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数理解析研究所講究録
第2036巻 2017年 92-99

EXAMPLES OF SIEGEL MODULAR FORMS IN THE
KERNEL OF THE THETA OPERATOR MOD p
SIEGFRIED BÖCHERERHIROTAKA KODAMASHOYU NAGAOKA

ABSTRACT. We construct many examples of Siegel modular forrns
in the kernel of the theta operator \mathrm{m}\mathrm{o}\mathrm{d} p by using theta series.

1. INTRODUCTION

Ramanujans $\theta$ operator is a familiar topic in the theory of elliptic
modular forms, defined by

f=\displaystyle \sum a(n)e^{2 $\pi$ inz}\mapsto $\theta$(f)=\frac{1}{2 $\pi$ i}f'=\sum na(n)e^{2 $\pi$ inz}
For Siegel modular forms of degree n the Fourier expansion runs,

positive semidefinite half‐integral matrices of size \mathrm{n} and we
can define several analogues of the Ramanujan $\theta$ operator: For 1\leq
overr\leq n we may introduce

F=\displaystyle \sum_{T}a(T)e^{2 $\pi$ itr(TZ)}\mapsto$\Theta$^{[r]}(F)=\displayst le\frac{1}(2$\pi$ )^{r}(\frac{\partial}{\partial_{ij})^{[r]}F= \displaystyle \sum_{T}T^{[r]}a(T)e^{2 $\pi$ itr(TZ)},
where, for a matrix A of size n,

we denoteby A^{[r]} the matrix of all

the determinants of its submatrices of size \mathrm{r} and \partial_{ij}:=\displaystyle \frac{1}{2}(1+$\delta$_{ij})\frac{\partial}{\partial zij}.

In general, $\Theta$^{[r]}F is no longer a modular form, but it is a modularform \mathrm{m}\mathrm{o}\mathrm{d} p (even a p ‐adic modular form), vector‐valued if r<n,see[5].

Our aim in this note is to explore the existence and explicit con‐struction of Siegel modular forms which are in the kernel of such

93$\Theta$ ‐operators mod p, we focus on the scalar‐valued casej=n for,details and the general case we refer to [7].
Obvious candidates for such modular forms are theta series$\theta$_{s}^{n}$\theta$_{s}^{n}(Z)=\displaystyle\sum_{X\in\mathrm{Z}^{(n )} ,e^{$\pi$itr(X^{t}SXZ)},
attached to even integral positive quadratic forms S of rank n andlevel being a positive power of p ; we will also consider variants ofthis involving a hamonic polynomial:$\theta$_{S,\det}^{n}(Z)=\displaystyle \sum_{X\in \mathrm{Z}^{(n )} ,\det(X)e^{ $\pi$ itr(X^{t}SXZ)}.
Looking at the Fourier expansion, evidently such theta series arein the kernel of $\Theta$^{[n]}\mathrm{m}\mathrm{o}\mathrm{d} pare.

On the other hand, these theta series

not of level one and one has to do a level change to level one

by increasing the weight by a suitable multiple of p-1 This isalways possible, see [5]. This method works only for even degree.

n,because we otherwise enter into the realm of modular forms ofhalf‐integral weight, but a somewhat weaker variant of our method
is also possible for the case of odd n.
On the other hand, for even degree, our method provides plenty ofexamples for level one forms F (of weight in an arbitrary congru‐
ence class modulo p-1 ) which satisfy$\Theta$^{[n]}F\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} p the only
obstruction comes from the arithmetic of quadratic forms, which,puts some constraints on the existence of positive definite forms

with the requested properties (e.g. for n=2 there do not exist pos‐

itive definite quadratic forms of level p if p\equiv 1\mathrm{m}\mathrm{o}\mathrm{d} 4 ).
,We have to make an important comment on what we mean by ex‐plicit construction here: The kernel of $\Theta$^{[n]}\mathrm{m}\mathrm{o}\mathrm{d} p is a notion which
depends only on modular forms \mathrm{m}\mathrm{o}\mathrm{d} p therefore the weight of theconstructed modular form is only of interest \mathrm{m}\mathrm{o}\mathrm{d} (p-1) We callour procedure weak construction if we are only interested in theweight \mathrm{m}\mathrm{o}\mathrm{d} p-1 On the other hand, one is also interested in ex‐
plicit small uue weights for which we can get modular forms in
the kernel \mathrm{m}\mathrm{o}\mathrm{d} p In this note we focus on the latter strong con‐,.

.

.

struction

Indeed, in some cases our strong construction is the best

possible in the sense that the level one form F obtained from the94

theta series has the smallest possible weight among all forms con‐gruent to it, i.e. the weight of F is equal to its filtration in thesense of Serre andSwinnerton‐Dyer, see [6].

