Regularized inner products of meromorphic modular forms and higher Green's functions

TL;DR

This paper introduces a new regularized inner product for meromorphic modular forms, enabling the evaluation of higher Green's functions through generalized quadratic form Poincaré series and theta lifts.

Contribution

It develops a novel regularized inner product for meromorphic modular forms, facilitating the computation of higher Green's functions from quadratic form Poincaré series.

Findings

Defined a new regularized inner product for meromorphic modular forms

Connected quadratic form Poincaré series with higher Green's functions

Provided methods for evaluating higher Green's functions via theta lifts

Abstract

In this paper we study generalizations of quadratic form Poincar\'e series, which naturally occur as outputs of theta lifts. Integrating against them yields evaluations of higher Green's functions. For this we require a new regularized inner product, which is of independent interest.