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The moduli space of Enriques surfaces and the fake monster Lie superalgebra.
\hfill 27 October 1994,
corrected 27 January 1995.
Topology vol. 35 no. 3, 699-710, 1996.
Richard E. Borcherds,\footnote{$^*$}{ Supported by NSF grant
DMS-9401186.}
Mathematics department,
Evans Hall \#3840,
University of California at Berkeley,
CA 94720-3840
U. S. A.
e-mail: reb@math.berkeley.edu
\bigskip
MSC numbers: 14J28, 11F55.
\bigskip
We show that the moduli space of complex Enriques surfaces is an
affine variety with a copy of the affine line removed. We do this by
using the denominator function of a generalized Kac-Moody superalgebra
(associated with superstrings on a 10-dimensional torus) to construct a
non-vanishing section of an ample line bundle on the moduli space.
\proclaim Contents.
1. Introduction.
Notation.
2.~The lattice $M$ and the moduli space of Enriques surfaces.
3.~Construction of the automorphic form $\Phi$.
\proclaim
1.~Introduction.
The moduli space $D^0$ of Enriques surfaces is known to be the
quotient $D$ of a 10-dimensional hermitian symmetric space $\Omega$ by
a discrete group $O_M(\Z)$, with a divisor $H_d$ removed. The
symmetric space $\Omega$ has a $O_M(\Z)$-line bundle $P$ over it such
that the sections of $P^n$ are essentially automorphic forms of weight
$n$. The line bundle $P$ is ample on $D$ and some power defines an
embedding of $D=\Omega/O_M(\Z)$ into projective space, which makes
$D$, and hence $D^0$, into a quasiprojective variety. We will prove
\proclaim Theorem 1.1. The moduli space $D^0$ is a quasiaffine
variety.
To prove this we will show that the ample bundle $P^4$ is trivial when
restricted to the complement of the divisor $H_d$, so that the trivial
bundle over the moduli space is ample and therefore the moduli space
is quasiaffine (and not just quasiprojective). Sections of $P^4$ are
essentially the same as automorphic forms of weight $4$, so to show
that $P^4$ is trivial we construct an automorphic form $\Phi$ (theorem
3.2) of weight 4 on the hermitian symmetric space $\Omega$ which has a
zero of order 1 along the divisors $H_d$ and has no other zeros. This
automorphic form $\Phi$ then defines a trivialization of $P^4$
restricted to the moduli space $D^0$.
The function $\Phi$ is constructed in [B92] as a twisted denominator
function of the fake monster Lie algebra, associated to an
automorphism of order 2 of the Leech lattice fixing an 8-dimensional
subspace. The fact that $\Phi$ is an automorphic form should follow
from a generalization of the results of [B95] from the level 1 case
covered there to higher levels. As this generalization has not yet
been done, we prove that $\Phi$ is an automorphic form (in section 3)
by an ad hoc argument using the fact that $\Phi$ can be written as
either an infinite product or an infinite sum. We then show that
$\Phi$ has no zeros other than the hyperplanes $H_d$.
It is not hard to describe precisely how $D^0$ differs from an affine
variety: it is an affine variety with a copy of the affine line $\C^1$
removed. This follows from the description of the Baily-Borel
compactification of $D$ given by Sterk in [S, 4.5, 4.6, 4.7]. For the
readers convenience we briefly recall Sterk's results. The Baily-Borel
compactification consists of $D$ together with 2 points corresponding
to the 2 orbits of primitive isotropic vectors in $M$, and two
1-dimensional pieces isomorphic to $\C$ and $\C^*$ corresponding to
the two orbits of primitive isotropic rank 2 sub-lattices of $M$. The
closure of the divisor $H_d$ of $D$ contains one of the points and the
1-dimensional component $\C$. The complement of the closure of $H_d$
is an affine variety, and this affine variety is just the union of
$D^0$, a copy of $\C^*$, and a point. The copy of $\C^*$ is
constructed as the quotient of the upper half plane by the group
$\Gamma_0(2)=\{{ab\choose cd}\in SL_2(\Z)|c\equiv 2 \bmod 0\}$ and the
point is then just one of the cusps, so the union of $\C^*$ and this
point is just a copy of the affine line $\C$. Hence the moduli space
$D^0$ is an affine variety with a copy of the affine line $\C$
removed.
I. Dolgachev pointed out to me that the form constructed in theorem
3.2 might be one case of an infinite family of forms as follows. Let
$R$ denote one of the following four division algebras ${\bf R},{\bf
C},{\bf H},{\bf O}$ of real, complex, Hamiltonian, or Cayley numbers.
Let $S_n(R)$ denote Hermitian $n\times n$ matrices with entries in $R$
and let $S_n(R)^+$ be the cone of positive definite matrices. Consider
the tube domain $ S_n(R)+iS_n(R)^+$. Except when $R = {\bf O}$ and $n
> 3$ it is a symmetric bounded domain. When $n = 2,R = {\bf O}$ we get
the domain $\Omega$ whose quotient $D$ with a divisor $H_d$ removed
parameterizes Enriques surfaces. Let $\Gamma(R)$ be the arithmetic
group $Sp(2n,\Z)$ ($R = \R$), $GL(2n,\Z[i])$ ($R = \C$), $SU(2n,{\bf
\cal O}_{\bf H})$ ($ R = {\bf H}$), $O_M(\Z)$ ($R = {\bf O}, n = 2)$.
There might be a similar $\Gamma$-modular form on each of these
spaces. The complement of its zeroes should be the period space of a
family of (possibly non simply connected) Calabi-Yau manifolds of
dimension $n$. The existence of a such a form is known for $n = 2$,
$R\ne {\bf H}$. If $R = \R$, $ S_2(\R)+iS_2(\R)^+$ is the
3-dimensional Siegel space $Z_2$, and the family is the family of
Kummer surfaces. In the case $\C$, the domain $ S_2(\C)+iS_2(\C)^+$is
4-dimensional of type $I_{2,2}\cong IV_4$ and the family is the family
of K3-surfaces which are nonsingular models of branched covers of the
plane ramified over the union of six lines (see [M] for the
construction of the corresponding form). The case $R={\bf O}$ is
theorem 3.2. If $n$ is arbitrary and $R = \C$ Dolgachev conjectures
that the family is the family of Calabi-Yau $n$-folds which are
obtained by a resolution of double covers of ${\bf P}^n$ branched
along $2n+2$ hyperplanes in general position.
