Modular lattices from finite projective planes

Tathagata Basak

doi:10.5802/jtnb.867

Tathagata Basak ¹

¹Department of Mathematics Iowa State University Ames, IA 50011

Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 269-279

Abstract (Original language)
Résumé

Using the geometry of the projective plane over the finite field $𝔽_{q}$ , we construct a Hermitian Lorentzian lattice $L_{q}$ of dimension $(q^{2} + q + 2)$ defined over a certain number ring $𝒪$ that depends on $q$ . We show that infinitely many of these lattices are $p$ -modular, that is, $p L_{q}^{'} = L_{q}$ , where $p$ is some prime in $𝒪$ such that ${| p |}^{2} = q$ .

The Lorentzian lattices $L_{q}$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q \equiv 3 mod 4$ is a rational prime such that $(q^{2} + q + 1)$ is norm of some element in $ℚ [\sqrt{- q}]$ , then we find a $2 q (q + 1)$ dimensional even unimodular positive definite integer lattice $M_{q}$ such that $Aut (M_{q}) \supseteq PGL (3, 𝔽_{q})$ . We find that $M_{3}$ is the Leech lattice.

En utilisant la géométrie du plan projectif sur un corps fini $𝔽_{q}$ , nous construisons un réseau hermitien de type Lorentz $L_{q}$ de dimension $(q^{2} + q + 2)$ defini sur un certain anneau d’entiers $𝒪$ dépendant de $q$ . Nous montrons qu’une infinité de ces réseaux sont $p$ -modulaires, c’est-à-dire que $p L_{q}^{'} = L_{q}$ , où $p$ est un premier de $𝒪$ tel que ${| p |}^{2} = q$ .

Les réseaux lorentziens $L_{q}$ mènent parfois à la construction de réseaux définis positifs intéressants. En particulier, si $q \equiv 3 mod 4$ est tel que $(q^{2} + q + 1)$ est la norme d’un élément de $ℚ [\sqrt{- q}]$ , alors nous obtenons un réseau entier unimodulaire $M_{q}$ défini positif et de dimension paire $2 q (q + 1)$ tel que $Aut (M_{q}) \supseteq PGL (3, 𝔽_{q})$ . Nous prouvons que $M_{3}$ est le réseau de Leech.

Article information
Cite this article

Received: 2012-12-30
Accepted: 2013-09-02
Published online: 2015-03-09

MR

DOI: 10.5802/jtnb.867

Classification: 11H56, 51E20, 11E12, 11E39

Author's affiliations:

Tathagata Basak ¹

¹ Department of Mathematics Iowa State University Ames, IA 50011

Tathagata Basak. Modular lattices from finite projective planes. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 269-279. doi: 10.5802/jtnb.867

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