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{\bf Automorphic forms on $O_{s+2,2}(\R)$ and infinite products.}
Invent. Math. 120, p. 161-213 (1995)
Richard E. Borcherds.
Mathematics department,
University of California at Berkeley,
CA 94720-3840
U. S. A.
e-mail: reb@math.berkeley.edu
18 August 1994, corrected 9 Dec.
\bigskip
The denominator function of a generalized Kac-Moody algebra is often
an automorphic form for the group $O_{s+2,2}(\R)$ which can be written
as an infinite product. We study such forms and construct some
infinite families of them. This has applications to the theory of
generalized Kac-Moody algebras, unimodular lattices, and reflection
groups. We also use these forms to write several well known modular
forms, such as the elliptic modular function $j$ and the Eisenstein
series $E_4$ and $E_6$, as infinite products.
\bigskip
{\bf Contents.}
1. Introduction.
Notation
2. Automorphic forms and Jacobi forms
3. Classical theory
4. Hecke operators for Jacobi groups.
5. Analytic continuation.
6. Vector systems and the Macdonald identities.
7. The Weierstrass $\wp$ function.
8. Generators for $O_M(\Z)^+$.
9. The positive weight case.
10. The zero weight case.
11. The negative weight case.
12. Invariant modular products
13. Heights of vectors.
14. Product formulas for modular forms.
15. Generalized Kac-Moody algebras.
16. Hyperbolic reflection groups
17. Open problems.
\bigskip
{\bf 1. Introduction.}
\bigskip
The main result of this paper is a method for constructing automorphic
forms on $O_{s+2,2}(\R)^+$ as infinite products. For example, a
special case of theorem 10.1 states that if $24|s$ and we define
$c(n)$ by $\eta(\tau)^{-s}=q^{-s/24}\prod_{n>0}(1-q^n)^{-s}=\sum_n
c(n)q^n$ and $\rho$ is a certain vector then $$\Phi(v)=e^{-2\pi i
(\rho, v)}\prod_{r>0}(1-e^{-2\pi i (r,v)})^{c(-(r,r)/2)}$$ is an
automorphic form for the discrete subgroup $O_{II_{s+2,2}}(\Z)^+$ of
$O_{II_{s+2,2}}(\R)^+$ (or rather, its analytic continuation is an
automorphic form, as the infinite product does not converge
everywhere). We first describe some applications of this method, and
then describe the proof.
The simplest application is a product formula for the elliptic modular
function $j(\tau)$. More precisely,
$$j(\tau)=q^{-1}\prod_{n>0}(1-q^n)^{3a(n^2)}$$ where the $a(n)$'s are
the coefficients of a certain nearly holomorphic (``holomorphic except
for poles
at cusps'') modular form $3q^{-3}+744q+\cdots $ of weight $1/2$ (see
example 2 of section 14 for a precise description of the $a(n)$'s).
There are similar product formulas for many other modular forms and
functions, for example the Eisenstein series $E_4$, $E_6$, $E_8$,
$E_{10}$ and $E_{14}$ and the modular function $j(\tau)-1728$. The
usual product formula $\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}$ is the
simplest case of these product formulas.
More generally, theorem 14.1 gives an isomorphism between
a certain additive group of nearly holomorphic modular forms of weight $1/2$
and a multiplicative
group of meromorphic modular forms all of whose zeros and poles
are either cusps or imaginary quadratic irrationals. In particular,
as an immediate corollary of
theorem 14.1,
we find a
product formula for the classical modular polynomial
$$\prod_{[\sigma]}
(j(\tau)-j(\sigma))=q^{-H(-D)}\prod_{n>0}(1-q^n)^{c_0(n^2)}$$ where $\sigma$
runs through a complete set of representatives modulo $ SL_2(\Z)$
for the imaginary
quadratic numbers which are roots of an equation of the form
$a\sigma^2+b\sigma+c=0$ ($a,b,c\in \Z$) of some fixed discriminant
$b^2-4ac=D<0$ (except that
when $\sigma $ is a conjugate of one of the elliptic fixed points
$i$ or $(1+i\sqrt 3)/2$ we have to replace the corresponding factor
$j(\tau)-1728$ or $j(\tau)$ by $(j(\tau)-1728)^{1/2}$ or
$j(\tau)^{1/3}$). The exponents $c_0(n^2)$ are coefficients of the
unique nearly holomorphic weight $1/2$ modular form
for $\Gamma_0(4)$ whose power series $\sum_{n\in \Z} c_0(n)q^n$ is
of the form $q^{D}+O(q)$ and whose coefficients $c_0(n)$ vanish
unless $n\equiv 0,1\bmod 4$ (Kohnen's ``plus space'' condition).
