Modular lattices from finite projective planes
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, pL 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 q3mod4 is a rational prime such that (q 2 +q+1) is norm of some element in [-q], then we find a 2q(q+1) dimensional even unimodular positive definite integer lattice M q such that Aut(M q )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 pL 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 q3mod4 est tel que (q 2 +q+1) est la norme d’un élément de [-q], alors nous obtenons un réseau entier unimodulaire M q défini positif et de dimension paire 2q(q+1) tel que Aut(M q )PGL(3,𝔽 q ). Nous prouvons que M 3 est le réseau de Leech.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.867
Classification: 11H56,  51E20,  11E12,  11E39
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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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.867/

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