Kondo [K] has recently proved that the moduli space of Enriques
surfaces is rational.
It is also possible to construct automorphic forms related to the
moduli spaces of polarized K3 surfaces using similar methods (as A.
Todorov suggested to me). For example, the form constructed in example
4 of section 16 of [B95] associated with the Dynkin diagram $E_7$ is
an automorphic form for the group $O_{II_{2,18}\oplus
\langle-2\rangle}(\R)$ which vanishes exactly on the hyperplanes of
norm $-2$ vectors, and is very closely related to the moduli space of
K3 surfaces with a polarization of degree 2.
I thank I. Dolgachev, A. Torodov, N. I. Shepherd-Barron, and the referee for
explaining moduli spaces of K3 and Enriques surfaces to me and for
suggesting several improvements and corrections.
\proclaim Notation and terminology.
All varieties are defined over the complex numbers. The bilinear forms
on lattices have the opposite signs to those in [B95]; this is because the
sign conventions in algebraic geometry are the opposite of those
in the theory of Lorentzian lattices.
\item{${}'$} If $M$ is a lattice then $M'$ means the dual of $M$.
\item{${}^+$} If $G$ is a subgroup of $O_M(\Z)$ then $G^+$
is the subgroup of $G$ of elements not interchanging the two
components of $\Omega$.
\item{$c(n)$} $\sum_nc(n)q^n=\eta(\tau)^{-8}\eta(2\tau)^8\eta(4\tau)^{-8}$.
\item{$C$} An open solid cone in $L\otimes \R$.
\item{$\C$} The complex numbers.
\item{$\Delta(\tau)$} $=\eta(\tau)^{24}$.
\item{$d$} A norm $-2$ vector of $M$.
\item{$D$} The complex space $\Omega/O_M(\Z)$.
\item{$D^0$} The moduli space of Enriques surfaces, which
is $D$ with the divisor $H_d$ removed.
\item{$E_4(\tau)$} $=1+240 \sum_{m>0,n>0} m^3q^{mn}$.
\item{$E_8$} The $E_8$ lattice. If $n$ is an integer
then $E_8(n)$ means $E_8$ with the values of the bilinear form
multiplied by $n$.
\item{$\eta(\tau)$} $=q^{1/24}\prod_{n>0} (1-q^n)$.
\item{$\Phi$} An automorphic form of weight 4.
\item{$f(\tau)$} $=\eta(\tau)^{-8}\eta(2\tau)^8\eta(4\tau)^{-8}$.
\item{$\Gamma_1$} The subgroup of $O_M(\Z)$ generated by reflections
of $R_2$ and by $-1$.
\item{$\Gamma_2$} The subgroup of $O_M(\Z)$ generated by reflections
of $R_0\cup R_2$ and by $-1$.
\item{$\Gamma_3$} A finite index subgroup of $O_M(\Z)$ defined in lemma 2.5.
\item{$H_d$} The points of $\Omega$ which are orthogonal to
the norm $-2$ vector $d\in M$.
\item{$\Im(y)$} The imaginary part of $y$.
\item{$II_{m,n}$} The even unimodular Lorentzian lattice of
dimension $m+n$ and signature $m-n$.
\item{$I_5$} A modified Bessel function.
\item{$L$} The lattice $E_8(-2)\oplus II_{1,1}$. The element $(v,m,n)\in L$
with $v\in E_8(-2)$, $m,n\in \Z$ has norm $v^2+2mn$.
\item{$\lambda$} A vector in $L'$.
\item{$M$} The lattice $L \oplus II_{1,1}(2)$.
The element $(v,m,n)\in M$
with $v\in L $, $m,n\in \Z$ has norm $v^2+4mn$.
\item{$\mu$} A vector in $L$.
\item{$m,n$} Integers.
\item{$O_M(\Z)$} The group of all automorphisms of the lattice $M$.
\item{$\Pi^+$} The set of positive vectors of $L$, i.e.,
the vectors which have positive inner product with $\rho$ or
are positive multiples of $\rho$.
\item{$q$} $e^{2\pi i \tau}$
\item{$\Q$} The rational numbers.
\item{$\rho,\rho'$ } The norm 0 vectors $\rho=(0,0,1)$
and $\rho'=(0,1,0)$ of the lattice $E_8(-2)\oplus II_{1,1}$.
\item{$r$} A norm $-2$ vector of $M$.
\item{$R_0,R_2$} The sets of norm $-2$ vectors of $M$
which have inner product 0 or 2 with $u$.
\item{$S$} The surface of points $y\in iC$ with $(y,y)=-1$.
\item{$\tau$} A complex number with positive imaginary part.
\item{$u$} The norm 0 vector $(0,0,1)\in L\oplus II_{1,1}(2)= M$.
\item{$v$} A vector of $L$.
\item{$W$} The reflection group of the lattice $L=E_8(-2)\oplus II_{1,1}$
generated by the reflections of norm $-2$ vectors.
\item{$\chi$} A homomorphism of $O_M(\Z)^+$ to $\{\pm 1\}$
taking reflections of norm $-2$ vectors to $-1$ and reflections
of norm $-4$ vectors to 1.
\item{$y$} A vector in $L\otimes \R+iC$.
\item{$\Psi(y)$} $=\Phi(y+(\rho-\rho')/2)$.
\item{$\Omega$} The hermitian symmetric space (with 2 components)
associated with the lattice $M$, consisting of all points $\omega\in
P(M\otimes \C)$ such that $(\omega,\omega)=0$ and
$(\omega,\bar\omega)>0$.
\proclaim 2.~The lattice $M$ and the moduli space of Enriques surfaces.
In this section we recall some facts about the moduli space of Enriques
surfaces, and the associated lattice and symmetric space. Many
of these results can be found in [B-P-V chapter VIII].
