数理解析研究所講究録
第 1826 巻 2013 年 180-185

180

SIEGEL EISENSTEIN SERIES, HECKE OPERATORS, AND
FOURIER EXPANSIONSLYNNE H. WALLING
ABSTRACT. We discuss the action of Hecke operators on Siegel Eisenstein series in the case of degree 2 and square-free level.

1. INTRODUCTION
Determining representation numbers of quadratic forms is a classical problem in number theory, and elliptic modular forms have been used to great
advantage in studying this problem. The number of times a positive definite
quadratic form represents an integer is given by the tth Fourier coefficient of the theta series attached to , and this theta series is one of ourbasic examples of a modular form. It is well-known that the average theta
series lies in the space spanned by Eisenstein series (this weighted average istaken over the genus of , which consists of all quadratic forms that locallyeverywhere are isometric to ). In the case of integral weight, the Fourier
expansions for the Eisenstein series are well-known (see, for instance, [15]);
then, realising the average theta series as a linear combination of Eisenstein series, one obtains closed-form formulas for the average representation
numbers (see, for instance, [17]).
Siegel introduced generalised theta series to study how often a given quadratic form represents any other quadratic form ; these generalised theta
series are our prototypical examples of Siegel modular forms. Currently, thestudy of Siegel modular forms is a very active area of research, and there
are many different approaches used, both to provide new proofs of known
results, and to obtain new insights, tools, and of course theorems.
Two fundamental problems that have not been completely solved are thatof finding explicit Fourier series expansions for all Siegel Eisenstein series,
and that of determining the action of Hecke operators on all Siegel Eisenstein series. In the case of elliptic modular forms, for any (integral) weight,
level and character, as we know the Fourier expansions of a basis for the
space of Eisenstein series, we can use these to determine the action of the
Hecke operators on Eisenstein series. However, in the case of Siegel modular forms, closed-form formulas for Fourier coefficients of Eisenstein series
are only known in certain cases: In [9], formulas for the degree 2, level 1
Eisenstein series are developed. These are also developed in [4] (chapter II)
using the Fourier expansion for the level l, index 1 Jacobi Eisenstein series(see [4] chapter I), and the connection between Jacobi forms and degree 2
$Q$

$t$

$Q$

$Q$

$Q$

$Q$

$T$

2010 Mathematics Subject Classification: Primary llF41

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LYNNE H. WALLING

Siegel forms revealed by the first proof of the Saito-Kurokawa correspondence (proved in the series of papers [11], [12], [13], [1], [19]). Then in[7], [8], Katsurada combines the induction formula for local densities witha functional equation to obtain formulas for the Fourier coefficients of theEisenstein series with Siegel degree at least 3 and level l. In [10], Kohnen
gives an explicit linear version of the Ikeda lift (this lift was conjectured
by Duke and Imamoglu); using this, he obtains formulas for the Fourier
coefficients of the Eisenstein series of any even Siegel degree and level 1.
Then in [2], Choie and Kohnen modify Kohnen’s approach to yield formulas
for the Fourier coefficients of the Eisenstein series of any odd Siegel degreeand level 1. In [14], Mizuno modifies the approach of [4], using (among
other things) a converse theorem of Imai [6] to obtain formulas for one of
the Eisenstein series of Siegel degree 2, odd square-free level, and primitive
character. Quite recently, in [16], Takemori uses -adic Siegel $mo$ dular forms
to develop formulas for one of the Eisenstein series of Siegel degree 2, arbitrary level, and primitive character. All these Fourier coefficient formulas
are rather complicated.
Recently [18], we determined the action of Hecke operators on all Eisenstein series of Siegel degree 2 and square-free level by intricate but elementary methods, without any use of known Fourier coefficients of Siegel Eisenstein series. The idea of the approach is described below; it relies merelyon the definition of the Eisenstein series and the explicit set of matrices described in [5] that give the action of the Hecke operators. Via this concrete
approach, we find that the natural basis for the space of degree 2 Eisensteinseries of square-free level
and character consists of eigenforms for all
Hecke operators $T(p),$ $T_{j}(p^{2})(1\leq j\leq n)$ where is a prime not dividing, we obtain Hecke relations
and we compute the eigenvalues. For primes
among these Eisenstein series when
; we use these to diagonalise the, and webasis to obtain a basis consisting of eigenforms for all $T(p),$compute the eigenvalues (see Proposition 2.1 below). Additionally, we notethat these Hecke relations can be used with known Eisenstein series Fourier
coefficients to generate the Fourier coefficients of other Eisenstein series. In
particular, when $\chi=1$ , we note that one can use the Fourier expansion
for the level 1 Eisenstein series to generate the Fourier expansions for thebasis of Eisenstein series of square-free level
and trivial character; thishas recently been carried out by Martin Dickson [3].
Currently we are in the process of extending this work to arbitrary Siegel, thedegree. For trivial character, we are again finding that for primes$T(q),$
many
action of
Hecke relations to allow us toyield sufficiently
generatre the Fourier expansions of all basis elements with square-free levelfrom the Fourier expansion of the level 1 Eisenstein series.
$p$$\mathcal{N}$$\chi$$\mathcal{N},$$p$$q|\mathcal{N}$$\chi^{2}\neq 1$$T_{j}(p^{2})$$\mathcal{N}$$q|\mathcal{N}$$T_{j}(q^{2})$$\mathcal{N}$