The product on the left, as a function
of $j(\tau)$, is just the classical
modular polynomial for discriminant $D$,
whose degree is the
Hurwitz class number $H(-D)$.
This formula can
be compared with the Gross-Zagier formula ([G-Z], formula 1.2 and
theorem 1.3)
$$\prod_{[\tau_1],[\tau_2]}(j(\tau_1)-j(\tau_2))^{4/w_1w_2} = \pm
\prod_{x\in Z,n,n'>0, x^2+4nn'=d_1d_2}n^{\epsilon(n')} $$ where the
first product is over representatives of equivalence classes of
imaginary quadratic irrationals of discriminants $d_1$, $d_2$, $w_1$
and $w_2$ are the number of roots of 1 in the orders of discriminants
$d_1$, $d_2$, and $\epsilon(n')=\pm 1$ is defined in [G-Z]. It is
also related to the denominator formula
$$j(\sigma)-j(\tau)=p^{-1}\prod_{m>0,n\in\Z}(1-p^mq^n)^{c(mn)}$$ of
the monster Lie algebra (where $p=e^{2\pi i \sigma}$, $q=e^{2\pi i
\tau}$, and $j(\tau)-744 =\sum_nc(n)q^n=q^{-1}+196884q+\cdots$).
These 3 product formulas
for $\prod(j(\sigma)-j(\tau))$ cover the cases when both, one, or
neither of $\sigma$ and $\tau$ run over representatives of imaginary
quadratic numbers of fixed discriminant, while the others can be
arbitrary complex numbers with large imaginary part. In spite of the
similarity of the left hand sides, there does not seem to be any
obvious way to deduce any of these 3 formulas from the others.
There are several strange results about Niemeier lattices (even
24-dimensional unimodular lattices) and the Leech lattice, which were
proved by Niemeier, Venkov, and Conway [C-S]. For example, every
Niemeier lattice either has no roots or the root system has rank 24,
the number of roots is divisible by 24, and the Leech lattice is the
Dynkin diagram of $II_{25,1}$. We will find generalizations of these
results for all $24n$-dimensional even unimodular lattices $K$ in
section 12. For example, $K$ either has no vectors of norm at most
$2n$ (i.e., it is extremal) or the vectors of norm at most $2n$ span
the vector space $K\otimes \R$, and if $\theta_K(\tau)$ is the theta
function of $K$, then the constant term of
$\theta(\tau)/\Delta(\tau)^n$ is divisible by 24. We also use
properties of one automorphic form to give a very short proof of the
existence and uniqueness of the Leech lattice.
Many examples of automorphic forms on $O_{s+2,2}(\R)$ which are
modular products are the denominator formulas of generalized Kac-Moody
algebras. The simplest example is the product formula for the
denominator function $j(\sigma)-j(\tau)$ of the monster Lie algebra
given above. This function obviously transforms under a group of the
form $(SL_2(\Z)\times SL_2(\Z)).(\Z/2\Z)$, which is isomorphic to the
congruence subgroup $O_{II_{2,2}}(\Z)^+$ of $O_{II_{2,2}}(\R)$.