We define the lattice $M$ to be $L\oplus II_{1,1}(2)$, where
$L=E_8(-2)\oplus II_{1,1}$, $E_8$ is the $E_8$ lattice, and $II_{1,1}$
is the 2-dimensional even indefinite unimodular lattice. If $L$ is
any lattice and $n$ is an integer then $L(n)$ means $L$ with the
bilinear form multiplied by $n$. We write vectors of $M=L\oplus
II_{1,1}(2)$ as $(v,m,n)$ with $v\in L$, $m,n\in \Z$, so that this
vector has norm $v^2+4mn$. Similarly we write vectors of
$L=E_8(-2)\oplus II_{1,1}$ as $(v,m,n)$ with $v\in E_8(-2)$, $m,n\in
\Z$, so that this vector has norm $v^2+2mn$.
The norm 0 vector $u$ of $M$ is defined to be the vector $(0,0,1)\in
M$, and similarly we define $\rho=(0,0,1)\in L$ and $\rho'=(0,1,0)\in
L$. We write $O_M(\Z)$ for the automorphism group of the lattice $M$.
We will say that a vector of a lattice has even type if it has even
inner product with all vectors, and we will say it has odd type
otherwise. There are two orbits of primitive norm 0 vectors of $M$
under $O_M(\Z)$, which can be distinguished by whether they have even
or odd type. There are also two orbits of primitive norm 0 vectors in
$L$ which can be distinguished in the same way.
We define the hermitian symmetric space $\Omega$ of $M$ to be the set
of vectors $\omega\in P(M\otimes \C)$ such that $(\omega,\omega)=0$
and $(\omega,\bar\omega)>0$ (where $P$ means the projective space of a
vector space). The space $\Omega$ has two components, and we write
$O_M(\Z)^+$ for the subgroup of index 2 of $O_M(\Z)$ of elements that
do not exchange these two components. There is a second model for
$\Omega$ which we will use in section 3. The positive norm vectors of
$L\otimes \R$ form two open cones, and we choose one of them and call
it $C$. Then one of the two components of $\Omega$ can be identified
with $L\otimes \R+iC$ by identifying the point $v\in L\otimes \R+
iC$ with the point of $\Omega$ represented by $(v,1/2,-v^2/2)\in
M\otimes \C$.
Any automorphism of $M\otimes \R$ induces an automorphism of $\Omega$,
and if it does not exchange the two components of $\Omega$ this
induces an automorphism of $L\otimes \R+iC$. We describe these
automorphisms in a few cases which we will need later. If $\sigma$ is
an automorphism of $M=L\oplus II_{1,1}(2)$ fixing all vectors of
$II_{1,1}$ then it induces an automorphism of $L$ and hence of
$L\otimes \R+iC$ in the obvious way. If $\lambda\in L'$ then there is
an automorphism of $M$ taking $(v,m,n)$ to
$(v+2m\lambda,m,n-(v,\lambda)-m\lambda^2)$, and the induced
automorphism of $L\otimes \R+iC$ takes $y$ to $y+\lambda$. Finally
reflection in the hyperplane of the vector $(0,1,-1/2)$ of $M\otimes \Q$ takes
$(v,1/2,-v^2/2)$ to $(v,-v^2,1/4)$ and therefore induces the
automorphism of $L\otimes \R+iC$ taking $y$ to $-y/2(y,y)$.
Remark. The group $O_M(\Z)$ seems to be defined slightly differently
from the group $\Gamma$ in [B-P-V] but it follows from [N remark 1.15]
that these two groups are the same. Similarly proposition VIII.20.6
of [B-P-V] states that there are finitely many equivalence classes of
norm $-2$ vectors of $M$ under $O_M(\Z)$, but it follows from [N] (or
from lemma 2.3 below) that there is in fact only one orbit of norm
$-2$ vectors. In particular the divisor $\bigcup_dH_d/O_M(\Z)$ on
$\Omega/O_M(\Z)$ is irreducible, and not just a finite union of
irreducible divisors.
If $d$ is a norm $-2$ vector of $M$ we write $H_d$ for the divisor of
points of $\Omega$ represented by points orthogonal to $d$. Then it
follows from 20.5, 21.2, 21.4, of [B-P-V VIII], or from [N, 1.14] that
the moduli space of Enriques surfaces is
$$D^0=\big(\Omega\backslash(\bigcup_dH_d)\big)/O_M(\Z).$$
In the rest of this section we prove some auxiliary results about
the subgroups of $O_M(\Z)$ generated by various subsets.
\proclaim Lemma 2.1.
Suppose that $v$ is a vector in $L\otimes \Q$ but not in the dual $L'$
of $L$, and suppose that $x$ is any real number. Then we can find a
vector $\mu\in L$ with $\mu^2\equiv 2\bmod 4$ such that
$|(\mu-v)^2-x|<2$.
Proof. As $v$ is not in $L'$ we can find a primitive norm 0 vector
$\rho$ of $L$ of odd type such that $(\rho,v)$ is not an integer. This
is because the primitive norm 0 vectors of odd type span $L$. As
$O_L(\Z)$ acts transitively on such norm 0 vectors we can assume that
$\rho=(0,0,1)\in E_8(-2)\oplus II_{1,1}$. Then $v=(\lambda,a,b) $
with $a$ not an integer. We will find some $\mu $ of the form
$\mu=(0,m,n)\in II_{1,1}$, with $m$ and $n$ both odd so that
$\mu^2\equiv 2\bmod 4$. So we have to find odd integers $m$ and $n$
satisfying $|(\mu-v)^2-x| = |\lambda^2+2(a-m)(b-n)-x|<2$. As $a$ is
not an integer we can find some odd $m$ with $0<|a-m|<1$. Then
whenever we add 2 to $n$ we change $2(a-m)(b-n)$ by a nonzero number
less than 4, so we can chose some odd integer $n$ so that
$2(a-m)(b-n)$ is at a distance of less than 2 from any given real number
$x-\lambda^2$. This proves lemma 2.1.
\proclaim Lemma 2.2. Suppose that $R_2$ is the set of
norm $-2$ vectors of $M$ having inner product 2 with $u=(0,0,1)$, and
$\Gamma_1$ is the subgroup of $O_M(\Z)$ generated by the reflections
of vectors of $R_2$ and the automorphism $-1$. Then any vector $r\in M$
is conjugate under $\Gamma_1$ to a vector of the form $(v,m,n)\in M$
with either $m=0$ or $v/m\in L'$ and $m>0$.
