%Plain tex
\magnification=\magstep1
\vbadness=10000
\hbadness=10000
\tolerance=10000
\def\C{{\bf C}} % complex numbers
\def\Q{{\bf Q}} % rational numbers
\def\R{{\bf R}} % real numbers
\def\Z{{\bf Z}} % integers
\def\F{{\bf F}} % finite field
\def\Sp{\hbox{\rm Sp}}
\def\GL{\hbox{\rm GL}}
\def\Aut{\hbox{\rm Aut}}
\def\Ma{\hbox{\rm mass}}
\def\OO{{\rm O}}
\def\pii{\pi\hbox{\rm i}}
\proclaim
A Siegel cusp form of degree 12 and weight 12.
J. reine angew. Math 494 (1998) 141-153.
Richard E. Borcherds,
\footnote{$^*$}{ Supported by a Royal Society
professorship.}
D.P.M.M.S.,
16 Mill Lane,
Cambridge,
CB2 1SB,
England.
reb@dpmms.cam.ac.uk
\bigskip
E. Freitag,
Universit\"at Heidelberg,
Im Neuenheimer Feld 288,
D-69120 Heidelberg, Germany.
freitag@mathi.uni-heidelberg.de
\bigskip
R. Weissauer,
Universit\"at Mannheim,
D-68131 Mannheim, Germany.
weissauer@math.uni-mannheim.de
\medskip\noindent
It has been conjectured by Witt [Wi] (1941)
and proved later (1967) independently by Igusa [I]
and M.\ Kneser [K] that
the theta series with respect to the two unimodular even
positive definite lattices of rank 16 are linearly dependent in
degree $\le 3$
and linearly independent in degree 4.
In this paper we consider the next
case of the 24 Niemeier lattices of rank 24. The associated
theta series are linearly dependent in degree $\le 11$ and linearly
independent
in degree 12. The resulting Siegel cusp form of degree 12 and weight
12 is a Hecke eigenform which seems to have
interesting properties. We would like to thank G.\ H\"ohn for
helpful comments and hints.
\proclaim Construction of Siegel cusp forms by theta series.
\noindent
Let $\Lambda$ be an even unimodular positive definite
lattice, i.e.\ a free abelian group equipped with
a positive definite symmetric bilinear form $(x,y)$,
such that $\Lambda$ coincides with its dual and such that
$$Q(x):={1\over 2}(x,x)$$
is integral. By reduction mod $2$ we obtain
a quadratic form
$$q:E:=\Lambda/2\Lambda\longrightarrow\Z/2\Z,
\qquad q(a+2\Lambda)=Q(a)\hbox{ \rm mod }2.$$
on the $\Z/2\Z$-vector space $E$.
The standard theta series of degree $n$ with respect to
$\Lambda$ is
$$\vartheta_{\Lambda}(Z)=\sum_{g\in \Lambda^n}\exp\pi\hbox{\rm i}
\sigma(T(g)Z)\qquad(\sigma=\hbox{trace}),$$
where
$$T(g):=\bigl((g_i,g_j)\bigr)_{1\le i,j\le n}\qquad(g=(g_1,\dots,g_n)).$$
The variable $Z$ varies on the Siegel upper half plane
of degree $n$.
This is a modular form with respect to the full Siegel
modular group $\Sp(2n,\Z)$, but is not a cusp form.
The weight is $m/2$ if $m$ denotes the rank of $\Lambda$,
and $m$ is divisible by $8$.
To obtain a cusp from we modify this definition.
\smallskip
Assume that a function $\epsilon(F)$ is given which
depends on subspaces $F\subset E$.
For $g\in \Lambda^n$ we denote by $F(g)$ the image of
$\Z g_1+\cdots+\Z g_n$ in $E$.
For an arbitrary degree $n$ we define
$$f^{(n)}(Z):=\sum_{g\in\Lambda^n}\epsilon(F(g))\exp{\pii\over2}
\sigma(T(g)Z)\qquad(\sigma=\hbox{\rm trace}).$$
In general this will not be a modular form with
respect to the full modular group.
\smallskip
To construct a suitable function $\epsilon(F)$
we use the orthogonal group $\OO(E)$ of
the vector space $E$. It consists of all elements from
the general linear group $\GL(E)$ which preserve the quadratic form $q$.
