Eisenstein series of weight $k \geq 2$ and integral binary quadratic forms

Andreas Mono

Published 2020-10-21Version 1

In a recent paper arXiv:2003.12354v2, Matsusaka investigated parabolic, elliptic, and hyperbolic Eisenstein series in weight $2$. He provided the analytic continuation to $s=0$ in the elliptic case, and conjectured an expression describing the same continuation in the hyperbolic case. We extend Matsusakas setting to general weight $k \geq 2$, and embed his Eisenstein series into a framework based on discriminants of integral binary quadratic forms. Lastly, we compute the Fourier expansion of our Eisenstein series in the hyperbolic case by adapting Zagiers method, and using results of Andersen, Duke.