Proof. We can assume that $r=(v,m,n)$ has the property that
$|(r,u)|=|2m|$ is minimal among all conjugates of $r$ under
$\Gamma_1$. If $m=0$ we are done, so we can assume that $m\ne 0$, and
we wish to show that $v/m\in L'$.
Suppose that $v/m\not\in L'$. By lemma 2.1 we can find a vector
$\mu\in L$ with $|(\mu-v/m)^2+(-4n/m-v^2/m^2)|<2$ and $\mu^2\equiv
2\bmod 4$. But if we calculate the inner product $(r',u)$, where $r'$
is the reflection of $r$ in the hyperplane of $(\mu, 1,
(-\mu^2-2)/4)\in R_2 $, we find that $(r',u)$ has absolute value
$|(r',u)|=|(r,u+2(\mu,1,(-\mu^2-2)/4))|=
|m((\mu-v/m)^2+(-4n/m-v^2/m^2))|<|2m|=|(r,u)|$,
which is not possible because we assumed that $|(r,u)|=|2m|$ was
minimal. Hence $v/m\in L'$. We can obviously then assume that $m>0$
by using the automorphism $-1$. This proves lemma 2.2.
We write $R_0$ or $R_2$ for the sets of norm $-2$ vectors
of $M$ which have inner products $0$ or $2$ with $u$. We let $\Gamma_1$
be the group generated by $-1$ and by the reflections of elements of
$R_2$, and we let $\Gamma_2$ be the group generated by $-1$
and the reflections of elements of $R_0\cup R_2$.
\proclaim Lemma 2.3. Any norm $-2$
vector of $M$ is conjugate to an element of $R_0\cup R_2$ under the group
$\Gamma_1$. In particular the group $\Gamma_2$ is the group generated by
$-1$ and the reflections of all norm $-2$ vectors of $M$.
Proof. Put $r=(v,m,n)$, so that by lemma 2.2 we can assume that either
$m=0$ or $v/m\in L'$ and $m>0$. If $m=0$ then $r$ is
orthogonal to $u$ so this case is trivial.
Now suppose that $v/m\in L'$ (so that $(v/m,v/m)\in \Z$) and $m>0$.
Then $-2=(r,r)= m^2(v/m,v/m)+4mn$ is divisible by $(m^2,4m)$, so
$m=1$. Hence $r=(v,1,n)$ is an element of $R_2$. This proves lemma 2.3.
Remark. Lemma 2.3 can be used to give an elementary proof of Namikawa's
result [N, 2.13] that $O_M(\Z)$ acts transitively on the norm $-2$
vectors of $M$, which implies that the complement of $D^0$ in $D$ is
an irreducible divisor. This can be done by noting
that $R_0$ and $R_1$ are acted on transitively by the
groups in (1) and (2) of lemma 2.4, and then showing
that there is some automorphism of $M$ (e.g., reflection in a norm $-4$ vector)
mapping some vector of $R_0$ into $R_1$.
\proclaim Lemma 2.4. Two primitive norm 0 vectors $z$ and $z'$ of $M$
of even type are in the same orbit of $\Gamma_2$ if and only if they
are congruent mod $ 2M$. In particular there are only a finite number
of orbits of such norm 0 vectors under $\Gamma_2$.
Proof. By lemma 2.3 the group $\Gamma_2$ is the group generated by $-1$ and all
reflections of norm $-2$ vectors and hence is normal in $O_M(\Z)$. We
can find a norm 0 vector $u\in 2M'$ such that $(u,z)\not\equiv 0\bmod
4$ because $z$ is primitive, and as $\Gamma_2$ is normal in $O_M(\Z)$
we may assume that $u=(0,0,1)$. Any conjugate of $z$ under $\Gamma_2$
is congruent to $z\bmod 2M$ and therefore has inner product with $u$
not divisible by 4, and in particular is not orthogonal to $u$. By
lemma 2.2 this implies that we may assume that $z=(mv,m,n)$ for some
$v\in L'$, $m>0$. As $z$ has norm 0, we see that $m^2v^2=-4mn$, so
$mv^2=-4n$ as $m\ne 0$. The vector $2v$ is in $L$, so if $m$ is
divisible by some odd number $p$ then $z/p\in M$. As $m=(z,u)/2$ is
odd and $z$ is primitive this shows that $m= 1$.
Hence we can assume that $z=(v,1,n)$ and $z'=(v',1,n')$ with $v\equiv
v'\bmod 2L$. If $r$ is a norm $-2$ vector of $L$ then the products of
the reflections of $(r,0,0)$ and $(r,0,1)$ is the automorphism taking
$(v,1,n)$ to $(v+2r,1,n-(v,r)+2)$. The lattice $L$ is generated by
its norm $-2$ vectors $r$, so these automorphisms can be used to map
$z$ to $z'$. This proves lemma 2.4.
\proclaim Lemma 2.5. The group $\Gamma_3$ generated by the
following sets of automorphisms has finite index in $O_M(\Z)^+$.
\item{(1)} The automorphisms in $O_L(\Z)^+$ (extended to automorphisms
of $M$ by letting them act trivially on $II_{1,1}(2)$).
\item{(2)} The group of automorphisms taking $(v,m,n)$ to
$(v+2m\lambda,m,n-(v,\lambda)-m\lambda^2)$ for $\lambda\in L'$. This
is the group of all automorphisms of $M$ fixing $u$ and all vectors of
$M/\langle u\rangle$.
\item{(3)} An automorphism given by reflection of a norm $-2$
vector $r$ of $M$ which has inner product 2 with $u$. (The group in
(2) above acts transitively on the set of such norm $-2$ vectors $r$,
so it does not matter which we choose.)
\item{(4)} The automorphism $-1$.
