Shintani theta lifts of harmonic Maass forms

TL;DR

This paper introduces a regularized Shintani theta lift that transforms harmonic Maass forms into (sesqui-)harmonic Maass forms, linking Fourier coefficients to CM value traces and cycle integrals, and connects to the Millson theta lift.

Contribution

It defines a new regularized Shintani theta lift for harmonic Maass forms and explores its properties and connections to other lifts and special functions.

Findings

Fourier coefficients are traces of CM values and cycle integrals.

The lift relates via the ξ-operator to the Millson theta lift.

Constructs ξ-preimages of Zagier's series and Ramanujan's mock theta functions.

Abstract

We define a regularized Shintani theta lift which maps weight $2k+2$ ($k \in \Z, k \geq 0$) harmonic Maass forms for congruence subgroups to (sesqui-)harmonic Maass forms of weight $3/2+k$ for the Weil representation of an even lattice of signature $(1,2)$. We show that its Fourier coefficients are given by traces of CM values and regularized cycle integrals of the input harmonic Maass form. Further, the Shintani theta lift is related via the $\xi$-operator to the Millson theta lift studied in our earlier work. We use this connection to construct $\xi$-preimages of Zagier's weight $1/2$ generating series of singular moduli and of some of Ramanujan's mock theta functions.