数理解析研究所講究録
第 1907 巻 2014 年 145-153

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On the computation of Siegel modular formsof degree 2 with Sage
Sho Takemori
Abstract
In this paper, we introduce a package of Sage [9] for the calculationof Siegel modular forms of degree 2.

1Introduction

Sage [9] is a free and open software for various areas of mathematics. WithSage, we can compute many number theoretical objects including modularforms of one variable i.e. elliptic modular forms. But we cannot computemodular forms of several variables such as Siegel modular forms with built-infunctions of Sage. The author wrote a package [10] for Siegel modular formsof degree two. In this paper, we introduce the package by computing Hecke
eigenforms. This paper does not contain any new mathematical results.

2Definitions

In this section, we recall the definition and related topics of Siegel modularforms.

2.1
LetbyDefinition of Siegel modular forms of degree
$n$

$n$

be a positive integer and define the Siegel modular group of degree$\Gamma_{n}:=\{g\in GL_{2n}(\mathbb{Z})|tgw_{n}g=w_{n}\}.$

$n$

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Herespace$w_{n}=(\begin{array}{ll}0_{n} -1_{n}1_{n} 0_{n}\end{array})$$\mathfrak{H}_{n}$. Note that$\Gamma_{1}=SL_{2}(\mathbb{Z})$. Define Siegel upper halfby$\mathfrak{H}_{n}:=\{Z=X+iY|X,$$Y\in Sym_{n}(\mathbb{R})$, $Y$ is positive $definite\}.$the the set of holomorphic functions
For a non-negative integer , let$F$ onsatisfying the following condition:$k$$M_{k}(\Gamma_{n})$$\mathfrak{H}_{n}$$F((AZ+B)(CZ+D)^{-1})=\det(CZ+D)^{k}F(Z)$ ,$\forall(\begin{array}{ll}A BC D\end{array})\in\Gamma_{n}.$a Siegel
If $n=1$ , we add the cusp condition. We call an element ofismodular form of degree and weight (and level 1). If $n=1,$
equal to the space of elliptic modular forms of weight . It is known thatis a finite dimensional vector space over$M_{k}(\Gamma_{n})$

$k$

$n$

$M_{k}(\Gamma_{1})$$k$$\mathbb{C}.$$M_{k}(\Gamma_{n})$2.2Let$\mathfrak{H}_{2}$Fourier expansion of Siegel modular forms of degree two$F\in M_{k}(\Gamma_{2})$. Then$F$be a Siegel modular form of degree 2. We put$Z=(\begin{array}{ll}\tau zz \omega\end{array})\in$

has the following Fourier expansion:$F( (\begin{array}{ll}\tau zz \omega\end{array}))=n, m,4nm-r^{2}\geq 0\sum_{n,r’ m\in \mathbb{Z}}a((n, r, m), F)e(n\tau+rz+m\omega)$

where $e(z)=e(2\pi iz)$ forSiegel operator :$\Phi$$z\in \mathbb{C}$,. With the notation above, we define the$M_{k}(\Gamma_{2})arrow M_{k}(\Gamma_{1})$by$\Phi(F):=\sum_{n=0}^{\infty}a((n, 0,0), F)e(nz)$We define the space of cusp forms$S_{k}(\Gamma_{2})$.

of degree 2 by

$S_{k}(\Gamma_{2}):=ker\Phi\subseteq M_{m}(\Gamma_{2})$

.

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2.3

Hecke polynomials

Let $n=1$ or 2. For, let $T(m)\in End_{\mathbb{C}}(M_{k}(\Gamma_{n}))$ the mth Hecke
operator. We omit the definition of $T(m)$ . See [2] for the definition. For aprime and, define a polynomial $Q_{p}^{(n)}(F;X)$ as follows.
$m\in \mathbb{Z}_{\geq 1}$$p$$F\in M_{k}(\Gamma_{n})$1. If $n=1$ , then we define$Q_{p}^{(1)}(F;X)=1-\lambda(p)X+p^{k-1}X^{2}$

2. If $n=2$ , then we define$Q_{p}^{(2)}(F;X)=1-\lambda(p)X$$+(\lambda(p)^{2}-\lambda(p^{2})-p^{2k-4})X^{2}-\lambda(p)p^{2k-3}X^{3}+p^{4k-6}X^{4}$

Remark 1.