It is a basic fact for our construction that $\OO(E)$ admits
a subgroup of index $2$. It is the kernel of the
so-called {\it Dickson invariant}.
We refer to [B] for some details.
To define the Dickson invariant we chose a basis
$e_1,\dots e_{m}$ of $E$ such that $q$ is of the form
$$q\Bigl(\sum_{i=1}^{m} x_ie_i\Bigr)=\sum_{j=1}^{m/2}x_jx_{m/2+j},$$
which is possible because all even unimodular lattices are equivalent
over $\Z/p\Z$ for any natural number $p$.
The orthogonal group $\OO(E)$ now appears as a subgroup of the symplectic
group $\Sp(m,\Z/2\Z)$. It consists of all symplectic
matrices $M=\left(A B\atop C D\right)$
such that the diagonals of $A'C$ and $B'D$ are
zero. This is the image of the so-called theta group.
It is easy to check that
$$D:\OO(E)\longrightarrow\Z/2\Z,\quad D(M)=\sigma(C'B),$$
is a homomorphism. This is the Dickson invariant.
It is non-trivial because if $a\in E$ is an element with
$q(a)\ne 0$ then the ``transvection''
$x\mapsto x-(a,x)a$ has non-zero Dickson invariant.
\smallskip
A subspace $F\subset E$ is called {\it isotropic\/}
if the restriction of $q$ to $F$ vanishes.
We now consider maximal isotropic subspaces of $E$.
Their dimension
is $m/2$. The orthogonal group $\OO(E)$ acts transitively on the
set of these spaces. But under the kernel of the Dickson invariant $D$
there are two orbits. Two spaces $F_1$ and $F_2$ are in the same
orbit if and only if their intersection has even dimension.
We select one of the two orbits and call it the first orbit
and call the other the second orbit.
\smallskip
We now define a special $\epsilon(F)$ as follows.
It is different from $0$ if and only if
$F$ is maximal isotropic. It is $1$ on the first orbit and $-1$
on the second one.
\smallskip
In the following we consider the system of modular forms $f^{(n)}$
constructed by means of this special $\epsilon(F)$.
\smallskip
Our first observation is that the functions $f^{(n)}(Z)$ have period $1$
in all variables and hence admit a Fourier expansion
$$f^{(n)}(Z)=\sum_T a_n(T)\exp(\pii\sigma(TZ)),$$
where $T$ runs over all integral symmetric matrices with even
diagonal.
Our next observation is that
the coefficients $a_n(T)$ are invariant under unimodular
substitutions $T\mapsto U'TU$, where $U\in\GL(n,\Z)$.
Let $L$ be an arbitrary even lattice of rank $n$. The
Gram matrix $T=\bigl((e_i,e_j)\bigr)$ with respect to a
basis of the lattice is determined up to unimodular
equivalence. We can define
$$a(L):=a_n(T).$$
An easy computation gives
$$a(L)=\#\Aut(L)\sum_{M}\epsilon(M/2M),$$
where the sum is over all $M$ such that
\item{1.} $M$ is a $n$-dimensional sublattice of $\Lambda$.
\item{2.} $M$ is isomorphic to $L(2)$.
($L(2)$ denotes the doubled lattice $L$.
It has the same underlying group as $L$ but the
norms $(x,x)$ are doubled.)
\item{3.} $M/2M$ is maximal isotropic in $\Lambda/2\Lambda$.
\smallskip
The group $\Aut(\Lambda)$ acts on the set of all $M$.
It acts also on the subspaces $F\subset E$. We later need to know
that this group preserves the Dickson invariant.
This is the case if $\Aut(\Lambda)$ is contained
in the special orthogonal group.
For this one has to use that
the composition of the natural homomorphism
$\Aut(\Lambda)\to\OO(E)$ with $(-1)^D$ is the determinant [B].
\smallskip
In the following we assume that all automorphisms of $\Lambda$
have determinant $+1$.
Otherwise all $f^{(n)}$ vanish. So we have to exclude
all lattices $\Lambda$ which contain a vector of norm $2$.