(Strictly speaking, this is an automorphic function rather than an
automorphic form because it has weight 0 and is not holomorphic at the
cusps.) A second example is the denominator formula for the fake
monster Lie algebra $$\Phi(v)=\sum_{w\in W}\sum_{n>0}\det(w)\tau(n)
e^{-2\pi i n(w(\rho),v)}= e^{-2\pi i (\rho,v)}\prod_{r>0}(1-e^{-2\pi i
(r,v)})^{p_{24}(1-r^2/2)}.$$ This function is obviously antiinvariant
under the group $O_{II_{25,1}}(\Z)^+$, but also turns out to be an
automorphic form of weight 12 under the group $O_{II_{26,2}}(\Z)^+$.
This is equivalent to saying that $\Phi$ satisfies the functional
equation $$\Phi(2v/(v,v)) =-((v,v)/2)^{12}\Phi(v).$$ We can construct
many new examples of generalized Kac-Moody algebras from automorphic
forms on $O_{s+2,2}(\R)$, and conversely we can find many examples of
such automorphic forms using generalized Kac-Moody algebras.
There are close connections between automorphic forms on
$O_{s+2,2}(\R)$ and hyperbolic reflection groups. For any such
automorphic form with a modular product we will define its ``Weyl
vectors''. These often turn out to be the Weyl vectors for some
hyperbolic reflection group. One example given in section 16
corresponds to the reflection group of the even sublattice of
$I_{21,1}$; this is the largest dimension in which the reflection
group of a hyperbolic lattice has finite index in the automorphism
group. Similarly all the reflection groups of the lattices $I_{n,1}$
for $n\le 19$ that were investigated by Vinberg have automorphic forms
associated with them.
We now discuss how to construct automorphic forms as infinite
products. This construction depends on 3 results, given in sections 4,
5, and 6. The first result (theorem 5.1) states that under mild
conditions a modular product can be analytically continued as a
meromorphic function to the whole of the Hermitian symmetric space $H$
of $O_{s+2,2}(\R)$, and its poles and zeros can only lie on certain
special divisors, called quadratic divisors. (A modular product is,
roughly speaking, an infinite product whose exponents are given by the
coefficients of some nearly holomorphic modular forms; see section 5.)
The proof of this uses the Hardy-Ramanujan-Rademacher asymptotic
series for the coefficients of a nearly holomorphic modular form.
The second result (theorem 6.5) is a generalization of the Macdonald
identities from affine root systems to ``affine vector systems''. This
generalization states (roughly) that an infinite product over the
vectors of an affine vector system is a Jacobi form. (For affine root
systems the usual Macdonald identities follow easily from this using
the fact that any Jacobi form can be written as a finite sum of
products of theta functions and modular forms.) The third result we
use (section 4) is a description of Hecke operators $V_\l$ for certain
parabolic subgroups (``Jacobi subgroups'') of discrete subgroups of
$O_{s+2,2}(\R)$.
If we put these three results together, we sometimes find that an
expression of the form $\exp(\rho+\sum_{\l\ge 0}\phi|V_\l)$, where
$\phi$ is a nearly holomorphic Jacobi form, is an automorphic form on
$O_{s+2,2}(\R)$. We prove this by showing that it transforms like an
automorphic form under 2 parabolic subgroups $J(\Z)^+$ and $F(\Z)^+$,
and then checking (in theorem 8.1) that these two subgroups generate a
discrete subgroup of $O_{s+2,2}(\R)^+$ of finite covolume. The
invariance under the Jacobi group $J(\Z)^+$ follows from the results
on Hecke operators on Jacobi forms that we recall in sections 2 to 4,
and the invariance under the Fourier group $F(\Z)^+$ follows by
calculating the Fourier coefficients explicitly and checking that
these are invariant under $F(\Z)^+$.
When the Jacobi form $\phi$ is holomorphic, this is similar to a
method for constructing automorphic forms on $Sp_4(\R)$ found by Maass
[M], and generalized to $O_{s+2,2}(\R)$ by Gritsenko [G], who showed
that $\sum_{\l\ge 0}\phi|V_\l$ was an automorphic form. The two main
extra complications we have to deal with when $\phi$ is not
holomorphic are firstly that this sum no longer converges everywhere
and so has to be analytically continued, and secondly that the ``Weyl
vector'' $\rho$ has to be chosen correctly.