2. DEFINITIONS AND RESULTS
For $n\in z_{+}$ , the symplectic group(written in the form ofblocks)$Sp_{n}(\mathbb{Z})$is the set of $2n\cross 2n$ matrices$n\cross n$$\{(\begin{array}{ll}A BC D\end{array})\in SL_{2n}(\mathbb{Z}):$$A$ $tB,$$CtD$ are symmetric, $AtD-B^{t}C=I\}.$

182
SIEGEL EISENSTEIN SERIES, HECKE OPERATORS, AND FOURIER EXPANSIONSFor$\mathcal{N}\in \mathbb{Z}_{+},$$\Gamma_{0}(\mathcal{N})$is the subgroup of

consisting of those matrices$Sp_{n}(\mathbb{Z})$is congruent to modulo
where the block
consisting of those matrices where the block-dimensional cusp for the Siegel upper half-space
$C$

$0$

$\mathcal{N};\Gamma_{\infty}$$Sp_{n}(\mathbb{Z})$$C$

is the subgroup ofis equal to . Each
$0$

$0$

$\mathcal{H}_{(n)}=\{X+iY:X, Y\in \mathbb{R}_{sym}^{n,n}, Y>0\}$and each Eisenstein series in the natural basis for the subspace ofSiegel Eisenstein series corresponds to an element of$\Gamma_{\infty}\backslash Sp_{n}(\mathbb{Z})/\Gamma_{0}(\mathcal{N})$For
is defined by$\gamma_{0}\in Sp_{n}(\mathbb{Z})$, the weight$k$$\Gamma_{0}(\mathcal{N})-$.

Eisenstein series corresponding to$\Gamma_{\infty}\gamma_{0}\Gamma_{0}(\mathcal{N})$$E_{\gamma 0}(\tau)=\sum_{\gamma}\overline{\chi}(\det D_{\gamma})1(\tau)|\gamma0\gamma$

where$\Gamma_{\infty}\gamma_{0}\gamma$

varies over the$\Gamma_{0}(\mathcal{N})$-orbit of$\Gamma_{\infty}\gamma_{0}$, and for $\gamma=(_{M}*$$N*)\in$; hereFrom now on, suppose is square-free. We can show that the elements ofcorrespond to factorisations of
as a product of $n+1$$Sp_{n}(\mathbb{Z}),$$1(\tau)|\gamma=\det(M\tau+N)^{-k}$$\tau\in \mathcal{H}_{(n)}.$$\mathcal{N}$$\mathcal{N}$$\Gamma_{\infty}\backslash Sp_{n}(\mathbb{Z})/\Gamma_{0}(\mathcal{N})$positive integers as follows: With$\Gamma_{\infty}$in the same$(_{M^{*}}, N^{*\prime})$$\mathcal{N}_{0}\cdots \mathcal{N}_{n}=\mathcal{N}$$\Gamma_{0}(\mathcal{N})$, we have$\Gamma_{\infty}$$(_{M^{*}}$