3$Q_{p}^{(n)}(F,p^{-s})^{-1}$is the Euler factor of spinor -function of $F.$
$L$

Structure theorem for the ring of Siegel
modular forms of degree 2In this section, we recall the structure theorem for the ring of Siegel modular
forms of degree 2 proved by Igusa [4]. The structure theorem and the explicit
formula for Siegel Eisenstein series of degree 2 enable us to compute Siegel
modular forms of degree 2 explicitly.
Let$M( \Gamma_{2})=\bigoplus_{k\in \mathbb{Z}_{\geq 0}}M_{k}(\Gamma_{2})$be the ring of Siegel modular forms of degree 2. Put$x_{10}:=E_{4}E_{6}-E_{10},$$x_{12} :=3^{2}\cdot 7^{2}E_{4}^{3}+2\cdot 5^{3}E_{6}^{2}-691E_{12},$

whereis the Siegel-Eisenstein series of degree 2 and weight . Thenandare Siegel cusp forms of weight 10 and 12 respectively. For $k=10$ , 12,
we put$E_{k}$$k$$x_{12}$$X_{k}:= \frac{1}{a((1,1,1),x_{k})}x_{k}.$The following theorem was proved by Igusa [4].

$x_{10}$148
Theorem 1.
$X_{35}$1. There exists a weight 35 cusp form

$X_{35}$(we normalize

so that $a((2, -1,3), X_{35})=1$ ).

2.

$E_{4},$ $E_{6},$ $X_{10},$ $X_{12}$3.

$E_{4},$ $E_{6},$ $X_{10}$andgenerate

as a -algebra.

and$X_{35}$$X_{12}$are algebraically independent over

$M(\Gamma_{2})$$\mathbb{C}$$\mathbb{C}.$The Fourier coeﬃcients of Siegel-Eisenstein series of degree 2 was knownof weightby Kaufhold [5]. Aoki and Ibukiyama [3] proved that cusp form$X_{35}$35 can be written by a polynomial of Siegel-Eisenstein and its diﬀerentials.
can be written by the polynomials
Thus the generators of the ringof Siegel-Eisenstein series of degree 2 and its diﬀerentials. Therefore we canexplicitly.
compute the Fourier coeﬃcients of an element of$M(\Gamma_{2})$$M(\Gamma_{2})$4Computation of elliptic $mo$ dular forms

In this section, we compute Hecke polynomial of elliptic cusp forms by using
built-in functions of Sage.
R. $<x>=$ PolynomialRing(QQ, 1, order ’neglex’)def euler $factor_{-}of_{-}1$ $(f, p)$ :wt $=f$ . weight $()$
return $1-f[p]/f[1]*x+p^{-}(wt-1)*x^{arrow}2$
wts of $one_{-}\dim=\backslash$$[k$ forin range $(12,30)$
if CuspForms(l, k).dimension $()$ $==1$ ]
$=$

$k$

with
In the code above, we compute $Q_{p}^{(1)}(f;X)$ for $p=2$ and$\dim S_{k}(\Gamma_{1})=1$ . The function $euler_{-}$factor
o $f1$ takes an eigenform anda prime and returns $Q_{p}^{(1)}(f;X)$ . wts–of one–dim is the list of thepositive integers such that $12\leq k<30$ and $\dim S_{k}(\Gamma_{1})=1.$$f\in S_{k}(\Gamma_{1})$

$p$

$k$

euler $factor_{-}at_{-}2=$
in wts $of_{-}one_{-}dim$ :for$f=$ CuspForms $(l, k).$ basis ( $)[O]$
euler $factor_{-}at_{-}2[k]=$ euler $factor_{-}of_{-}1(f, 2)$$\{\}$$k$

The python’s dictionary $euler-factor_{-}at_{-}2$ is a dictionary such that, which value is as follows:$Q_{2}^{(1)}(f_{k};X)$$k\mapsto$

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sage: euler $factor_{-}at_{-}2${12: $1+24*x+2048*x^{\sim}2,$
16: $1-216*x+32768*x^{\sim}2,$
18: $1+528*x+131072*x^{arrow}2,$
20: $1-456*x+524288*x^{arrow}2,$
22: $1+288*x+2097152*x^{-}2,$
26: $1+48*x+33554432*x^{arrow}2\}$

For example the $Q_{2}^{(1)}(\triangle;X)=1+24X+2048X^{2}$ , wheredelta.

