The Singular Theta Correspondence, Lorentzian Lattices and Borcherds-Kac-Moody Algebras

Abstract

This dissertation answers some of the questions raised in Borcherds' papers on Moonshine and Lorentzian reflection groups. We prove (assuming an open conjecture of Burger, Li and Sarnak) that a Lorentzian reflection group with Weyl vector is associated to a vector-valued modular form. This result allows us to establish a folklore conjecture that the maximal dimension of a Lorentzian reflection group with Weyl vector is 26. In the case of elementary lattices, we show that these vector-valued forms can be obtained by inducing scalar-valued forms. This allows us to explain the critical signatures which occur in Borcherds' work. Many of the structures which occur at these critical signatures are especially beautiful and have appeared independently throughout the literature. We investigate Borcherds-Kac-Moody (BKM) algebras with denominator formulas that are singular weight automorphic forms. The results of these investigations suggest that all such BKM algebras are related to orbifold constructions of vertex algebras and elements of the Monster sporadic group. Finally, we show how work in this dissertation combined with results of Bruinier gives a new insight into the arithmetic mirror symmetry conjecture of Gritsenko and Nikulin.