Modular lattices from finite projective planes

TL;DR

This paper constructs a family of Lorentzian lattices from finite projective planes over F_q, explores their modular properties, and links them to notable structures like the Leech lattice and the monster group.

Contribution

It introduces a new construction of Lorentzian lattices from finite projective geometries and reveals their modularity and connections to exceptional groups and lattices.

Findings

Infinitely many lattices are p-modular with prime p related to q.

The q=3 case relates to the monster simple group.

Constructed lattices include the Leech lattice for q=3.

Abstract

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 $\cO$ 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 $\cO$ such that |p|^2 = q. The reflection group of the Lorentzian lattice obtained for q = 3 seems to be closely related to the monster simple group via the presentation of the bimonster as a quotient of the Coxeter group on the incidence graph of P^2(F_3). The Lorentzian lattices L_q sometimes lead to construction of interesting positive definite lattices. In particular, if q is a rational prime that is 3 mod 4, and (q^2 + q + 1) is norm of some element in Q[\sqrt{-q}], then we find a 2q(q+1) dimensional even unimodular positive definite integer lattice M_q such that Aut(M_q) contains PGL(3,F_q). We find that M_3 is the Leech lattice.