$N*)$

and

-orbit if and only if $rank_{q}M=rank_{q}M’$(here $rank_{q}M$ denotes the rank of $M$ overfor each prime). Thus
we can parameterise our basis of the space of Eisenstien series by thesefactorisations of , labeling the baeis elements as
. (If
for
any prime
is not well-defined.
$0<i<n$ , the above series forIf we try to build
from-Eisenstein series whenforsome
in our basis$0<i<n$ , we get . So we only havewhenprimesfor all
Now consider the case that $n=2$ , and let
where. Consider a prime not dividing the level ; the sum
forinvolves terms such as$q|\mathcal{N}$$\mathbb{Z}/q\mathbb{Z}$$\mathcal{N}$$\chi_{q}^{2}=1$$\mathbb{E}_{(\mathcal{N}_{0},\ldots N_{n})}$$q|\mathcal{N}_{i},$$E_{(\mathcal{N}_{0},\ldots N_{n})}$$\Gamma(\mathcal{N})$$\mathbb{E}_{(\mathcal{N}_{0,\ldots r}\wedge\Gamma_{n})}$$0$$q|\mathcal{N}_{i},$$\chi_{q}^{2}\neq 1$$\mathbb{E}_{(\mathcal{N}_{0},\ldots,\mathcal{N}_{n})}$$q|\mathcal{N}_{1}\cdots \mathcal{N}_{n-1}.)$$\chi_{q}^{2}=1$$\rho=(\mathcal{N}_{0},\mathcal{N}_{1},\mathcal{N}_{2})$$\mathcal{N}_{0}\mathcal{N}_{1}\mathcal{N}_{2}=\mathcal{N}$$\mathcal{N}$$p$$\mathbb{E}_{\rho}(\tau)|T(p)$$p^{k-3}\chi(p)\overline{\chi}_{\rho}(M, N)$. $\det(M(1 p)G^{-1}(\tau tG^{-1}+(u 0))(^{1/p}$$1)+N)^{-k}$$=p^{k}p^{k-3}\chi(p)\overline{\chi}_{\rho}(M, N)$. $\det(M(l p)G^{-1}\tau+(M(u 0)+N)tG(p 1))^{-k}$
where

$u$

varies over$(\mathbb{Z}/p\mathbb{Z})^{\cross},$$G$

varies over$SL_{n}(\mathbb{Z})/\mathcal{K},$$\mathcal{K}=\{\gamma\in SL_{n}(\mathbb{Z}):\gamma\equiv(_{*}^{*} 0*)(p)\},$

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LYNNE H. WALLING

and $\chi_{\rho}(M, N)=\chi(\det D_{\gamma})$ where$N*)=\Gamma_{\infty}\gamma 0\gamma$$\Gamma_{\infty}(_{M}*$for$\gamma\in\Gamma_{0}(\mathcal{N})$.

The other terms can be similarly massaged to be of the shape$p^{2k-3}\chi(p^{2})\overline{\chi}_{\rho}(M, N)\det(pM\tau+N)^{-k},$and$p^{2k-3}\overline{\chi}_{\rho}(M, N)\det(M\tau+MY+pN)^{-k}$withvarying modulo . We demonstrate how we proceed withthe latter terms.
Suppose; decompose
aswhere eachis supported
on pairs $(MN)$ with $rank_{p}M=i$ . To demonstrate our technique, consider-equivalence class representative
the case $rank_{p}M=1$ . Adjust the$(MN)$ to assume that
divides the lower row of $M$ ; set$Y\in \mathbb{Z}_{sym}^{2,2}$$p$$p\nmid \mathcal{N}$$E_{i}$$\mathbb{E}_{0}+\mathbb{E}_{1}+\mathbb{E}_{2}$$\mathbb{E}_{\rho}$$SL_{2}(\mathbb{Z})$$p$$(M’N’)=(1 \frac{1}{p})(MMY-\vdash pN)$ .

