Explicit formula for the $(\text{GL}_2, \text{GL}_2)$ theta lift via Bruhat decomposition

TL;DR

The paper derives an explicit formula for the type II $( ext{GL}_2, ext{GL}_2)$ theta lift using Kronecker-Eisenstein series, extending classical formulas to CM fields without smoothing.

Contribution

It provides a new explicit formula for the theta lift in the $( ext{GL}_2, ext{GL}_2)$ setting, applicable to CM fields, without the need for smoothing techniques.

Findings

Explicit formula for the $( ext{GL}_2, ext{GL}_2)$ theta lift

Extension of the formula to CM fields

Application of Kronecker-Eisenstein series in constructing currents

Abstract

Using combinations of weight-1 and weight-2 of Kronecker-Eisenstein series to construct currents in the distributional de Rham complex of a squared elliptic curve, we find a simple explicit formula for the type II $(\text{GL}_2, \text{GL}_2)$ theta lift without smoothing, analogous to the classical formula of Siegel for periods of Eisenstein series. For $K$ a CM field, the same technique applies without change to obtain an analogous formula for the $(\text{GL}_2(K),K^\times)$ theta correspondence.