In the final section we also show that some of the known exam‐

ples of congruences for degree two Siegel modular forms can beexplained by our methods.
2. PRELIMINARIESFor standard facts about Siegel modular forns we refer to [1, 8, 10].

The group Sp(n,\mathbb{R}) acts on the upper half space \mathbb{H}_{n} in the usual way
and on functions f : \mathbb{H}_{n}\rightar ow \mathbb{C} by the standard slash‐operator.
We will mainly be concerned with modular forms for congruencesubgroups of type$\Gam a$_{0}^{n}(p):=\{M=\left(\begin{ar ay}{l AB\ CD \end{ar ay}\right)|C\equiv0\mathrm{m}\mathrm{o}\mathrm{d}p\},where p is an odd prime. The nebentypus character will be trivial
or thequadratic character$\chi$_{p}(*):=(^{\underline{(-1)^{\frac{p-1}{2} p} *1

for an odd prime p.

3. EXAMPLES
We exibit here some examples of degree 2, which should be ex‐plained by our construction.
3.1. k= 12, p=23

Let $\Delta$_{12} be Ramanujans cusp form of weight.

12 and [$\Delta$_{12}] the Klingen‐Eisenstein series of degree 2 attached toit. It was shown in [2] that this is not congruent zero \mathrm{m}\mathrm{o}\mathrm{d} 23 but,$\Theta$^{[2]}([$\Delta$_{12}]\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} 23
3.2. k=16, p=31 Mizumoto [12] found a similar example, byconsidering the unique normalized eigen cups form $\Delta$_{16} of weight
16 and its Klingen‐Eisenstein [$\Delta$_{16}\mathrm{J};^{-} he showed that.

$\Theta$^{[2]}([$\Delta$_{16}]\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} 31

953.3. k= 35, p = 23.

Let $\chi$_{35} be the wellknown cusp form of de‐gree 2 and weight 35, then it was shown in [9] by using explicit
computations and a Smrm bound that$\Theta$^{[2]}($\chi$_{35})\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} 23.
A similar example was exhibited in [11] for weight 47 and a con‐gruence \mathrm{m}\mathrm{o}\mathrm{d} 31 We understand that in the meantime S.Takemori
has found many more numerical examples of similar type in degree.

2.
It will follow from our considerations that all these examples canbe explained by our constructions, using binary quadratic forms.
3.4.

Siegel Eisenstein series: By quite different methods the lastauthor showed that for even degree \mathrm{n} and primes p with p>n+3
and p\equiv(-1)^{\frac{n}{2} \mathrm{m}\mathrm{o}\mathrm{d} 4 the Siegel Eisenstein series of degree n andweight \displaystyle \frac{n}{2}+t\cdot\frac{p-1}{2} is in the kemel of $\Theta$ if t is an odd positive integer,see\mathfrak{l}3.5. A very simple construction. We have to mention that we canget examples by startming from an arbitrary Siegel modular.form fof level one, we may.then consider g(z) :=f(pz) ; this is a modular
form for the congruence subgroup $\Gamma$_{0}(p) and therefore g is congru‐
ent to a level one form G ; this G is in the kemel \mathrm{m}\mathrm{o}\mathrm{d} p of the theta

operator.
This construction shows that (at least for even degree) for all weights
mod p‐l we can get modular forms in the kernel òf Theta operators\mathrm{m}\mathrm{o}\mathrm{d} p.
4. MAIN RESULTS
The theorems below produce a lot of examples of level one Siegel

modular forms of explicit ĩow weight in the kernel of $\Theta$^{[n]}\mathrm{m}\mathrm{o}\mathrm{d} p.

Note however that the arithmetic of quadratic forms puts constraints
on the existence of thepositive even integral quadratic forms used.

Theorem 4.1. Let S be an even integral positive definite quadratic

form of even rank n with determinant p^{2} and level p with p\geq n if96p\equiv 1\mathrm{m}\mathrm{o}\mathrm{d} 4 and p\geq n+3 in general.
Then there exists a modular form F of level one and weight

k=\displaystyle \frac{n}{2}+(p-1)
such that F\equiv$\theta$_{s}^{n}(Z).

Theorem 4.2. Let S be an even integral positive definite quadratic
form of even rank n with determinant p and level p where p is aprime with p\geq 2n+3.
Then there exists a modular form F of level one and weight k\displaystyle \frac{n}{2}+\frac{p-1}{2} such that F\equiv$\Psi$_{s}(Z)=.

