Fourier coefficients of Siegel-Eisenstein series
introduction
$ \newcommand\supparen[1]{^{(#1)}} \newcommand\suppn{\supparen n} \newcommand\Enk{E_k\suppn} \newcommand\bs{\backslash} \newcommand\Gamn{\Gamma_{\!n}} \newcommand\UHP{\mathcal H} \newcommand\UHPn{\UHP_n} \newcommand\smallmat[4]{\left[\begin{smallmatrix} {#1}&{#2}\\{#3}&{#4}\end{smallmatrix}\right]} \newcommand\smallmatabcd{\smallmat abcd} \newcommand\SpnZ{\Sp n\Z} \newcommand\Sp[2]{\operatorname{Sp}_{#1}(#2)} \newcommand\siX[1]{{\mathcal X}_{#1}} \newcommand\Xn{\siX n} \newcommand\Xnsemi{\siX n^{\rm semi}} \newcommand\fc[2]{a(#1;#2)} \newcommand\e{\rm e} \newcommand\ip[2]{\langle #1,#2\rangle} \newcommand\suppnminusone{\supparen {n-1}} \newcommand\Enminusonek{E_k\suppnminusone} \newcommand\Eonek{E_k\supparen1} \newcommand\Zp{\Z_p} $ For any positive integer degree $n$ and even integer weight $k>n+1$, the Siegel Eisenstein series of weight $k$ and degree $n$ is $$ \Enk(z)=\sum_{\gamma\in P_\Z\bs\Gamn}j(\gamma,z)^{-k}. $$ Here $z$ lies in the Siegel upper half space $\UHPn$, and the summand $j(\gamma,z)^{-k}$ is $1$ for the Siegel parabolic subgroup $P_\Z=\{\smallmat ab0d\}$ of the integral symplectic group $\Gamn=\SpnZ$.
This Eisenstein series has the Fourier series representation $$ \Enk(z)=\sum_{t\in\Xnsemi}\fc t{\Enk}\,\e(\ip tz), $$ where $\Xnsemi$ denotes the set of semi-integral positive semidefinite $n$-by-$n$ matrices. The Siegel $\Phi$ map takes Eisenstein series to Eisenstein series, $\Phi\Enk=\Enminusonek$ and $\Phi\Eonek=1$, so it suffices to compute the Fourier coefficients of Eisenstein series for definite indices $t$; the set of such matrices is denoted $\Xn$. Eisenstein series are central to number theory, from Garrett's pullback formula to the Langlands program. The algorithmic computation of the Siegel Eisenstein series Fourier coefficients $\fc t\Enk$ began with C. L. Siegel and was completed by H. Katsurada. The Fourier coefficient formula for definite indices, to be elaborated below, is $$ \fc t\Enk= \dfrac{2^{\lfloor \frac{n+1}{2} \rfloor} \prod_{p}F_p(t,p^{k-n-1})} {\zeta(1-k)\prod_{i=1}^{\lfloor n/2\rfloor}\zeta(1-2k+2i)} \cdot\begin{cases} L(\chi_{D_t},1-k+n/2)&\text{$n$ even},\\ 1&\text{$n$ odd}. \end{cases} $$ The Fourier coefficient depends only on the genus of its index $t$. In fact the polynomial $F_p(t,X)\in\Z[X]$ depends only on the class of $t$ over $\Zp$.