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\Xsix{\siX 6} \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} \newcommand\Xm{\siX m} \newcommand\Qp{\Q_p} \newcommand\Qpx{\Qp^\times} \newcommand\GL[2]{\operatorname{GL}_{#1}(#2)} \newcommand\GLnZ{\GL n\Z} \newcommand\GLnZp{\GL n{\Zp}} \newcommand\Znn{\Z_{\ge0}} \newcommand{\ord}[2]{{\rm ord}_{#1}(#2)} \newcommand\inv{^{-1}} \newcommand\suppm{\supparen m} $ 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 $\Zp$.
$F_p$-polynomials
Polynomials $F_p(u,X)\in\Z[X]$ for $p$ prime and $u\in\Xm$ appear in the Siegel Eisenstein series Fourier coefficient formula. The first author of this paper wrote a program to compute these polynomials \cite{king03}, which has since been modified to accept higher degree input. We refer to \cite{katsurada99} for the definition of the $F_p$ polynomials; there Katsurada proved a functional equation for these polynomials, which was an important step in his establishment of their recurrence relations. We review this functional equation because it serves as a check on computations. The functional equation makes reference to the Hilbert symbol and to the Hasse invariant. To review, for $a,b\in\Qpx$ the Hilbert symbol $(a,b)_p$ is $1$ if $aX^2+bY^2=Z^2$ has nontrivial solutions in $\Qp^3$ and $-1$ if not. For $u\in\GL m\Qp^{\rm sym}$ the Hasse invariant of $u$ is $h_p(u)=\prod_{i\le j}(a_i,a_j)_p$ where $u$ is $\GL m\Qp$-equivalent to the diagonal matrix having entries $a_1,\cdots,a_m$. If $m$ is even then $(-1)^{m/2}\det(2u)$ takes the form $D_uf_u^2$ where $D_u$ is $1$ or the fundamental discriminant of a quadratic number field and $f_u$ is a positive integer; let $\chi_{D_u}$ denote the quadratic Dirichlet character of conductor $|D_u|$. For rank $m=0$, the empty matrix has determinant $1$ by convention and so $D_u=f_u=1$.
- theorem [Katsurada's Functional Equation]
Let $u\in\Xm$. Set $$ e_p(u)=\begin{cases} 2(\lfloor\frac{\ord p{\det(2u)}-1-\delta_{p,2}}{2}\rfloor+\chi_{D_u}(p)^2) &\text{if $m$ is even},\\ \ord p{\det(2u)/2}&\text{if $m$ is odd}. \end{cases} $$ Here $\delta_{p,2}$ is the Kronecker delta. Then $$ F_p(u,p^{-m-1} X\inv)= \pm(p^{(m+1)/2}X)^{-e_p(u)}F_p(u,X), $$ where if $m$ is even then the ``$\pm$ sign is positive, and if $m$ is odd then it is $$ \big(\det(u),(-1)^{(m-1)/2}\det(u)\big)_p\,(-1,-1)_p^{(m^2-1)/8}\,h_p(u) $$ with $(\cdot,\cdot)_p$ the Hilbert symbol and $h_p$ the Hasse invariant as described above.
%Katsurada used this functional equation to write down complicated
%(especially for $p=2$) recursion relations for the $F_p$-polynomials
%in terms of the local invariants of $u$, data that identify the
%$\GLnZp$-equivalence class of $u$.
%Hence, for a given $u$ all $F_p(u,X)$ can be computed from the
%genus symbol of $u$, the amalgamation of the local invariants.
%O. King \cite{king03} wrote a program to compute these polynomials,
%which has since been modified to accept higher degree input, and is now being made publically
%available.
Fourier Coefficient Formula
Let $n$ be a positive integer. For any $t\in\Xnsemi$ we have $t\sim u\oplus0_{n-m}$ under $\GLnZ$-equivalence, where $m={\rm rank}(t)\in\Znn$ and $u\in\Xm$. The following result may be found in \cite{katsurada99, katsurada10}.
- theorem Siegel Eisenstein Fourier Coefficient Formula
Let $n$ be a positive integer and $k>n+1$ an even integer. Let $t\in\Xnsemi$, and let $u$, $D_u$, $f_u$, and $\chi_{D_u}$ be as above. Let $c\suppm_k=2^{-\lfloor(m+1)/2\rfloor} \zeta(1-k)\prod_{i=1}^{\lfloor m/2\rfloor}\zeta(1-2k+2i)$. Then $$ \fc t\Enk={1}/{c_k\suppm}\cdot\begin{cases} L(\chi_{D_u},1-k+m/2)\prod_{p\mid f_u}F_p(u,p^{k-m-1}), &\text{$m$ even},\\ \prod_{p:\ord p{(1/2)\det(2u)}>0}F_p(u,p^{k-m-1}) &\text{$m$ odd}. \end{cases} $$
The Riemann zeta values and the quadratic $L$ value in the formula
have the form $\zeta(1-j)=-B_j/j$ and $L(\chi,1-j)=-B_j(\chi)/j$
with the $B_j$ basic or quadratic Bernoulli numbers,
and so they are known rational numbers:
if $f$ is the conductor of $\chi$, then
$\sum_{a=1}^f \chi(a) \frac{te^{at}}{e^{ft}-1} = \sum_{j=0}^{\infty}
B_j(\chi) \frac{t^j}{j!}$ (\cite{aik14}, page {53}).
The genus symbol of any $u\in\Xm$ is easy to compute,
and then our program gives $F_p(u,p^{k-m-1})$.
Thus Siegel Eisenstein series Fourier coefficients are tractable.
example
For example, consider the Fourier coefficient index \begin{equation*} t=\frac12\left[\begin{matrix} 2 & 1 & 1 & 0 & 1 & 2 \\ 1 & 4 & 2 & 2 & 0 & 1 \\ 1 & 2 & 4 & 2 & 0 & 0 \\ 0 & 2 & 2 & 4 & 2 & 2 \\ 1 & 0 & 0 & 2 & 4 & 2 \\ 2 & 1 & 0 & 2 & 2 & 8 \end{matrix}\right]\in\Xsix. \end{equation*} Our genus symbol program takes $2t$ as an input and returns the genus symbol $4^{-2}_4\, 3^{-1}$. Our $F_p$ polynomial program takes this genus symbol and the determinant $\det(2t)=48$ as input and returns the $F_p(t,X)$ polynomials for all~$p\mid2\det(2t)$, $$ \left[F_2(X),F_3(X)\right]=\left[1+24X+256X^2+3072X^3+16384X^4,1\right]. $$ These data make no reference to any particular Eisenstein series degree or weight. With these $F_p$ polynomials and the weight $k=16$ as input, our Eisenstein series Fourier coefficient program returns $$ \fc {t}{E^{(6)}_{16}} =\frac{ 9780154654408147370255260881715200} {13912726954911229324966739363569}\,. $$
computational resource
- https://drive.google.com/file/d/0B8XXo8Tve1cxMVhLZzhDSGF6QUk/view
- http://siegelmodularforms.org/pages/degree3/