# Modular lattices from finite projective planes

•  Department of Mathematics Iowa State University Ames, IA 50011
• Volume: 26, Issue: 2, page 269-279
• ISSN: 1246-7405

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## Abstract

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Using the geometry of the projective plane over the finite field ${𝔽}_{q}$, we construct a Hermitian Lorentzian lattice ${L}_{q}$ of dimension $\left({q}^{2}+q+2\right)$ 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 ${|p|}^{2}=q$.The Lorentzian lattices ${L}_{q}$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q\equiv 3\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$ is a rational prime such that $\left({q}^{2}+q+1\right)$ is norm of some element in $ℚ\left[\sqrt{-q}\right]$, then we find a $2q\left(q+1\right)$ dimensional even unimodular positive definite integer lattice ${M}_{q}$ such that $Aut\left({M}_{q}\right)\supseteq PGL\left(3,{𝔽}_{q}\right)$. We find that ${M}_{3}$ is the Leech lattice.

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