Both these theorems follow directly from Theorem 4 and Theorem

5 (corrected version) in [4].
The next result is concerned with similar results for theta series withharmonic polynomial:

Theorem 4.3. Let S be an even integral positive definite quadratic

form with determinant p Then there exists a cusp form F of level.

one ofwithweightF\equiv$\theta$_{S,\det}^{n}\mathrm{m}\mathrm{o}\mathrm{d} p.

k=\displaystyle \frac{n}{2}+1+3\cdot\frac{p-1}{2}Remark 4.4. Even if such S exists, one has to impose two addi‐
tional conditions on the integral automorphisms of S to assure that$\theta$_{S,\det}^{n} is not congruent zero \mathrm{m}\mathrm{o}\mathrm{d} p : There should not exist improper

integral automorphisms of S and S should not have an integral au‐
tomorphism of order p.

Remark 4.5. The results above can be refined to give a lower boundfor the number of \mathrm{m}\mathrm{o}\mathrm{d} p linearly independent elements in the kernelof $\Theta$^{[n]}\mathrm{m}\mathrm{o}\mathrm{d} p.

5. ABOUT THE PROOFS

In most cases, the proof relies on finding a suitable modular form$\epsilon$ of level \mathrm{p} such that 8\equiv

1\mathrm{m}\mathrm{o}\mathrm{d}_{\backslash }p and such that $\epsilon$ has good97

properties mod \mathrm{p} in all cusps. Then one can hope that for a given
modular form f for $\Gamma$_{0}(p) the function

F :=trace(f\cdot$\epsilon$^{t})
has the desired properÍy \mathrm{m}\mathrm{o}\mathrm{d} p for a suitable power of $\epsilon$ Here the.

trace denotes a summation over representatives of $\Gamma$_{0}(p)\backslash Sp(n,\mathbb{Z})and in our special case f denotes the theta series in question. There
are several variantsof this basic procedure, see e.g. [4, 5]. In par‐ticular, for Theorem 4.3., a very delicate choice of $\epsilon$ is necess\backslash\mathrm{a}\mathrm{r}\mathrm{y}.
It is essential here to understand the p ‐adic properties of(f\cdot$\epsilon$^{t})|$\omega$_{j} (0\leq j\leq n)
where the matrices$\omega$_{j}\left(bgin{ary}l 0_{j}&0 -1_{j}&0\ &\mathr{l}_n-j&0 _{n-j}\ mathr{l}_j&0 _{j}&0\ &0_{n-j}&0 1_{n-j} \end{ary}\ight)

parametrize the (zero‐dimensional) cusps of Sp(n, \mathbb{Z})We emphasize that the choice of $\epsilon$ is easier, if we do not worry
about the weight of F.
.

6. THE EXAMPLES MOD 23
The class number of \mathbb{Q}(\sqrt{-23}) is 3, the three inequivalent binary

quadratic forms of discriminant -23 are given bys_{0}=\left(\begin{ar ay}{l} 2&1\ 1&12 \end{ar ay}\right),S_{\pm1}=\left(\begin{ar ay}{l} 4&\pm1\ \pm\mathrm{l}&6 \end{ar ay}\right)

where S_{1} and S_{-1} are improperly equivalent. By Theorem 4.2. themodular forms$\theta$_{s_{0} ^{2}\pm$\theta$_{s_{1} ^{2}are congruent \mathrm{m}\mathrm{o}\mathrm{d} 23 to level one forms

of weight 12=

1+\displaystyle \frac{23-1}{2}.

This should be compared with example 3.1 and with the case ofSiegel Eisenstein series from 3.4.
Furthermore, by Theorem 4.3. the weight 2 theta series$\theta$_{S_{1},\det}^{2}

98

is congruent \mathrm{m}\mathrm{o}\mathrm{d} 23 to a level one cusp form of weight

35=2+3\displaystyle \cdot\frac{23-1}{2}.
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[5] S.Böcherer, S.Nagaoka, On p‐adic properties of Siegel modular forms. In:
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[7] Böcherer,S., Kodama,H. Nagaoka,S.: On the kernel of theta operators modp. Preliminary manuscript 2016
E. Freitag, Siegelsche Modulfunktionen. Springer, Berlin(1983)[8][9] T Kikuta, H. Kodama, S. Nagaoka, Note on Igusas cusp form of weight 35,
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vol270, Springer‐Verlag 1985Siegfried Böcherer
Kunzenhof 4\mathrm{B}79117 Freiburg

Germanyboecherer@math.uni‐mannhetm.de

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Hirotaka Kodama

Academic Support Center

Kogakuin UniversityHachioji
Tokyo 192‐0015
kt13511@ns.kogakuin.ac.jp
Shoyu Nagaoka
Department of Mathematics
Kindai University
Higashi‐Osaka
Osaka 577‐8502

nagaoka@math.kindai.ac.jp