The upper row of $M$ is non-zero modulo , as is the lower row of $N$ ; so$(M’N’)$ is a coprime symmetric pair, with $rank_{p}M’\geq 1$ . Also, $p^{-k}\det(M’\tau+$$N’)^{-k}=\det(M\tau+MY+pN)^{-k}$ . We can choose; then $(M’N’)\equiv$$p$$Y\equiv 0(\mathcal{N})$$(1 1/p)(MpN)(\mathcal{N})$and so$p^{-k}\chi_{\mathcal{N}_{0}\mathcal{N}_{2}}(p)\cdot\overline{\chi}_{\rho}(M’, N’)\det(M’\tau+N’)^{-k}$$=\overline{\chi}_{\rho}(M, N)\det(M\tau+MY+pN)^{-k}.$Reversing, given a coprime symmetric pair $(M’N’)$ , we need to countthe number of times an element of $SL_{2}(\mathbb{Z})(M’N’)$ is generated through themodulo , wepreceeding process. Thus as we varyneed to count how often$E\in SL_{2}(\mathbb{Z}),$$Y\in \mathbb{Z}_{sym}^{2,2}$$p$$(MN)= (1 p)E(M’\frac{1}{p}(N’-M’Y))$

represents distinct, integral$SL_{2}(\mathbb{Z})$For $E\in SL_{2}(\mathbb{Z})$ , we have
$E\equiv(\begin{array}{ll}* 0* *\end{array})(p)$-equivalence classes with $rank_{p}M=1.$$(1 p)E(1 1/p)\in SL_{2}(\mathbb{Z})$(so using such $E$ does not change theclass of $(MN))$ . Thus we need to consider$E=(\begin{array}{ll}0 -11 0\end{array})$or$(\begin{array}{ll}1 \alpha 0 1\end{array}),$$\alpha$(a) Say $rank_{p}M’=1$ ; assume
$p$

$p$

$SL_{2}(\mathbb{Z})$$E\in \mathcal{K}\backslash SL_{2}(\mathbb{Z})$varying modulo

if and only if-equivalence; so we can take$p.$

divides row 2 of $M’$ . For 1 choice of $E,$

divides row 1 of $EM’$ , and then since $M=(1 p)EM’,$ $rank_{p}M\neq 1.$

So take any other choice ofchoose so that$E$

( choices). To have $N$ integral, we need to
$p$

$Y$

$E(N’-M’Y)\equiv(\begin{array}{ll}0 0* *\end{array}) (p)$;184
SIEGEL EISENSTEIN SERIES, HECKE OPERATORS, AND FOURIER EXPANSIONS

that satisfy this condition. Then we necessarily
there are choices forhave that $(MN)$ is a coprime pair. This gives us a contribution toof(b) Say $rank_{p}M’=2$ . To have $N$ integral we need
$Y$

$p$

$\mathbb{E}_{\rho}|T(p)$$p^{-k}\chi_{N_{0}N_{2}}(p)\cdot p^{2}\cdot p^{2k-3}\mathbb{E}_{1}.$$E(N’-M’Y)\equiv(_{*}^{0} 0*) (p)$ ,and to have $(MN)$ coprime we need$E(N’-M’Y)\not\equiv 0(p)$.

So we have $p+1$ choices for , and for each of these, $p-1$ choices for $Y.$
This gives us a contribution toof
We continue in this manner to evaluate the action of $T(p)$ and.
This gives us the following (Propositions 3.3 and 3.4 in [18]):$E$$\mathbb{E}_{\rho}|T(p)$$p^{-k}\chi_{\mathcal{N}_{0}\mathcal{N}_{2}}(p)\cdot(p^{2}-1)\cdot p^{2k-3}E_{2}.$$T_{1}(p^{2})$Proposition 2.1. For$p\nmid \mathcal{N}$, we have$\mathbb{E}_{(\mathcal{N}_{0}y_{1}y_{2})}|T(p)=\lambda(p)\mathbb{E}_{(\mathcal{N}_{0},\mathcal{N}_{1}y_{2})}$

where$\lambda(p)=(\chi_{N_{0}\mathcal{N}_{1}}(p)p^{k-1}+\chi_{\mathcal{N}_{2}}(p))(\chi_{N_{0}}(p)p^{k-2}+\chi_{\mathcal{N}_{1}\mathcal{N}_{2}}(p))$,

and$\mathbb{E}_{(\mathcal{N}oN_{1}N_{2})}|T_{1}(p^{2})=\lambda_{1}(p^{2})E_{(\mathcal{N}_{0}N_{1}N_{2})}$

where$\lambda_{1}(p^{2})=(p+1)(\chi_{\mathcal{N}_{0}}(p^{2})p^{2k-3}+\chi(p)p^{k-3}(p-1)+\chi_{N_{2}}(p^{2}))$.