\bigskip
{\bf Notation (in roughly alphabetical order).}
\bigskip
\item{${}^+$} If $G$ is a subgroup of a real orthogonal group then
$G^+$ means the elements of $G$ of positive spinor norm.
\item {${}'$} If $L$ is a lattice then $L'$ means the dual of $L$.
\item {${}^\perp$} If $u$ is a vector (or sublattice) of a lattice then
$u^\perp$ means the orthogonal complement of $u$.
\item{$\bar{}$} If $\lambda$ is a vector in a lattice then
$\bar\lambda$ is the orthogonal projection of $\lambda$ into some sublattice.
\item{$\alpha$} A coordinate of a vector in $M$.
\item{$a$} An entry of a matrix ${ab\choose cd}$ in $SL_2(\Z)$.
\item{$A(v)$} A Fourier coefficient of an automorphic form on $O_M(\R)^+$.
\item{$\beta$} A coordinate of a vector in $M$.
\item{$b$} An entry of a matrix ${ab\choose cd}$ in $SL_2(\Z)$.
\item{$B_k$} A Bernoulli number: $\sum_{k\in \Z}B_kt^k/k! =t/(e^t-1)$.
\item{$\gamma$} A coordinate of a vector in $M$.
\item{ $\Gamma_0(N)$} $ \{{ab\choose cd}\in SL_2(\Z)|c\equiv 0 \bmod N\}$
\item{$\Gamma(z)$} Euler's gamma function.
\item{$c$} An entry of a matrix ${ab\choose cd}$ in $SL_2(\Z)$.
\item{$c(v)$} The multiplicity of a vector in a vector system,
or a Fourier coefficient of a modular form or Jacobi form, or an exponent
of a modular product.
\item {\C} The complex numbers.
\item {$C$} The positive cone in a Lorentzian lattice.
\item{$\delta$} A coordinate of a vector in $M$.
\item{$\delta^m_n$} 1 if $m=n$, 0 otherwise.
\item {$\Delta$} The delta function,
$\Delta(\tau)=q\prod_{n>0}(1-q^n)$.
\item{$d$} The number of elements of a vector system,
an entry of a matrix ${ab\choose cd}$ in $SL_2(\Z)$.
\item{$D$} The discriminant of a quadratic divisor or an imaginary
quadratic irrational
or an imaginary quadratic field.
\item{$e^{\pm z}$} means $e^z$ if $\Re(z)<0$, $e^{-z}$ if $\Re(z)>0$.
\item{$e_n$} The Dynkin diagrams or lattices of $e_8$, $e_{10}$, and so on.
\item{$E_k$} An Eisenstein series of weight $k$, equal to $1-(2k/B_{k})\sum_{n>0}\sigma_{k-1}(n)q^n$.
\item {$\zeta$} $e^{2\pi i z}$. $\zeta^y=e^{2\pi i (y,z)}$.
\item {$f$} A function.
\item {$F$} A Fourier group. See section 2.
\item{$F(\tau)$} $= \sum_{n>0}\sigma_1(2n-1)q^{2n-1}$.
\item{${}_2F_1$} The hypergeometric function.
\item{$g$} An element of the group $G$, or a function.
\item {$G$} A group.
\item {$GL$} A general linear group.
\item {$GO$} A general orthogonal group.
\item {$\eta(\tau)$} $=q^{1/24}\prod_{n>0}(1-q^n)$.
\item{$h$} The height of a vector. See section 13.
\item{$h(\tau)$} $h(\tau)=h_0(4\tau)+h_1(4\tau)$
is a modular form of weight $1/2$.
\item{$H$, $H_u$} Hermitian symmetric spaces of $O_{n+2,2}(\R)$.
\item {$H(n)$} The Hurwitz class number of $n$. See section 13.
\item {$H_{N,j}(n)$} A generalization of the Hurwitz class number of $n$.
