Rationality of the Petersson Inner Product of Cohen's Kernels

TL;DR

This paper explicitly computes and analytically continues a twisted Eisenstein series to establish a formula for the Petersson inner product of Cohen's kernel and its twist, demonstrating its rationality and extending prior results.

Contribution

It provides a new explicit formula for the Petersson inner product of Cohen's kernels and their twists, extending previous work by Kohnen and Zagier.

Findings

Derived an explicit formula for the Petersson inner product

Proved the rationality of the inner product values

Extended previous results to new classes of Eisenstein series

Abstract

By explicitly calculating and then analytically continuing the Fourier expansion of the twisted double Eisenstein series $E_{s,k-s}^{*}(z,w; 1/2)$ of Diamantis and O'Sullivan, we prove a formula of the Petersson inner product of Cohen's kernel and one of its twists, and obtain a rationality result. This extends a result of Kohnen and Zagier.