5$\Delta$

is the Ramanujan’s

Computation of Siegel modular forms inSage

In this section, we compute Siegel modular forms of degree 2 by using thepackage [10]. The following code has been tested under Sage 6.11 and “degree2”’ (revision $706bfe$ ).

5.1

Computation of generators of$M(\Gamma_{2})$The generator
can be obtained by the function $x10_{-}$with prec(prec).
Here the argument prec is a positive integer and this function computes theFourier coeﬃcients offor$X_{10}$$X_{10}$$\{(n, r, m)|0\leq n,$ $m\leq$prec, $4nm-r^{2}\geq 0\}.$can be obtained by the function $x12_{-}with_{-}prec($ prec) and$x35_{-}with_{-}prec($ prec) . Siegel-Eisenstein series
can be obtained by thefunction $eisenstein_{-}series_{-}degree2(k,$ prec) . Here are examples.
$X_{12}$

and$X_{35}$$E_{k}$$||\begin{array}{lllllll}\overline{fromdegree2.allimp\circ rt*} =4prec of\#\mathbb{R}e cusp forms weight 10 and 12X10=x10_{-}with_{-}prec(prec) X12=x12_{-}with_{-}prec(prec) \end{array}|$150$\#$

Fourier coeﬃcient of $XlO$ at $(1, 1, 1)$ .

$X10[(1,1,1)]\#=>1$
Fourier coeﬃcient of $XlO$ at $(3, 5, 4)$ .
$X10[(3,5,4)]\#=>2736$
$\#$

5.2

Computation of Hecke polynomials

and a small weight . For
We calculate $Q_{p}^{(2)}(F;X)$ for $p=2,$$F=X_{10}$ and $p=2$ , we can compute $Q_{p}^{(2)}(F;X)$ as follows:$k$$F\in S_{k}(\Gamma_{2})$sage: X10 $=x10_{-}with_{-}$ prec ( $4)$
sage: X10. euler factor $of_{-}spinor_{-}1(2)$ . factor$()$$(-1+256*x)*(-1+512*x)*(1+528*x+131072*x^{arrow}2)$
The last factor is equal toThus we have$Q_{2}^{(1)}(f_{18}, x)$, where$f_{18}\in S_{18}(\Gamma_{1})$is an eigenform.

$Q_{2}^{(2)}(X_{10};X)=(1-2^{8}X)(1-2^{9}X)Q_{2}^{(1)}(f_{18}, X)$.

Here is an another example.

sage: Q12 X12. euler factor $of_{-}spinor_{-}1(2)$sage: $Ql2.$ factor ($=$$)$$(-1+1024*x)*(-1+2048*x)*(1+288*x+2097152*x^{arrow}2)$
Thus we also have$Q_{2}^{(2)}(X_{12};X)=(1-2^{10}X)(1-2^{11}X)Q_{2}^{(1)}(f_{22}, X)$.

. We
is related to an eigenform of
But not every eigenform of
whose Hecke eigenvalue of $T(2)$compute cuspidal eigenformis not related to elliptic modular
is equal to $-840960$ . The cusp form
forms.
$S_{k}(\Gamma_{1})$$S_{k}(\Gamma_{2})$$X_{20}\in S_{20}(\Gamma_{2})$$X_{20}$is the space of cusp forms of degree 2 and weight 20.
sage: S20 CuspFormsDegree2 $(20,$ prec $=4)$
sage: $S20.$ hecke charpoly(2).factor $()$$\# X20$ is an eigenformof weight 20 whose eigenvalue of$\# T(2)$ is $-840960.$
sage: X20 S20. eigenform $with_{-}eigenvalue_{-}t2(-840960)$$\# S20$$=$$=$

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sage: X20. euler factor $of_{-}spinor_{-}1(2)$ . factor$()$$1+840960*x+390238044160*x^{-}2+115580662311813120*x^{-}3$$+18889465931478580854784*x^{-}4$

5.3

Maass relation and Saito-Kurokawa lift

In this subsection, we explain our examples above by the theorem proved by
Maass, Andrianov and Zagier.
Before we state the theorem, we introduce the Maass relation and Maass
subspace.

