staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Kenji Okuda; Syouji Yano
A generalization of Voronoï’s Theorem to algebraic lattices
Journal de théorie des nombres de Bordeaux, 22 no. 3 (2010), p. 727-740, doi: 10.5802/jtnb.742
Article PDF | Reviews MR 2769341 | Zbl 1253.11072 | 1 citation in Cedram

Résumé - 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.

Bibliography

[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.  MR 3002 |  Zbl 0023.19905
[3] M.Koecher, Beitr${\rm \ddot{a}}$ge zu einer Reduktionstheorie in Positivit${\rm \ddot{a}}$tsbereichen. I. Math.Ann. 141 (1960), 384–432.  MR 124527 |  Zbl 0095.25301
[4] M.Koecher, Beitr${\rm \ddot{a}}$ge zu einer Reduktionstheorie in Positivit${\rm \ddot{a}}$tsbereichen. II. Math.Ann. 144 (1961), 175–182. MR MR0136771(25$\sharp $232)  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).