See section 13.
\item {$\H_{N,j}(\tau)$} A function with coefficients $H_{N,j}(n)$.
See section 13.
\item{$\theta$, $\theta_K$} Theta functions of lattices or cosets of lattices.
See section 3.
\item {$I_\nu$} A modified Bessel function.
\item {$II_{m,n}$} The even unimodular Lorentzian lattice of dimension
$m+n$ and signature $m-n$.
\item{$\Im$} The imaginary part of a complex number.
\item {$j$} The elliptic modular function $j(\tau)=q^{-1}+744+196884q+\cdots$.
\item {$J$} A Jacobi group. See section 2.
\item {$J_\l$} A double coset of $J(\Z)^+$.
\item {$\kappa$} A vector of $K$.
\item {$k$} The weight of an automorphic form or Jacobi form.
\item {$K$} An even positive definite lattice of dimension $s$.
\item {$K_\mu$} A modified Bessel function.
\item {$\lambda$} An element of $K$.
\item {$\Lambda$} The Leech lattice. See [C-S].
\item {$\l$} An integer, usually indexing Hecke operators.
\item {$L$} An even Lorentzian lattice of dimension $s+2$,
sometimes equal to $K\oplus II_{1,1}$.
\item {$\mu$} An element of $K$, or a real number.
\item {$m$} The index of a Jacobi form or vector system. See section 3 or 6.
\item {$M$} An even lattice of dimension $s+4$ and signature $s$,
sometimes equal to $L\oplus II_{1,1}$.
\item{$M[U]$} The lattice generated by $U$ and all vectors of $M$ having integral inner product with everything in $U$.
\item{$\nu$} A real number.
\item{$n$} An integer, often indexing the coefficients of a modular form.
\item{$N$} The level of a modular form.
\item {$O$} An orthogonal group.
\item{$O(q^n)$} A sum of terms of order at most $q^n$.
\item {$p$} $e^{2\pi i \sigma}$
\item{$p_m(n)$} The number of partitions of $n$ into parts of $m$ colors.
\item {$P$} A principle $\C^*$ bundle over $H$.
\item {$\wp^{(n)}$} $\wp^{(n)}(z,\tau)= {d\over dz}^n\wp(z,\tau)$, where
$\wp$ is the Weierstrass function. See section 7.
\item {$q$} $e^{2\pi i \tau}$
\item{$Q$} A quadratic form; $Q(v)=(v,v)/2$.
\item {$\Q$} The rational numbers.
\item {$\rho$} A Weyl vector. See section 6.
\item {$R$} A commutative ring, usually $\Z$, $\Q$, $\R$, or $\C$.
\item {$\Re$} The real part of a complex number.
\item {$r$} An element of the ring $R$, or a vector of $K$.
\item {$\R$} The real numbers.
\item {$\sigma$} A complex number with positive imaginary part.
\item{$\sigma_{k-1}(n)$} $=\sum_{d|n}d^{k-1}$ if $n>0$, $-B_k/2k$ if $n=0$.
\item{$s$} The signature of a lattice or Jacobi form.
\item{$sl_2(\R)$} The set of 2 by 2 real matrices of trace 0.
\item {$SL$} A special linear group.
\item {$SO$} A special orthogonal group.
\item {$\tau$} A complex number with positive imaginary part, or Ramanujan's
function $\tau(n)$.
\item {$t_{u,v}, t_r$} Automorphisms of a lattice. See section 2.
\item {$T_\l$} A Hecke operator.
\item {$T_{\l,\l}$} A Hecke operator.
\item {$u$} A norm zero vector in a lattice, often contained in $U$.
\item {$U$} A 2-dimensional null lattice, usually a sublattice of $M$.
\item {$U_\l$} A Hecke operator acting on Jacobi forms.
\item{$\phi$,$\phi_u$} Jacobi forms. See sections 2 and 3.
\item{$\Phi$} An automorphic form on $O_M(\R)^+$.
