, page 727-740
  • ISSN: 1246-7405

Abstract

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Let K be an algebraic number field and 𝒪 K the ring of integers of K . In this paper, we prove an analogue of Voronoï’s theorem for 𝒪 K -lattices and the finiteness of the number of similar isometry classes of perfect 𝒪 K -lattices.

How to cite

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Okuda, Kenji, and Yano, Syouji. "A generalization of Voronoï’s Theorem to algebraic lattices." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 727-740. <http://eudml.org/doc/116430>.

@article{Okuda2010,
abstract = {Let $K$ be an algebraic number field and $\mathcal\{O\}_\{K\}$ the ring of integers of $K$. In this paper, we prove an analogue of Voronoï’s theorem for $\mathcal\{O\}_\{K\}$-lattices and the finiteness of the number of similar isometry classes of perfect $\mathcal\{O\}_\{K\}$-lattices.},
affiliation = {Department of Mathematics, Graduate School of Science, Osaka-University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan},
author = {Okuda, Kenji, Yano, Syouji},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Voronoi's theorem; lattices on algebraic number fields; extreme, eutactic, perfect lattices},
language = {eng},
number = {3},
pages = {727-740},
publisher = {Université Bordeaux 1},
title = {A generalization of Voronoï’s Theorem to algebraic lattices},
url = {http://eudml.org/doc/116430},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Okuda, Kenji
AU - Yano, Syouji
TI - A generalization of Voronoï’s Theorem to algebraic lattices
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 3
SP - 727
EP - 740
AB - Let $K$ be an algebraic number field and $\mathcal{O}_{K}$ the ring of integers of $K$. In this paper, we prove an analogue of Voronoï’s theorem for $\mathcal{O}_{K}$-lattices and the finiteness of the number of similar isometry classes of perfect $\mathcal{O}_{K}$-lattices.
LA - eng
KW - Voronoi's theorem; lattices on algebraic number fields; extreme, eutactic, perfect lattices
UR - http://eudml.org/doc/116430
ER -

References

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  1. R.Coulangeon, Voronoï Theory over Algebraic Number Fields. Monographies de l’Enseignement Mathématique 37 (2001), 147–162. Zbl1139.11321MR1878749
  2. P.Humbert, Théorie de la réduction des formes quadratiques définies positives dans un corps algébrique K fini. Com. Math. Helv. 12 (1939–1940), 263–306. MR 2:148a. Zbl0023.19905MR3002
  3. M.Koecher, Beitr a ¨ ge zu einer Reduktionstheorie in Positivit a ¨ tsbereichen. I. Math.Ann. 141 (1960), 384–432. Zbl0095.25301MR124527
  4. M.Koecher, Beitr a ¨ ge zu einer Reduktionstheorie in Positivit a ¨ tsbereichen. II. Math.Ann. 144 (1961), 175–182. MR MR0136771(25 ♯ 232) Zbl0099.01701MR136771
  5. M.Laca, N.S.Larsen and S.Neshveyev, On Bost-Connes type systems for number fields. J.Number Theory 129 (2009), 325–338. Zbl1175.46061MR2473881
  6. A.Leibak, On additive generalization of Voronoï’s theory to algebraic number fields. Proc. Estonian Acad. Sci. Phys. Math. 54 (2005), no.4,195–211. Zbl1095.11022MR2190027
  7. J.Martinet, Perfect Lattices in Euclidean Spaces. Grundlehren der Mathematischen Wissenschaften 327, Springer Verlag, 2003. Zbl1017.11031MR1957723
  8. I.Satake, Nijikeishiki no Riron (Theory of Quadratic Forms), (in Japanese). Lectures in Mathematical Science The University of Tokyo, Graduate School of Mathematical Sciences. 22 (reprint 2003). 

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