Modular lattices from finite projective planes

Using the geometry of the projective plane over the finite field F-q, we construct a Hermitian Lorentzian lattice Lq 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 vertical bar p vertical bar(2) = q. The Lorentzian lattices L-q sometimes lead to construction of interesting positive definite lattices. In particular, if q 3 mod 4 is a rational prime such that (q(2) + q + 1) is norm of some element in Q[root-q], then we find a 2q(q+1) dimensional even unimodular positive definite integer lattice M-q such that Aut(M-q) superset of PGL(3,F-q). We find that M-3 is the Leech lattice.