A Singular Theta Lift and the Shimura Correspondence

Abstract

Modular forms play a central and critical role in the study of modern number theory. These remarkable and beautiful functions have led to many spectacular results including, most famously, the proof of Fermat's Last Theorem. In this thesis we find connections between these enigmatic objects. In particular, we describe the construction and properties of a singular theta lift, closely related to the well known Shimura correspondence.

We first define a (twisted) lift of harmonic weak Maass forms of weight 3/2-k, by integrating against a well chosen kernel Siegel theta function. Using this, we obtain a new class of automorphic objects in the upper-half plane of weight 2-2k for the group Gamma_(N). We reveal these objects have intriguing singularities along a collection of geodesics. These singularities divide the upper-half plane into Weyl chambers with associated wall crossing formulas. We show our lift is harmonic away from the singularities and so is an example of a locally harmonic Maass form. We also find an explicit Fourier expansion.

The Shimura/Shintani lifts provided very important correspondences between half-integral and even weight modular forms. Using a natural differential operator we link our lift to these. This connection then allows us to derive the properties of the Shimura lift. The nature of the singularities suggests we formulate all of these ideas as distributions and finally we consider the current equation encompassing them.

This work provides extensions of the theta lifts considered by Borcherds (1998), Bruinier (2002), Bruinier and Funke (2004), Hövel (2012) and Bringmann, Kane and Viazovska (2013).