\item{$\psi$} A Jacobi form. See sections 2 and 3.
\item{$\Psi$} A meromorphic modular form with a modular product expansion.
\item {$v$} A vector in a vector system. See section 6.
\item {$V$} A vector system. See section 6.
\item {$V_\l$} A Hecke operator acting on Jacobi forms. See section 4.
\item {$V_{\l,\l}$} A Hecke operator acting on Jacobi forms. See section 4.
\item {$W$} A Weyl chamber. See section 12.
\item{$x$} A real number or an element of $K\otimes \R$, often equal to
$\Re(z)$.
\item{$y$} A real number or an element of $K\otimes \R$, often equal to
$\Im(z)$.
\item{$z$} An element of the
complexification of $K$, or a complex number, often equal to $x+iy$.
\item {$\Z$} The integers.
Terminology.
\item{}{\bf Nearly holomorphic.} Meromorphic with all poles at the cusps.
\item{}{\bf Automorphic form.} See section 2.
\item{}{\bf Classical Jacobi form.} A function of several variables
transforming like a modular form in one of them and like a theta function
in the others. See section 3.
\item{}{\bf Fourier group.} A certain
parabolic subgroup of $O_M$. See section 2.
\item{}{\bf Height.} The height of a vector is the minimum inner product
with a Weyl vector. See section 13.
\item{}{\bf Index.} See section 6 for the index of a vector system, and
sections 2 and 3 for the index of a Jacobi form.
\item{}{\bf Jacobi form.} See sections 2,3.
\item{}{\bf Jacobi group.} A parabolic subgroup of $O_M$. See section 2.
\item{}{\bf Koecher principle.} An nearly holomorphic automorphic form on a simple group of
rank greater than 1 is automatically holomorphic at the cusps.
\item{}{\bf Modular product.} An infinite product whose exponents
are the coefficients of nearly holomorphic modular forms. See section 5.
\item{}{\bf Primitive sublattice.} A sublattice $Z$ of $M$ is primitive if $Z$ contains
any vector of $M$ a nonzero multiple of which is in $Z$.
\item{}{\bf Rational quadratic divisor.} The zero set of $a(y,y)+(b,y)+c$
where $a\in \Z, b\in L,c\in \Z$. See section 5.
\item{}{\bf Signature.} The signature of a Jacobi form
is the signature of any of the lattices associated to it, and is one less
than the number of variables the Jacobi form depends on.
\item{}{\bf Singular weight.} Weight $s/2$ or 0. See section 3.
\item{}{\bf Spezialschar.} (``Special space.'') A space of automorphic forms
whose Fourier coefficients satisfy certain relations. See section 9
and [M paper I].
\item{}{\bf Spinor norm.} A homomorphism from a real orthogonal group to
$\R^*/\R^{*2}$
taking reflections of vectors of positive or negative norm to $1$
or $-1 $ respectively.
\item{}{\bf Theta function.} A modular form or Jacobi form
depending on a lattice. See section 3.
\item{}{\bf Vector system.} A multiset of vectors in a lattice with some of
the properties of a root system. See section 6.
\item{}{\bf Weyl chamber.} A generalization of the Weyl chamber of
a root system to vector systems. See section 6.
\item{}{\bf Weyl vector.} A generalization of the Weyl vector of
a root system to vector systems. See sections 6, 12.
\bigskip
{\bf 2. Automorphic forms and Jacobi forms}
\bigskip
We summarize some general facts about automorphic forms on
$O_{s+2,2}(\R)$ and set up some notation for them. General references
for this section are [F] for automorphic forms and [E-Z] for Jacobi
forms. The book [F] covers modular forms on symplectic groups rather
than orthogonal groups, but most of the general results carry over
with only minor changes. Similarly the book [E-Z] covers only Jacobi
forms of signature 1, but many of the results can easily be
generalized to Jacobi forms of arbitrary signature.
