Orthogonal Eisenstein Series at Harmonic Points and Modular Forms of Singular Weight

TL;DR

This paper studies the behavior of orthogonal Eisenstein series at harmonic points using theta lifts, revealing poles and residues that produce boundary Eisenstein series, and explores conditions for their surjectivity.

Contribution

It introduces a new analysis of orthogonal Eisenstein series at harmonic points via theta lifts and establishes conditions for their surjectivity onto boundary Eisenstein series.

Findings

Orthogonal Eisenstein series have a simple pole at s=1 with residues forming boundary Eisenstein series.

Residues of theta lift representations yield holomorphic orthogonal modular forms.

Provides a sufficient condition for the surjectivity of the theta lift construction.

Abstract

We investigate the behaviour of orthogonal non-holomorphic Eisenstein series at their harmonic points by using theta lifts. In the case of singular weight, we show that the orthogonal non-holomorphic Eisenstein series that can be written as a theta lift have a simple pole at $s = 1$ whose residues yield holomorphic orthogonal modular forms that are Eisenstein series on the boundary and we give a sufficient condition on the surjectivity of this construction.