We can reformulate the formula
for the Fourier coefficients as
\proclaim Lemma 1. The Fourier coefficients $a(L)$ of
the functions $f^{(n)}$ are given by
$$a(L)=\#\Aut(\Lambda)\#\Aut(L)
\sum_M{\epsilon(F)\over\#\Aut(\Lambda,M)}\qquad(F=M/2M).$$
Here $M$ runs over a set of representatives of $\Aut(\Lambda)$-orbits
of sublattices of $\Lambda$ which are isomorphic to $L(2)$.
The group $\Aut(\Lambda,M)$ consists of all elements of
$\Aut(\Lambda)$ which preserve $M$ as a set.
\noindent
We want to prove now that $f^{(n)}$ is a modular form with
respect to the full modular group. More precisely $f:=(f^{(n)})$ is a
stable system of Siegel modular forms, i.e.\ $f^{(n)}$ can be obtained
from $f^{(n+1)}$ by applying the Siegel $\Phi$-operator.
It is known that every stable system can be written
in a canonical way as linear combination of the
standard theta functions $\vartheta_L$. This leads us to
the following construction of a linear combination of
standard theta series.
\smallskip
Let $F\subset E$ be a maximal isotropic space. We consider the inverse
image $\pi^{-1}(F)$ of $F$ under the natural projection
$\pi:\Lambda\to E$. The quadratic form $Q/2$ is even and unimodular
on $\pi^{-1}(F)$. In this way we obtain a new $m$-dimensional
even unimodular lattice $\Lambda_F$. This
is the so-called perestroika of $\Lambda$ with respect to $F$
in the notation of Koch and Venkov [KV].
\smallskip
We need some more notation. Let $\Lambda'$
be an even unimodular positive definite
lattice of dimension $m$. We introduce the mass and the
modified mass by
$$\eqalign{
\Ma(\Lambda')&=\sum_{\Lambda_F\cong
\Lambda'}{1\over\#\Aut(\Lambda,F)},\cr
\Ma^\epsilon(\Lambda')&=\sum_{\Lambda_F\cong
\Lambda'}{\epsilon(F)\over\#\Aut(\Lambda,F)},\cr}
$$
where $F$ runs over a system of representatives of $\Aut(\Lambda)$-orbits
of maximal isotropic subspaces of $E$ with
perestroika of type $\Lambda'$.
\smallskip
We fix a system $\Lambda_1,\dots,\Lambda_h$ of representatives
of isomorphism classes of such lattices $\Lambda'$ and write
$$\Ma(i):=\Ma(\Lambda_i),\quad\Ma^\epsilon(i):=\Ma^\epsilon(\Lambda_i).$$
\proclaim Theorem 2. We have
$$f=\#\Aut(\Lambda)\sum_{i=1}^h\Ma^\epsilon(i)\vartheta_{\Lambda_i}.$$
In particular the $f^{(n)}$ are modular forms with respect to the
full modular
group. The forms $f^{(n)}$ vanish for $n1$
this is not always true.
\smallskip
To obtain information about the Satake parameters
(at the prime $p=2$) we need the image of the operator $T(p)$
in the local Hecke algebra. This formula can be found in
[F]. We
choose a root $y_j=\sqrt{ x_j}$ for
each Satake parameter. A direct consequence of
formula [F], IV.3.14, a) and b) is
$${\lambda(p)^2\over x_0^{-2}x_1\cdots x_n}=\prod_{j=1}^{12}
\bigl(y_j+y_j^{-1}\bigr)^2\quad\hbox{\rm and}\quad
p^{{n(n+1)\over2}-12n}=x_0^{-2}x_1\cdots x_n.$$
The computed value $\lambda(2)$ now gives:
\proclaim Theorem 13. The Satake parameters $x_i=y_i^2$ of our
cusp form of degree $12$ and weight $12$
at the place $p=2$ satisfy
$$\left\vert\prod_{i=1}^{12}(y_i+y_i^{-1})\right\vert=
{3^{11}\cdot5\cdot17\cdot901141\over 2^{26}}.$$
Remark added September 23 1999: unfortunately
theorem 13 is wrong. For the correct
value see the preprint ``On a conjecture of Duke-Imamoglu'' by
Breulman and Kuss, or see the paper by Ikeda mentioned in the next section.
\smallskip\noindent
{\bf Corollary.} The Ramanujan conjecture $\vert x_i\vert=1$
is violated for $p=2$.
\proclaim Open problems.
\noindent
We list a few questions about the Siegel cusp form $f$.
