# Modular lattices from finite projective planes

Tathagata Basak^{[1]}

- [1] Department of Mathematics Iowa State University Ames, IA 50011

Journal de Théorie des Nombres de Bordeaux (2014)

- Volume: 26, Issue: 2, page 269-279
- ISSN: 1246-7405

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topBasak, Tathagata. "Modular lattices from finite projective planes." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 269-279. <http://eudml.org/doc/275779>.

@article{Basak2014,

abstract = {Using the geometry of the projective plane over the finite field $\mathbb\{F\}_q$, we construct a Hermitian Lorentzian lattice $L_\{q\}$ of dimension $(q^\{2\} + q + 2)$ defined over a certain number ring $\mathcal\{O\}$ that depends on $q$. We show that infinitely many of these lattices are $p$-modular, that is, $p L^\{\prime \}_\{q\} = L_\{q\}$, where $p$ is some prime in $\mathcal\{O\}$ such that $\vert p\vert ^\{2\} =q$.The Lorentzian lattices $L_q$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q \equiv 3 \bmod 4$ is a rational prime such that $(q^2 + q + 1)$ is norm of some element in $\mathbb\{Q\}[\sqrt\{-q\}]$, then we find a $2q(q+1)$ dimensional even unimodular positive definite integer lattice $M_\{q\}$ such that $\operatorname\{Aut\}(M_q) \supseteq \operatorname\{PGL\}(3,\mathbb\{F\}_q)$. We find that $M_3$ is the Leech lattice.},

affiliation = {Department of Mathematics Iowa State University Ames, IA 50011},

author = {Basak, Tathagata},

journal = {Journal de Théorie des Nombres de Bordeaux},

language = {eng},

month = {10},

number = {2},

pages = {269-279},

publisher = {Société Arithmétique de Bordeaux},

title = {Modular lattices from finite projective planes},

url = {http://eudml.org/doc/275779},

volume = {26},

year = {2014},

}

TY - JOUR

AU - Basak, Tathagata

TI - Modular lattices from finite projective planes

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2014/10//

PB - Société Arithmétique de Bordeaux

VL - 26

IS - 2

SP - 269

EP - 279

AB - Using the geometry of the projective plane over the finite field $\mathbb{F}_q$, we construct a Hermitian Lorentzian lattice $L_{q}$ of dimension $(q^{2} + q + 2)$ defined over a certain number ring $\mathcal{O}$ that depends on $q$. We show that infinitely many of these lattices are $p$-modular, that is, $p L^{\prime }_{q} = L_{q}$, where $p$ is some prime in $\mathcal{O}$ such that $\vert p\vert ^{2} =q$.The Lorentzian lattices $L_q$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q \equiv 3 \bmod 4$ is a rational prime such that $(q^2 + q + 1)$ is norm of some element in $\mathbb{Q}[\sqrt{-q}]$, then we find a $2q(q+1)$ dimensional even unimodular positive definite integer lattice $M_{q}$ such that $\operatorname{Aut}(M_q) \supseteq \operatorname{PGL}(3,\mathbb{F}_q)$. We find that $M_3$ is the Leech lattice.

LA - eng

UR - http://eudml.org/doc/275779

ER -

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