Fourier coefficients of Siegel-Eisenstein series
introduction
<math> \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} </math> For any positive integer degree <math>n</math> and even integer weight <math>k>n+1</math>, the Siegel Eisenstein series of weight <math>k</math> and degree <math>n</math> is
- <math>
\Enk(z)=\sum_{\gamma\in P_\Z\bs\Gamn}j(\gamma,z)^{-k}. </math> Here <math>z</math> lies in the Siegel upper half space <math>\UHPn</math>, <math>j(g,z)=\det (cz+d)</math>, and the summand <math>j(\gamma,z)^{-k}</math> is <math>1</math> for the Siegel parabolic subgroup <math>P_\Z=\{\smallmat ab0d\}</math> of the integral symplectic group <math>\Gamn=\SpnZ</math>.
This Eisenstein series has the Fourier series representation
- <math>
\Enk(z)=\sum_{t\in\Xnsemi}\fc t{\Enk}\,\e(\ip tz), </math> where <math>\Xnsemi</math> denotes the set of semi-integral positive semidefinite <math>n</math>-by-<math>n</math> matrices. The Siegel <math>\Phi</math> map takes Eisenstein series to Eisenstein series, <math>\Phi\Enk=\Enminusonek</math> and <math>\Phi\Eonek=1</math>, so it suffices to compute the Fourier coefficients of Eisenstein series for definite indices <math>t</math>; the set of such matrices is denoted <math>\Xn</math>. 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 <math>\fc t\Enk</math> began with C. L. Siegel and was completed by H. Katsurada. The Fourier coefficient formula for definite indices, to be elaborated below, is
- <math>
\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{</math>n<math> even},\\ 1&\text{</math>n<math> odd}. \end{cases} </math> The Fourier coefficient depends only on the genus of its index <math>t</math>. In fact the polynomial <math>F_p(t,X)\in\Z[X]</math> depends only on the class of <math>t</math> over <math>\Zp</math>.
<math>F_p</math>-polynomials
Polynomials <math>F_p(u,X)\in\Z[X]</math> for <math>p</math> prime and <math>u\in\Xm</math> 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 <math>F_p</math> 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 <math>a,b\in\Qpx</math> the Hilbert symbol <math>(a,b)_p</math> is <math>1</math> if <math>aX^2+bY^2=Z^2</math> has nontrivial solutions in <math>\Qp^3</math> and <math>-1</math> if not. For <math>u\in\GL m\Qp^{\rm sym}</math> the Hasse invariant of <math>u</math> is <math>h_p(u)=\prod_{i\le j}(a_i,a_j)_p</math> where <math>u</math> is <math>\GL m\Qp</math>-equivalent to the diagonal matrix having entries <math>a_1,\cdots,a_m</math>. If <math>m</math> is even then <math>(-1)^{m/2}\det(2u)</math> takes the form <math>D_uf_u^2</math> where <math>D_u</math> is <math>1</math> or the fundamental discriminant of a quadratic number field and <math>f_u</math> is a positive integer; let <math>\chi_{D_u}</math> denote the quadratic Dirichlet character of conductor <math>|D_u|</math>. For rank <math>m=0</math>, the empty matrix has determinant <math>1</math> by convention and so <math>D_u=f_u=1</math>.
- theorem [Katsurada's Functional Equation]
Let <math>u\in\Xm</math>. Set
- <math>
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 </math>m<math> is even},\\ \ord p{\det(2u)/2}&\text{if </math>m<math> is odd}. \end{cases} </math> Here <math>\delta_{p,2}</math> is the Kronecker delta. Then
- <math>
F_p(u,p^{-m-1} X\inv)= \pm(p^{(m+1)/2}X)^{-e_p(u)}F_p(u,X), </math> where if <math>m</math> is even then the ``<math>\pm</math> sign is positive, and if <math>m</math> is odd then it is
- <math>
\big(\det(u),(-1)^{(m-1)/2}\det(u)\big)_p\,(-1,-1)_p^{(m^2-1)/8}\,h_p(u) </math> with <math>(\cdot,\cdot)_p</math> the Hilbert symbol and <math>h_p</math> the Hasse invariant as described above.
%Katsurada used this functional equation to write down complicated
%(especially for <math>p=2</math>) recursion relations for the <math>F_p</math>-polynomials
%in terms of the local invariants of <math>u</math>, data that identify the
%<math>\GLnZp</math>-equivalence class of <math>u</math>.
%Hence, for a given <math>u</math> all <math>F_p(u,X)</math> can be computed from the
%genus symbol of <math>u</math>, 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 <math>n</math> be a positive integer. For any <math>t\in\Xnsemi</math> we have <math>t\sim u\oplus0_{n-m}</math> under <math>\GLnZ</math>-equivalence, where <math>m={\rm rank}(t)\in\Znn</math> and <math>u\in\Xm</math>. The following result may be found in \cite{katsurada99, katsurada10}.
- theorem (Siegel Eisenstein Fourier Coefficient Formula)
Let <math>n</math> be a positive integer and <math>k>n+1</math> an even integer. Let <math>t\in\Xnsemi</math>, and let <math>u</math>, <math>D_u</math>, <math>f_u</math>, and <math>\chi_{D_u}</math> be as above. Let <math>c\suppm_k=2^{-\lfloor(m+1)/2\rfloor} \zeta(1-k)\prod_{i=1}^{\lfloor m/2\rfloor}\zeta(1-2k+2i)</math>. Then
- <math>
\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{</math>m<math> even},\\ \prod_{p:\ord p{(1/2)\det(2u)}>0}F_p(u,p^{k-m-1}) &\text{</math>m<math> odd}. \end{cases} </math>
The Riemann zeta values and the quadratic <math>L</math> value in the formula
have the form <math>\zeta(1-j)=-B_j/j</math> and <math>L(\chi,1-j)=-B_j(\chi)/j</math>
with the <math>B_j</math> basic or quadratic Bernoulli numbers,
and so they are known rational numbers:
if <math>f</math> is the conductor of <math>\chi</math>, then
<math>\sum_{a=1}^f \chi(a) \frac{te^{at}}{e^{ft}-1} = \sum_{j=0}^{\infty}
B_j(\chi) \frac{t^j}{j!}</math> (\cite{aik14}, page {53}).
The genus symbol of any <math>u\in\Xm</math> is easy to compute,
and then our program gives <math>F_p(u,p^{k-m-1})</math>.
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 <math>2t</math> as an input and returns the genus symbol <math>4^{-2}_4\, 3^{-1}</math>. Our <math>F_p</math> polynomial program takes this genus symbol and the determinant <math>\det(2t)=48</math> as input and returns the <math>F_p(t,X)</math> polynomials for all~<math>p\mid2\det(2t)</math>,
- <math>
\left[F_2(X),F_3(X)\right]=\left[1+24X+256X^2+3072X^3+16384X^4,1\right]. </math> These data make no reference to any particular Eisenstein series degree or weight. With these <math>F_p</math> polynomials and the weight <math>k=16</math> as input, our Eisenstein series Fourier coefficient program returns
- <math>
\fc {t}{E^{(6)}_{16}} =\frac{ 9780154654408147370255260881715200} {13912726954911229324966739363569}\,. </math>
computational resource
- https://drive.google.com/file/d/0B8XXo8Tve1cxMVhLZzhDSGF6QUk/view
- http://siegelmodularforms.org/pages/degree3/
- https://math.berkeley.edu/~reb/papers/siegel/index.html