\smallskip
\item{1.} Are the coefficients of
the cusp form of weight 12 and degree 12 all integers when
normalized so that the coefficient of $D_{12}$ is 1?
\smallskip\item{}
One can prove that the coefficients of $f^{(m/2)}/\#\Aut(\Lambda)$
are contained in $\Z[1/2]$ for arbitrary $\Lambda$ and
in $(1/2)\Z$ in case of the Leech lattice. This means
that the denominators of the normalized coefficients divide
$2^7\cdot3^5\cdot5^2\cdot7$.
\smallskip
\item{2.} Why are the coefficients of $f$ similar to those
of the modular form above? Is there a similar relation for
the coefficients of lattices of determinant $1\bmod 8$?
Is it possible to write down some simple explicit
formula for the coefficients of $f$?
\smallskip\item{}
{}From [We] it follows that the standard $L$-function $L(f,s)$
of $f$
has a pole at $s=1$. This suggests that $L(f,s)=\zeta(s)L(s)$,
where $L$ belongs to a $24$-dimensional $l$-adic
Galois representation. This Galois
representation cannot be pure (theorem 13) and therefore
one might expect that its weight filtration
sheds light on the relationship with
$\eta(8 \tau)^{12} \theta(\tau)$.
Remark added September 23 1999: the preprint ``On the lifting of
elliptic modular forms to Siegel cusp forms of degree 2n'' by Tomotsu
Ikeda (dated September 22 1999) proves a general lifting theorem which
includes the cusp form of this paper as a special case. In particular
he shows how to calculate its coefficients in terms of those of the weight
$13/2$ cusp form related to $\Delta(\tau)$ by the Shimura
correspondence, and shows that the coefficients depend only on the
genus of the corresponding lattice. His results also seem to imply that the
coefficients of the cusp form in this paper are always integers.
\bigskip\noindent
%
\proclaim References.
\item{[A]} A.\ Andrianov.
On zeta-functions of Rankin type associated with Siegel
modular forms,
Modular Functions of One Variable VI, 325-338,
Lecture Notes in Mathematics {\bf 627}, Springer-Verlag
Berlin Heidelberg New York, 1977
\smallskip
\item{[B]} N.\ Bourbaki. \'El\'ements
de Math\'ematique,
Livre VI, Deuxi\'eme Partie, Groupes et Alg\'ebre de Lie,
1.\ Hermann, Actualit\'es Scientifique
et Industrielles, {\bf 1293}, 1962.
\smallskip
\item{[CS]} J.\ H.\ Conway, N.\ J.\ A.\ Sloane.
Sphere packings, lattices
and groups. Springer-Verlag New York,
Grundlehren der mathematischen
Wissenschaften {\bf 290}, 1988
\smallskip
\item{[DLMN]}
C.\ Dong, H.\ Li, G.\ Mason, S.\ P.\ Norton.
Associative subalgebras of the Griess algebra and related topics,
preprint, http://xxx.lanl.gov/abs/q-alg/9607013, 1996
\smallskip
\item{[F]} E.\ Freitag. Siegelsche Modulformen,
Springer-Verlag Berlin New York,
Grundlehren der
mathematischen Wissenschaften {\bf 254}, 1983
\smallskip
\item{[I]} J-I.\ Igusa.\ Modular forms and projective invariants,
Am.\ J.\ of Math. {\bf 89}, 817--855, 1967.
\smallskip
\item{[K]} M.\ Kneser. Lineare Relationen zwischen Darstellungsanzahlen
quadratischer Formen, Math.\ Annalen {\bf 168}, 31--36, 1967
\smallskip
\item{[KV]} H.\ Koch, B.B\ Venkov.\ \"Uber ganzzahlige unimodulare
euklidische Gitter,
J.\ Reine Angew.\ Math. {\bf 398}, 144--168, 1989
\smallskip
\item{[We]} R.\ Weissauer.
Stabile Modulformen und Eisensteinreihen,
Lecture notes in Mathematics {\bf 1219},
Springer-Verlag Berlin Heidelberg New York,
1986
\smallskip
\item{[Wi]} E.\ Witt. Eine Identit\"at zwischen
Modulformen zweiten Grades, Math.\ Sem.\ Hamburg
{\bf 14}, 323--337, 1941
\bye