Modular lattices from finite projective planes

Basak, Tathagata

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

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, $p{L}_q^{\text{'}}=L_q$, where $p$ is some prime in $𝒪$ such that ${\left|p\right|}^2=q$.

The Lorentzian lattices $L_q$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q\equiv 3mod4$ is a rational prime such that $(q^2+q+1)$ is norm of some element in $\mathbb{Q}\left[\sqrt{-q}\right]$, then we find a $2q(q+1)$ dimensional even unimodular positive definite integer lattice $M_q$ such that $Aut\left(M_q\right)\supseteq PGL(3,{𝔽}_q)$. 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 $p{L}_q^{\text{'}}=L_q$, où $p$ est un premier de $𝒪$ tel que ${\left|p\right|}^2=q$.

Les réseaux lorentziens $L_q$ mènent parfois à la construction de réseaux définis positifs intéressants. En particulier, si $q\equiv 3mod4$ est tel que $(q^2+q+1)$ est la norme d’un élément de $\mathbb{Q}\left[\sqrt{-q}\right]$, alors nous obtenons un réseau entier unimodulaire $M_q$ défini positif et de dimension paire $2q(q+1)$ tel que $Aut\left(M_q\right)\supseteq PGL(3,{𝔽}_q)$. Nous prouvons que $M_3$ est le réseau de Leech.

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