Proof. The group generated by the automorphisms in (1) and (2) above
is the group of all automorphisms in $O_M(\Z)^+$ that fix the
primitive norm 0 vector $u$ of even type, so to prove that $\Gamma_3$
has finite index in $O_M(\Z)^+$ it is sufficient to show that there
are only a finite number of orbits under $\Gamma_3$ of primitive norm
0 vectors of even type. But $\Gamma_3$ contains $\Gamma_2$ because the
group of automorphisms in (2) acts transitively on the set of norm
$-2$ vectors having inner product $2$ with $u$ and the group of automorphisms
in (1) contains all reflections of vectors of $R_0$, and by lemma 2.4
$\Gamma_2$ has only a finite number of orbits on the set of primitive
norm 0 vectors of even type. This proves lemma 2.5.
Unfortunately the group $\Gamma_3$ generated by the transformations
above is not the whole group $O_M(\Z)^+$, and this means that the
proof that $\Phi$ is an automorphic form for $O_M(\Z)^+$ in section 3
has to be indirect. For example, the transformations above all
preserve the set of vectors in $M$ of even type whose inner product
with $u$ is not divisible by 4, and it is easy to see that $u$ is not
in this set but is conjugate to a vector in this set under reflection
in a norm $-4$ vector having inner product $-2$ with $u$.
\proclaim 3.~Construction of the automorphic form $\Phi$.
In this section we construct the automorphic form $\Phi$ of weight 4
for $O_M(\Z)^+$ on one of the two components of $\Omega$ whose zeros
are exactly the divisors orthogonal to norm $-2$ vectors of $M$. We
will construct $\Phi$ as a function on $L\otimes \R+iC$.
We start by recalling from [B92 section 14, example 3] the twisted
denominator formula for an automorphism of the monster Lie algebra
coming from an involution of the Leech lattice with an 8-dimensional
fixed subspace and using it to define $\Phi$. Unfortunately there are
2 misprints the formulas given there: the final term in the first
formula on page 442 should be $(-1)^{m+n}|p_g((1-r^2)/2)|$, and the
factor $q^{1/2}$ in the next line should not be there. It should
also be noted that the sign conventions for Lorentzian lattices in [B92]
are the opposite to those used here. With these
changes the twisted denominator formula is $$\eqalign{
\Phi(y)
& = \sum_{w\in W}\det(w)e^{2\pi i(\rho,w(y))}
\prod_{n>0}(1-e^{2\pi i n(\rho,w(y))})^{(-1)^n8}\cr
&=e^{2\pi i(\rho,y)}
\prod_{r\in \Pi^+}(1-e^{2\pi i (r,y)})^{(-1)^{(r,\rho-\rho')}c((r,r)/2)}\cr
}$$
where the first equality is the definition of $\Phi$ and the second
equality only holds in the region of convergence of the infinite
product (see the remark after lemma 3.1). The vector $y$ is an
element of $L\otimes \C$ with $\Im(y)\in C$, where $C$ is the positive
open cone in $L\otimes \R$. The group $W$ is the subgroup of $O_L(\Z)$
generated by the reflections of the norm $-2$ vectors of $L$ (and has
infinite index in the full reflection group of $L$). It is also the
Weyl group of the fake monster Lie superalgebra. The vectors $\rho$
and $\rho'$ are the norm zero vectors $(0,0,1)$ and $(0,1,0)$ of
$L=E_8(-2)\oplus II_{1,1}$, and $\rho$ is also the Weyl vector of the
fake monster Lie superalgebra. The set $\Pi^+$ is the set of positive
roots of the fake monster Lie superalgebra, which consists of all
nonzero vectors $(v,m,n)$ of norm at least $-2$ such that $m>0$ or
$m=0$ and $n>0$. The numbers $c(n)$ are the coefficients of
$$\eqalign{ f(\tau)&=\sum_nc(n)q^n\cr
&=\eta(\tau)^{-8}\eta(2\tau)^8\eta(4\tau)^{-8}\cr &
=q^{-1}(1+q)^8(1-q^2)^{-8}(1+q^3)^8\cdots\cr &= q^{-1} + 8 + 36q +
128q^2 + 402q^3 + 1152q^4 + 3064q^5 + O(q^6)\cr }$$
\proclaim Lemma 3.1. The sequence $\log(c(n))$
is asymptotic to $2\pi \sqrt n $.
Proof. The circle method (see [R]) gives an asymptotic expansion for
the coefficients $c(n)$ of any meromorphic modular form of negative
weight with no poles in the upper half plane in terms of the poles at
cusps. The dominant term in this asymptotic expansion for $f(\tau)$
comes from the pole of order $1/4$ of $f(\tau)$ at the cusp 0, given
by
$\tau^{4}f(-1/\tau)=16\eta(\tau)^{-8}\eta(\tau/2)^8\eta(\tau/4)^{-8}=
16q^{-1/4}+\cdots$. This shows that $c(n)$ is asymptotic to $\pi
n^{-5/2}I_5(2\pi\sqrt n)$ where $I_5$ is a modified Bessel function
(see [B95 lemma 5.3]). As $I_5(x)$ is asymptotic to $e^x/\sqrt{2\pi
x}$ we get the result stated in the lemma by taking logs. This proves
lemma 3.1.
In particular this implies that the infinite product for $\Phi$
converges whenever $y$ has an imaginary part in the open region
bounded by the hypersurface $S$ of points $y\in iC$ with $(y,y)=-1/2$
(see the proof of lemma 3.3 below).
Remark. It is easy to check lemma 3.1 by computing a few cases
numerically; for example, the values of $\pi n^{-5/2}I_5(2\pi\sqrt n)$
for $n=-1,0,1,2,3,4,5$ are $1.17$, $8.01$, $35.59$, $128.02$,
$402.80$, $1151.95$, and $3062.48$, which can be compared with the
values of the $c(n)$'s given above.
\proclaim Theorem 3.2. The function
$\Phi(y)$ is an automorphic form on $L\otimes \R+iC$ with respect to
the discrete subgroup $O_{M}(\Z)^+$ and the character $\chi$ of
$O_M(\Z)^+$ (defined below).
The proof of this theorem will take most of
the rest of this section. We first note
the following two obvious transformation laws for $\Phi$, which
follow immediately from the definition of $\Phi$.