, we proceed as before, but now the the comparison of
For a prime
andcontributes character sums on . Whenwe
get the Hecke relation$q|\mathcal{N}$$\chi_{\rho}(M’, N’)$$\chi_{\rho}(M, N)$$q|\mathcal{N}_{0}$$\chi_{q}$$\mathbb{E}_{(\mathcal{N}_{0}\lambda_{1}^{(}N_{2})}|T(q)=\lambda(q)E_{(\mathcal{N}0_{r}/V_{1}N_{2})}$$+a\mathbb{E}_{(\mathcal{N}_{0}/q,q\mathcal{N}_{1}N_{2})}+b\mathbb{E}_{(\mathcal{N}_{0}/qN_{1},q\mathcal{N}_{2})},$

where $a\neq 0$ iff $\chi_{q}=1,$ $b\neq 0$ iff$\chi_{q}^{2}=1$, and$\lambda(q)=\chi_{N_{1}}(q)\chi_{\mathcal{N}_{2}}(q^{2})$For$q|\mathcal{N}_{1}$.

, we have$\mathbb{E}_{(\mathcal{N}_{0}N_{1}N_{2})}|T(q)=\lambda(q)E_{(\mathcal{N}_{0}\chi_{1}y_{2})}+cE_{(\mathcal{N}_{0}N_{1}/q,q\mathcal{N}_{2})}$where $c\neq 0$ iff $\chi_{q}=1$ and$\lambda(q)=\chi_{N_{0}\mathcal{N}_{2}}(q)q^{k-1}$

for$q|\mathcal{N}_{2}$;, we have$\mathbb{E}_{(\mathcal{N}_{0}N_{1_{\gamma}}N_{2})}|T(q)=\lambda(q)\mathbb{E}_{(\mathcal{N}_{0}N_{1r}\mathcal{N}_{2})}$

where$\lambda(q)=\chi_{N_{0}}(q^{2})\chi_{\mathcal{N}_{1}}(q)q^{2k-3}.$With$q|\mathcal{N}$, we have similar statements for$E_{(\mathcal{N}oN_{1},\mathcal{N}_{2})}|T_{1}(q^{2})$, whereLYNNE H. WALLING$\lambda_{1}(q^{2})=\{\begin{array}{ll}\chi_{\mathcal{N}_{2}}(q^{2})(q+1) if q|\mathcal{N}_{0},\chi_{\mathcal{N}_{0}}(q^{2})q^{2k-3}+\chi_{\mathcal{N}_{2}}(q^{2})q if q|\mathcal{N}_{1},\chi_{\mathcal{N}_{0}}(q^{2})(q+1)q^{2k-3} if q|\mathcal{N}_{2}.\end{array}$Using these results, we construct a basis$\{\tilde{\mathbb{E}}_{(\mathcal{N}_{0},\mathcal{N}_{1},\mathcal{N}_{2})}:\mathcal{N}_{0}\mathcal{N}_{1}\mathcal{N}_{2}=\mathcal{N}, \chi_{\mathcal{N}_{1}}^{2}=1\}$

for
consisting of eigenforms for the full Hecke algebra; their eigengiven above.
values are the$\mathcal{E}_{k}^{(2)}(\mathcal{N}, \chi)$$\lambda,$$\lambda_{1}$REFERENCES[1] A.N. Andrianov, “Modular descent and the Saito-Kurokawa conjecture”, Inv. Math.
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[15] A. Ogg, Modular Forms and Dinchlet Senes, W.A. Benjamin, New York, 1969.
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