If $M$ is any even integral lattice (with associated quadratic form
$Q(v)=(v,v)/2$) we write $O_M$ for the algebraic group of rotations of
$M$, so that $O_M(R)$ is the group of rotations of $M\otimes R$
preserving the quadratic form $Q$ of $M\otimes R$. We write
$GL_M$ and $SL_M$ for the general and special linear groups of $M$,
$SO_M$ for
the special orthogonal group of $M$, and $GO_M$ for the general
orthogonal group (or conformal group) consisting of the linear
transformations multiplying the quadratic form by an invertible
element. We think of $SO_M$, $O_M$, $GL_M$, and so on as being
algebraic groups defined over $\Z$, so for example $O_M(\Z)$ is the
group of automorphisms of the lattice $M$.
There is a ``spinor
norm'' homomorphism from $O_{M}(R)$ to $R^*/R^{*2}$, which
which has the property that a reflection of a vector of
norm $Q(v)$ has spinor norm $Q(v)\in R^*/R^{*2}$. In this paper
$R^*/R^{*2} $ can usually be identified with the group $\{1,-1\}$
of order 2, and the reflection of a vector of positive or negative norm
then has spinor norm $+1$ or $-1$ respectively.
If $G$ is a subgroup of $O_{M}(R)$
we write $G^+$ for the subgroup of $G$ of elements of spinor
norm $1\in R^*/R^{*2}$. The elements of
$O_{M}(\R)$ with determinant 1 and positive spinor norm form the
connected component $SO_M(\R)^+$ of the identity. If $M$ is positive definite
the spinor norm on $O_M(\R)$ is
always positive, if $M$ is negative definite
it coincides with the determinant, and if
$M$ is indefinite then $SO_M(\R)^+$
has index 4 in $O_{M}(\R)$.
If $M$ is Lorentzian then the rotations of positive spinor norm
are exactly those that preserve rather than interchange the
two cones or negative norm vectors of $M\otimes\R$.
The group
$O_M(R)^+$ is the image of the pin group $Pin_M(R)$ induced by the
natural homomorphism from $Pin_M$ to $O_M$, and similarly $SO_M(R)^+$
is the image of the spin group $Spin_M(R)$. Notice that the map from
$Pin_M$ to $O_M$ is an epimorphism of algebraic groups, but the map
from $Pin_M(R) $ to $O_M(R)$ is not necessarily an epimorphism of
groups. It would really be more natural to use the pin and spin groups
throughout this paper rather than the orthogonal and special
orthogonal groups, but this is not (yet) essential and we will stick
to $O_M$ and $SO_M$ to save having to describe the construction of
$Pin_M$
and $Spin_M$.
From now on we assume that $M$ is a nonsingular even lattice of
signature $s$ and dimension $s+4$. We assume that we have chosen
a ``spin orientation'' on $M$, by which we mean a choice of
orientation on each 2-dimensional negative definite
subspace of $M\otimes \R$ which varies continuously.
There are 2 possible spin orientations on $M$, and they are
interchanged
by any rotation of negative spinor norm.
We construct a model for the Hermitian symmetric space of $O_M(\R)$.
We let $P$ be the vectors $z=x+iy\in M\otimes \C$ such that
$z^2=0$, $x^2<0$, and $(x,y) $ is a positively oriented base
of the 2-dimensional vector space spanned by $x$ and $y$. This space
$P$ is acted
on by $\C^*$ in the obvious way, and we define $H$ to be the quotient
of $P$ by this $\C^*$ action. Then $P$ and $H$ both have natural
complex structures, $H$ is an Hermitian symmetric space, and
$P$ is a principle $\C^*$ bundle over $H$. There is a natural
compactification
of $H$ which is the closure of $H$ in the projective space of
$M\otimes \C$.
The space $P$ is acted on transitively by $GO_M(\R)^+$,
the group of all conformal transformations of $M$ of positive spinor
norm. The subspace of $P$ of all vectors $x+iy$ with $x^2=-1$ is
acted on transitively by $O_M(\R)^+$ and is a principle $S^1$ bundle
over $H$.