If $\sigma\in O_L(\Z)^+$ then
$$\Phi(\sigma(y))= \chi(\sigma)\Phi(y)$$
where $\chi$ is a character of $ O_L(\Z)^+$
taking reflections of norm $-2$ vectors to $-1$ and taking reflections of
norm $-4$ vectors to $1$. If $\lambda\in L'$ then
$$\Phi(y+\lambda)=\Phi(y).$$
The next lemma is essentially a special case of theorem 5.1 of [B95].
\proclaim Lemma 3.3.
If we define $\Psi(y)=\Phi(y+(\rho-\rho')/2)$ then $\Psi(y)$ vanishes
whenever $y$ lies on the surface $S\subset iC$ of points $y_0\in iC$
with $(y_0,y_0)=-1/2$.
Proof. We can assume that $y_0\in i(L\otimes\Q)$ because rational points
are dense in $S$. If $y_0$ is a point in $S\cap i(L\otimes \Q)$ then we
look at the function $g(\tau)=-\log(\Psi(\tau y_0/i))\exp(-2\pi \tau
(\rho, y_0))$, defined for $\Im(\tau)$ large. This can be expanded as a
power series $g(\tau)=\sum_{n\in \Q}a(n)q^{n}$ in some rational power
of $q=e^{2\pi i\tau}$.
First we show that the coefficients of $g$ are non negative.
If we look at the infinite product expansion
$$
\Psi(y)
=-e^{2\pi i(\rho,y)}\prod_{r\in \Pi^+}
(1-(-1)^{(r,\rho-\rho')}e^{2\pi i (r,y)})^{(-1)^{(r,\rho-\rho')}c((r,r)/2)}
$$
we can see that all the Fourier coefficients $a(r)$ of $-\log(-\Psi(y)/e^{2\pi i
(\rho,y)})=\sum_ra(r)e^{2\pi i (r,y)}$ are non negative. This is because
if $r$ is a primitive vector then the Fourier coefficients $a(nr)$ $(n>0)$
are given by the Fourier coefficients of
$$-\log \prod_{m>0}(1-e^{2\pi im(r,y)})^{c((mr,mr)/2)}$$
if $(r,\rho-\rho')$ is even, and by
$$\eqalign{
&-\log \prod_{m>0}(1-(-1)^me^{2\pi im(r,y)})^{(-1)^mc((mr,mr)/2)}\cr
=&-\log \prod_{m>0,m{\;\rm odd}}(1-e^{2\pi im(r,y)})^{c((mr,mr)/2)}\cr
& -\log \prod_{m>0,m{\;\rm even}}(1-e^{2\pi im(r,y)})^{c((mr,mr)/2)-c((mr/2,mr/2)/2)}\cr
}$$
if $(r,\rho-\rho')$ is odd,
and in both cases we see that all the Fourier coefficients are
nonnegative because $0\le c(n)\le c(4n)$ for any $n$.
This implies that the coefficients in the
series for $g$ are non negative because $g$ is the restriction
of $-\log(-\Psi(y)/e^{2\pi i(\rho,y)})$
to a line so that its coefficient $a(n)$ are given by
$a(n)=\sum_{(r,y_0)=n}a(r)$.
By using lemma 3.1 we can check that
$\limsup_{n\rightarrow+\infty}\log(a(n))/n=2\pi$. We get $2\pi $ as
an upper bound for the $\limsup$ because in the sum $a(n)=\sum_{(r,y_0)=n}a(r)$
the number of terms is bounded by a polynomial (of degree 9) in $n$,
and in each term $(r,r)$ is at most $n^2/|y_0^2|=2n^2$,
and $a(r)$ is not much bigger than $c((r,r)/2)$, whose log
is about $2\pi \sqrt{(r,r)/2}\le 2\pi n$ by lemma 3.1.
We can prove that $2\pi$ is a lower bound for the $\limsup$ in a similar way,
by observing that all the coefficients $a(r)$ are positive so
that $a(n)$ is bounded below by the largest of them,
and that there are infinitely many $n$ such that the largest $a(r)$
in the sum is very roughly $e^{2\pi \sqrt{(n^2/|y_0^2|)/2}}= e^{2\pi n}$.
Therefore the series for $g$ has radius of convergence $|q|=\limsup
|a(n)|^{-1/n}=e^{-2\pi}$.
Hence $g$ has a singularity at $e^{2\pi i \tau}=q=e^{-2\pi}$ because a
power series with non negative coefficients with radius of convergence
$e^{-2\pi}$ has a singularity at $e^{-2\pi}$. (This is why we have to
replace $\Phi$ by $\Psi$: the coefficients of $-\log(\Phi(y)/e^{2\pi i
(\rho,y)})$ do not all have the same sign.) This means that
$g(\tau)=-\log(\Psi(\tau y_0/i)\exp(-2\pi \tau(\rho,y_0)))$ has a
singularity at $\tau=i$, so that $\log(\Psi(y))$ has a singularity at
$y=y_0$. However $\Psi(y)$ is holomorphic at $y=y_0$, so the only way that
$\log(\Psi(y))$ can have a singularity at $y=y_0$ is if $\Psi$ vanishes at
$y_0$. This shows that $\Psi$ vanishes on the surface $S$ and proves
lemma 3.3.
The main step in the proof of theorem 3.2 is the proof of the
following extra transformation law for $\Phi$ (or rather for $\Psi$).
\proclaim Lemma 3.4. $$\Psi(-y/2(y,y)) = -16(y,y)^{4} \Psi(y).$$
Proof. It is sufficient to prove this for purely imaginary values of
$y$ because then the result is true for all $y$ by analytic
continuation. The cone $iC$ has a pseudo-Riemannian metric induced by
the bilinear form on $L\otimes \R$ and has an associated wave operator
given by the Laplacian of its pseudo-Riemannian metric. On the space
$iC$, $\Psi$ is a solution of the wave equation because each of the
terms $\exp((w(n\rho),y))$ in the sum defining $\Phi$ is a solution of
the wave equation (as each of the vectors $w(n\rho)$ has norm 0). This
implies that $(y,y)^{10/2-1}\Psi(-y/16(y,y))$ is also a solution of
the wave equation by the transformation of the wave operator under the
conformal transformation $y\rightarrow -y/2(y,y)$ of $iC$. (For this
special conformal transformation this is easy to check directly as it is
just the fact that if $\Psi(y)$ is any solution to the wave equation in
$n$ dimensions then so is $(y,y)^{n/2-1}\Psi(-y/c(y,y))$ for any positive
constant $c$. The quickest way to prove this is to choose orthogonal
coordinates so that $y=(x_1,\ldots,x_n)$ and calculate $({\partial^2
\over \partial x_1^2}+\cdots +{\partial^2 \over \partial
x_{n-1}^2}-{\partial^2 \over \partial x_n^2})
(y,y)^{n/2-1}\Psi(-y/c(y,y))$ explicitly to show that it vanishes at
$y$. Because of the invariance of everything under
Lorentzian transformations, it is only necessary to check vanishing
at points $y$ of the form $(0,\ldots,0,r)$.)
