A generalization of Voronoï’s Theorem to algebraic lattices

Kenji Okuda; Syouji Yano

doi:10.5802/jtnb.742

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, Tome 22 (2010) no. 3, pp. 727-740.

Résumé
Abstract

Soient $K$ un corps de nombres et $𝒪_{K}$ l’anneau des entiers de $K$ . Dans cet article, nous prouvons un analogue du théorème de Voronoï pour les $𝒪_{K}$ -réseaux, et la finitude du nombre de classes de $𝒪_{K}$ -réseaux parfaits, à similitude près.

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.

MR Zbl

DOI : 10.5802/jtnb.742

Affiliations des auteurs :

Kenji Okuda ; Syouji Yano ¹

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

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     title = {A generalization of {Vorono{\"\i}{\textquoteright}s} {Theorem} to algebraic lattices},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {727--740},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
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     year = {2010},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.742/}
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Kenji Okuda; Syouji Yano. A generalization of Voronoï’s Theorem to algebraic lattices. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 727-740. doi : 10.5802/jtnb.742. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.742/

Bibliographie
Cité par

[1] R.Coulangeon, Voronoï Theory over Algebraic Number Fields. Monographies de l’Enseignement Mathématique 37 (2001), 147–162. | MR | Zbl

[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. | EuDML | MR | Zbl

[3] M.Koecher, Beitr $\ddot{a}$ ge zu einer Reduktionstheorie in Positivit $\ddot{a}$ tsbereichen. I. Math.Ann. 141 (1960), 384–432. | EuDML | MR | Zbl

[4] M.Koecher, Beitr $\ddot{a}$ ge zu einer Reduktionstheorie in Positivit $\ddot{a}$ tsbereichen. II. Math.Ann. 144 (1961), 175–182. MR MR0136771(25 $♯$ 232) | EuDML | MR | Zbl

[5] M.Laca, N.S.Larsen and S.Neshveyev, On Bost-Connes type systems for number fields. J.Number Theory 129 (2009), 325–338. | MR | Zbl

[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. | MR | Zbl

[7] J.Martinet, Perfect Lattices in Euclidean Spaces. Grundlehren der Mathematischen Wissenschaften 327, Springer Verlag, 2003. | MR | Zbl

[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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