The Shimura-Shintani correspondence via singular theta lifts and currents

TL;DR

This paper constructs a singular theta lift for the orthogonal group SO(2,1), producing locally harmonic Maass forms with singularities along geodesics, and explores their relation to the Shimura-Shintani correspondence.

Contribution

It introduces a new singular theta lift for SO(2,1) and analyzes its properties, extending previous theta lift frameworks by Hoevel and Bringann-Kane-Viazovska.

Findings

Constructed a singular theta lift for SO(2,1).

Produced locally harmonic Maass forms with singular sets along geodesics.

Derived properties connecting these forms to the Shimura-Shintani correspondence.

Abstract

We describe the construction and properties of a singular theta lift for the orthogonal group $\SO(2,1)$. We obtain locally harmonic Maass forms in the sense of Bringmann-Kane-Kohnen with singular sets along geodesics in the upper half plane. We consider these forms as currents and derive properties of the Shimura-Shintani correspondence. This work provides extensions of the theta lifts considered by Hoevel and Bringann-Kane-Viazovska.