Dear GMS,
I hope you are all doing well.
We will have a seminar on 3rd of December, Wednesday in 123 St. Paul's between 12:30 and 13:30. Our speaker is @Sueda Senturk Avcimailto:senturks@myumanitoba.ca. This is going to be the last talk of this semester. The details about her talk are below.
Title: Computing Fourier Coefficients of Non-Holomorphic Eisenstein Series
Abstract: The classical (holomorphic) Eisenstein series with the form \begin{equation*} E_k(z) = \frac{1}{2} \displaystyle\sum_{\substack{m,n \in \mathbb{Z} \ (m,n) \neq (0,0)}} (mz + n)^{-k} \end{equation*} for $z \in \mathbb{H}$ has the Fourier coefficients \begin{equation*} E_k(z) = \zeta(k) + \frac{(2\pi)^k (-1)^{k/2}}{(k - 1)!} \sum_{N=1}^\infty \sigma_{k-1}(N) q^N, \quad \text{where } q = e^{2\pi i z}. \end{equation*} We see that the sum-of-divisors function $\sigma$ pops up in the non-constant coefficient part, which can be obtained via an elementary examination.
When we change our focus to the non-holomorphic Eisenstein series with the form \begin{equation*} E_s(z) = \displaystyle\sum_{\substack{c,d \in \mathbb{Z} \ \gcd(c,d) = 1}} \frac{y^s}{|cz+d|^{2s}}, \quad \text{for } z \in \mathbb{H} \text{ and } \Re(s) > 1, \end{equation*} the coefficients inevitably change. More explicitly, their Fourier expansion is \begin{equation*} E_s(z) = y^s + \frac{\xi(2s-1)}{\xi(2s)} y^{1-s} + \frac{2y^{\frac{1}{2}}}{\xi(2s)} |m|^{s-\frac{1}{2}} \sigma_{1-2s}(m) K_{s-\frac{1}{2}}(2\pi |m|y) e^{2\pi i mx}. \end{equation*} However, we keep seeing $\sigma$ in the non-constant coefficient part, and it comes from a more complicated but rigorous series of computations over local fields. In this talk, we will attempt to explain the background setup of these computations. Finally, if time permits, we will talk about the Whittaker functions that appear throughout our computations for a moment.
The seminar can also be accessed from Zoom: https://umanitoba.zoom.us/j/61904749902?pwd=NBzY0WN8ItdlMq2fZiKCpb1uPf8L8i.1 Meeting ID: 619 0474 9902 Passcode: 846191
See you all in the seminar! GMS Websitehttps://sites.google.com/view/umgradmathsociety/ / Instagramhttps://www.instagram.com/umgradmathsociety/
Sincerely, Berkanthttps://cnnk.xyz GMS Executive