Definition 1 (Maass relation). For a cusp form
following condition.

$F\in S_{k}(\Gamma_{2})$, we consider the$a(n, r, m)= \sum_{d>0,d|gcd(n,r,m)}d^{k-1}a(1, r/d, mn/d^{2})$for all$n,$ $m,$ $4nm-r^{2}\geq

0$,(5.1). Here we put$a(n, r, m)=a((n, r, m), F)$ .

We denote by
the set of Siegel cusp forms
satisfying thecondition above. We callthe Maass subspace.
Forand $(n, r, m)$ , we can check the equation (5.1) by themethod $satisfies_{-}maass_{-}relation_{-}for.$sage: X10.satisfies maass $relation_{-}for(2,1,2\rangle$$F\in S_{k}(\Gamma_{2})$$S_{k}^{*}(\Gamma_{2})$$S_{k}^{*}(\Gamma_{2})$$F\in M_{k}(\Gamma_{2})$Truesage: X12. satisfies maass $relation_{-}for(2,1, 2)$True

sage: X20. satisfies maass $relation_{-}for(2,1, 2)$False

The following theorem was proved by Maass [6] [7] [8], Andrianov [1] andZagier [11]. And this theorem explains the relation between(resp.
)(resp.
).
and$X_{10}$$f_{18}$$X_{12}$$f_{22}$

Theorem 2. Let be an even number. The Maass subspaceis stable
under the action of Hecke operators. There exists one to one correspondence between an eigenformand an eigenform $f\in S_{2k-2}(\Gamma_{1})$given by$k$

$S_{k}^{*}(\Gamma_{2})$$a$$F\in S_{k}^{*}(\Gamma_{2})$$Q_{p}^{(2)}(F;X)=(1-p^{k-2}X)(1-p^{k-1}X)Q_{p}^{(1)}(f;X)$.

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Remark 2. The existence of the lift $f\mapsto F$ was conjectured by H. Saitoand Kurokawa independently.

Sho Takemori
Department of Mathematics,
Kyoto University
Kitashirakawa-Oiwake-Cho, Sakyo-Ku,
Kyoto, 606-8502, JapanE–mail: takemori@math.kyoto-u.ac.jp

References[1] A. N. Andrianov, Modular descent and the Saito-Kurokawa conjecture,
Inventiones mathematicae 53 (1979), no. 3, 267-280.
[2] A. N. Andrianov and V. G. Zhuravlev, Modular forms and Hecke operators, no. 145, American Mathematical Soc., 1995.

[3] H. Aoki and T. Ibukiyama, Simple graded rings of Siegel modular forms,
diﬀerential operators and Borcherds products, International Journal ofMathematics 16 (2005), no. 03, 249-279.
modular forms of genus two, American Journal of
Mathematics (1962), 175-200.

[4] J. Igusa, On$\mathcal{S}iegel$[5] G. Kaufhold, Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades, Mathematische Annalen 137 (1959),
no. 5, 454-476.
[6] H. Maass, \"Uber eine Spezialschar von Modulformen zweiten Grades, Inventiones mathematicae 52 (1979), no. 1, 95-104.
[7] –, \"Uber eine Spezialschar von Modulformen zweiten Grades (II),Inventiones mathematicae 53 (1979), no. 3, 249-253.

[8] –, \"Uber eine Spezialschar von Modulformen zweiten Grades (III),
Inventiones mathematicae 53 (1979), no. 3, 255-265.
[9] W. A. Stein et al., Sage Mathematics Software (Version 6.1.1), The Sage
Development Team, 2014, http: $//www$ . sagemath. org.

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[10] S. Takemori, degree2, https: $//$ github. com/stakemori/degree2.
[11] D. Zagier, Sur la conjecture de Saito-Kurokawa (d’apres H. Maass),
Seminar on Number Theory, Paris, vol. 80, 1979, pp. 371-394.