The singular theta correspondence, Lorentzian lattices and Borcherds-Kac-Moody algebras

This dissertation answers some of the questions raised in Borcherds' papers on Moonshine and Lorentzian reflection groups. Firstly, we show that the pseudo-cusp forms studied by Hejhal in connection with the Riemann hypothesis can be constructed using the singular theta correspondence. We prove (assuming an open conjecture of Burger, Li and Sarnak about the automorphic spectra of orthogonal groups) 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 addition, 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 symmetric 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 finite group. If this conjecture were true it would give a nice, simple classification of an important class of BKM algebras. These BKM algebras are interesting from a purely Lie algebraic point of view as they can be considered natural generalizations of finite and affine Lie algebras: The Weyl groups for finite dimensional Lie algebras are spherical reflection groups, for affine Lie algebras are planar reflection groups and for these BKM algebras are hyperbolic reflection groups. Since BKM algebras also appear in the string theory literature as algebras associated to BPS states, we expect a classification will be of further interest. 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.