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 \(\UHPn\), \(j(g,z)=\det (cz+d)\), 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/
- https://math.berkeley.edu/~reb/papers/siegel/index.html