Now we check that that $-16(y,y)^{4}\Psi(-y/2(y,y))$ and $\Psi(y)$
both have the same partial derivatives of order at most 1 on the
surface $S$. They both vanish on $S$ by lemma 3.3 and therefore have
the same constant term, and for the same reason their first partial
derivatives in any direction tangent to $S$ both vanish, so it is only
necessary to check that they have the same first partial derivatives
in the direction normal to $S$. But if $\Psi$ is any smooth function on
$iC$ whose partial derivative normal to $S$ at a point $s\in S$ is
$x$, then $\Psi(-y/2(y,y))$ has a partial derivative normal to $S$ at $s$ of
$-x$. (This follows by restricting
$\Psi$ to the line through $0$ and $s$ and using the elementary
fact that if a differentiable function $h$ is defined for positive reals $y$
then the derivatives of $h(y)$
and $-h(y/2y^2)=-h(1/2y)$ are equal at $y=1/\sqrt 2$.)
The function $-2(y,y)$ is 1 on $S$, so the partial derivative normal to $S$ of
$(-2(y,y))^n\Psi(-y/2(y,y))$ is $-x$ for any integer $n$ because
$\Psi(-y/2(y,y))$ vanishes for $y\in S$. Hence
$-16(y,y)^{4}\Psi(-y/2(y,y))$ and $\Psi(y)$ both have the same partial
derivatives of order at most 1 on the surface $S$. These two
functions both satisfy the wave equation and have the same partial
derivatives of order at most 1 on the non characteristic surface $S$,
so by the uniqueness part of the Cauchy-Kovalevsky theorem they must
be equal on $iC$. This proves lemma 3.4.
\proclaim Lemma 3.5. $\Phi$ is an automorphic form of weight 4
for the finite index subgroup $\Gamma_3$ of $O_M(\Z)$ (defined in lemma 2.5).
Proof. Lemma 3.4 shows that $\Psi$ transforms like an automorphic form
under reflection in the norm $-2$ vector $(0,1,-1/2)$ of $M\otimes \Q$
(which is not in $M$). We obtain $\Psi$ from $\Phi$ by applying the
automorphism of $M\otimes \Q$ taking $(v,m,n)$ to $(v+2\lambda m,
m,n-(v,\lambda)-\lambda^2m)$ where $\lambda=(\rho-\rho')/2\in L\otimes
\Q$. This automorphism takes $(0,1,-1/2)$ to the norm $-2$ vector
$(\rho-\rho', 1, 0)\in L$. Hence $\Phi$ transforms like an
automorphic form under reflection in the norm $-2$ vector
$(\rho-\rho',1,0)$ having inner product 2 with $u$. We have
therefore verified that $\Phi$ transforms an automorphic under all the
transformations of lemma 2.5, which proves lemma 3.5.
\proclaim Lemma 3.6. The form $\Phi$ vanishes (to order 1)
along all the divisors of norm $-2$ vectors of $M$.
Proof. We know by lemma 2.3 that any norm 2 vector is conjugate to a
vector in either $R_0$ or $R_2$, and the groups in (1) and (2) of
lemma 2.5 act transitively on these two sets, so it is sufficient to
prove that $\Phi$ vanishes along the divisors of one vector in $R_0$
and one vector in $R_2$. But $\Phi$ vanishes along the divisor of the
vector $(\rho-\rho',1,0)\in R_2$ by lemma 3.3 (see the proof of lemma
3.5), and $\Phi$ vanishes along the hyperplane orthogonal to any
vector in $R_0$ because the functional equation
$\Phi(\sigma(y))=\chi(\sigma)\Phi(y)$ implies that $\Phi$ changes sign
under reflection in this hyperplane. This proves lemma 3.6.
\proclaim Lemma 3.7. There exists an automorphic form of weight 16632/2
for the lattice $II_{2,10} =E_8(-1)\oplus II_{1,1} \oplus II_{1,1}$
whose zeros are exactly the hyperplanes of norm $-4$ vectors of
$II_{2,10}$.
Proof. The meromorphic modular form
$$E_4(\tau)^5/\Delta(\tau)^2-1248E_4(\tau)^2/\Delta(\tau) =
q^{-2}+16632+O(q)$$
has weight $-4$ and level 1 and no poles on the upper half plane, so
applying theorem 10.1 of [B95] shows the existence of an automorphic
form with the required properties. This proves lemma 3.7.
\proclaim Lemma 3.8. The only zeros of $\Phi$
lie on the hyperplanes of norm $-2$ vectors of $M$.
Proof. The group $II_{2,10}/2II_{2,10}$ has order $2^{12}$ and its
elements have a well defined norm mod 4. Under the group
$O_{II_{2,10}}(\Z)$ its elements split into 3 orbits: the zero
element, an orbit of size $2^{11}-2^5= 2016$ of elements of norm
congruent to $2\bmod 4$, and an orbit of size $2^{11}+2^5-1= 2079$ of
nonzero elements whose norm is congruent to $0\bmod 4$. We note that
every norm $-4$ vector $v$ of $II_{2,10} $ gives a unique norm $0\bmod
4$ nonzero element $v$ of $II_{2,10}/2II_{2,10}$, and this partitions
the norm $-4$ vectors of $II_{2,10}$ into 2079 disjoint classes.
