The Shimura–Shintani correspondence via singular theta lifts and currents

Abstract

We describe the construction and properties of a singular theta lift for the orthogonal group [Formula: see text]. We obtain locally harmonic Maass forms in the sense of Bringmann, Kane and 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 Hövel and Bringmann, Kane and Viazovska.