A generalization of Voronoï’s Theorem to algebraic lattices
Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 727-740.

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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.742
@article{JTNB_2010__22_3_727_0,
     author = {Kenji Okuda and Syouji Yano},
     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},
     number = {3},
     year = {2010},
     doi = {10.5802/jtnb.742},
     zbl = {1253.11072},
     mrnumber = {2769341},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.742/}
}
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/

[1] R.Coulangeon, Voronoï Theory over Algebraic Number Fields. Monographies de l’Enseignement Mathématique 37 (2001), 147–162. | MR 1878749 | Zbl 1139.11321

[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 138751 | MR 3002 | Zbl 0023.19905

[3] M.Koecher, Beitra ¨ge zu einer Reduktionstheorie in Positivita ¨tsbereichen. I. Math.Ann. 141 (1960), 384–432. | EuDML 160820 | MR 124527 | Zbl 0095.25301

[4] M.Koecher, Beitra ¨ge zu einer Reduktionstheorie in Positivita ¨tsbereichen. II. Math.Ann. 144 (1961), 175–182. MR MR0136771(25232) | EuDML 160888 | MR 136771 | Zbl 0099.01701

[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 2473881 | Zbl 1175.46061

[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 2190027 | Zbl 1095.11022

[7] J.Martinet, Perfect Lattices in Euclidean Spaces. Grundlehren der Mathematischen Wissenschaften 327, Springer Verlag, 2003. | MR 1957723 | Zbl 1017.11031

[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).

Cité par document(s). Sources :