For each of the 2079 nonzero vectors of norm $0 \bmod 4$ in
$II_{2,10}/2II_{2,10}$, an inverse image of this vector in $II_{2,10}$
together with $2II_{2,10}$ generates a copy of $ M(2)$. For each of
these 2079 copies of $ M(2)$ we take a copy of the form $\Phi$
corresponding to it (with its argument rescaled by a factor of $\sqrt
2$) and we multiply these 2079 automorphic forms together to get a
function $\Theta$. (It is not yet clear that $\Theta$ is uniquely
defined by this, because we have not yet proved that $\Phi$ is an
automorphic form for the whole of $O_M(\Z)^+$, but this does not
matter.) By lemma 3.5 $\Theta$ is an automorphic form for some finite
index subgroup of $O_{II_{2,10}}(\Z)$ of weight $4\times 2079$. The
hyperplane of any norm $-4$ vector $v$ of $II_{2,10}$ is a zero of the
factor of $\Phi$ corresponding to the vector $v\in
II_{2,10}/2II_{2,10}$ (which corresponds to a norm $-2$ vector in the
copy of $M$), so $\Theta$ vanishes on all the hyperplanes of all norm
$-4$ vectors of $II_{2,10}$. Therefore we can divide $\Theta$ by the
automorphic form of lemma 3.7 to obtain an automorphic form of weight
$4\times 2079-16632/2=0$ which is holomorphic at cusps by the Koecher
boundedness principle. This quotient must therefore be a constant, so
it has no zeros, and therefore the form $\Theta$ has no zeros other
than those corresponding to norm $-4$ vectors of $II_{2,10}$. But
this implies that $\Phi$ has no zeros other than those corresponding
to norm $-2$ vectors of $M$, otherwise these would give rise to other
zeros of $\Theta$. This proves lemma 3.8.
We can now complete the proof of theorem 3.2. By the Koecher
boundedness principle an automorphic form on $\Omega$ is determined up
to multiplication by a constant by its zeros on $\Omega$, because if
$f$ and $g$ are two forms with the same zeros then $f/g$ and its
inverse $g/f$ are both automorphic forms so they must both be
constant. The transform of $\Phi$ under any element of $O_M(\Z)^+$
has the same zeros as $\Phi$ because the zeros of $\Phi$ just
correspond to the norm $-2$ vectors of $M$. Hence the transform of
$\Phi$ under any element of $O_M(\Z)$ is equal to $\Phi$ multiplied by
some nonzero constant. This proves theorem 3.2.
Remark: The proof that $\Phi$ has no extra zeros relies on a strange
numerical coincidence. If we assume that $\Phi$ is an automorphic form
for the group $O_M(\Z)^+$ then we can give a more conceptual proof of
this as follows. (Unfortunately the proof that $\Phi$ is an
automorphic form for $O_M(\Z)^+$ uses the fact that $\Phi$ has no
extra zeros, so this argument is of no use unless someone finds a
different proof that $\Phi$ is automorphic under $O_M(\Z)^+$!) By
theorem 5.1 of [B95] any zero of $\Phi$ must be the hyperplane of some
primitive vector $v$ of $M$ (which is a subset of the hermitian
symmetric space called a rational quadratic divisor in [B95]). We have
to prove that $v$ has norm $-2$. By lemma 3.9 below there is some
primitive norm 0 vector orthogonal to $v$. As $\Phi$ transforms like
an automorphic form under $O_M(\Z)^+$, we can assume that this is the
norm zero vector $u$. But then the divisor of $v$ intersects the
region of convergence of the infinite product defining $\Phi$, which
is only possible if it is a zero of one of the factors in the infinite
product. But the only factors in the infinite product with zeros are
those of the form $1-\exp(2\pi i (x,y))$ with $x$ a vector of norm
$-2$. This shows that $v$ is a vector of norm $-2$ and hence shows
that the zeros of $\Phi$ are exactly the hyperplanes of norm $-2$
vectors of $M$.
\proclaim Lemma 3.9. Any vector of $M$ is orthogonal to
a conjugate of $u$ under $O_M(\Z)$.
Proof. The lattice $M$ contains a 2-dimensional primitive isotropic
sublattice $U$ such that every vector in $U$ has even type, so that
every primitive vector in $U$ is conjugate to $u$ under $O_M(\Z)^+$.
As $U$ has dimension greater than 1, there is some primitive vector in
$U$ orthogonal to $v$, which has the required properties. This proves
lemma 3.9.
Remark. It is not true that any vector is conjugate under the group
$\Gamma_3$ to a vector orthogonal to $u$; for example, this is not
true for a vector of even type having inner product with $u$ not
divisible by 4. It is also not true that any vector $v$ of $M$ is
orthogonal to a primitive isotropic vector $u$ of odd type. In fact it
is not hard to check that if $v$ has this property then $v$ has norm
$(v,v)$ divisible by 4. The proof of lemma 3.9 breaks down for this
case because lattices $U$ in the other orbit of 2-dimensional
primitive isotropic sublattices still have some primitive vectors of
even type.
\proclaim References.
\item{[B92]}{R. E. Borcherds,
Monstrous moonshine and monstrous Lie superalgebras, Invent. Math.
109, 405-444 (1992).}
\item{[B95]} R. E. Borcherds, Automorphic forms on $O_{s+2,2}(R)$
and infinite products. Accepted by Invent. Math. August 1994.
\item{[B-P-V]} W. Barth, C. Peters, A. Van de Ven,
``Compact complex surfaces'', Springer Verlag, 1984.
\item{[K]} S. Kondo, The rationality of the moduli space of
Enriques surfaces, Compositio Math. 91 (1994) no. 2, 159-173.
\item{[M]} K. Matsumoto, Theta functions on the bounded symmetric
domain of type $I_{2,2} $ and the period map of a 4-parameter family
of K3 surfaces. Math. Ann. 295, 383-409, (1993).
\item{[N]} Y. Namikawa, Periods of Enriques surfaces,
Math. Ann 270, p.201-222 (1985).
\item{[R]}{H. Rademacher, ``Topics in analytic number theory'',
Die Grundlehren der mathematischen Wissen\-schaften in Einzeldarstellungen
Band 169, Springer Verlag Berlin, Heidelberg, New York 1973.}
\item{[S]} H. Sterk, Compactifications of the period space of Enriques
surfaces, Math. Z. 207 p.1-36 (1991)
\bye