Complex conjugation in $M\otimes\C$ maps $P$ and $H$ to their complex
conjugates $\bar P$ and $\bar H$. If we identify $\bar P$ and $\bar H$
with their complex conjugates using complex conjugation (which
commutes with $GO_M(\R)$) then we get an action of $GO_M(\R)$ on $H$
and $P$ such that elements of negative spinor norm act as
antiholomorphic transformations. This is similar to the extension
of the usual action of $GL_2(\R)^+$ (the subgroup of elements of
positive determinant) on the upper half plane extended to
an action of $GL_2(\R)$ on the complex plane with the real line
removed. If we identify the upper and lower half planes using complex
conjugation, then we get an action of $GL_2(\R)$ on the upper half
plane, with the elements of negative determinant acting as antiholomorphic
transformations.
The group $O_M(\Z)^+$ is a discrete subgroup of $O_M(\R)^+$. We will
say that a function $\Phi$ on $P$ is a nearly holomorphic automorphic
form of weight $k\in
\Z$ for $O_M(\Z)^+$ if it has the following properties.
\item {1} $\Phi$ is holomorphic on $P$.
\item {2} $\Phi$ is homogeneous of degree $-k$, i.e., $\Phi(vz)=z^{-k}\Phi(v)$
for $z\in \C$.
\item {3} $\Phi$ is invariant under $O_M(\Z)^+$, i.e., $\Phi(\gamma v)=\Phi(v)$
for $\gamma\in O_M(\Z)^+$. More generally, we also allow $\Phi(\gamma v)
=\det(\gamma)\Phi(v)$ and call such forms antiinvariant under
$O_M(\Z)^+$.
If $\Phi$ is also ``holomorphic at the cusps'' (see below) then we
call $\Phi$ a holomorphic automorphic form, or automorphic form for short.
For $s\ge 1$ any nearly holomorphic form is automatically
holomorphic by the
Koecher
boundedness principle, which also holds for $s=0$ provided the lattice
$M$ does not have square determinant (by the Koecher boundedness
principle for Hilbert modular forms).
(The Koecher boundedness principle states that any automorphic form on
an Hermitian symmetric space associated to a group of real rank greater
than 1 is automatically holomorphic at all cusps if it is holomorphic on
the symmetric space. See the article on pp. 296-300 by Baily in [B-M].)
Homogeneous functions of degree $-k$ on $P$ can be identified
with sections of the line bundle $P^k$ over $H$, so nearly
holomorphic automorphic forms
of weight $k$ are just invariant (or antiinvariant) holomorphic
sections of $P^k$.
We can restrict $\Phi$ to the subspace of $P$ with $x^2=y^2=-1$
and then lift it to a function on $O_M(\R)^+$ (or better, to a function
on $Pin_M(\R)$). The conditions on $\Phi$
then say that this lift is
left invariant under $O_M(\Z)^+$ and transforms under right
multiplication by the elements of a maximal compact subgroup according
to some representation (described by $k$) of this compact subgroup.
Hence our definition is equivalent to a special case of the usual
definition of an automorphic form on a reductive Lie group.
We can also define forms of half integral weight either by using
the double cover of the line bundle $P$ instead of $P$, or by
allowing $\Phi$ to be a 2-valued holomorphic function, or
by using functions on the pin group. A form of weight $k$ on
the group $O_M(\R)^+$ becomes a form of weight $2k$
on $Pin_M(\R)$. This is because the weight $k$ indexes
representations of an $S^1$ subgroup of $O_M(\R)^+$
or $Pin_M(\R)$, and the map between the corresponding $S^1$ subgroups
is 2 to 1, so the integer parameterizing irreducible representations
has to be doubled. Forms of half integral weight
on $O_M$ correspond to ordinary modular forms of integral weight
rather than half integral weight, because the double cover
$Spin_M\rightarrow SO_M$ corresponds to the double cover
$SL_2\rightarrow PGL_2$ rather than the metaplectic double cover
of the special linear group.
In particular if $M$ has dimension 3 then automorphic forms
on $O_M(\R)^+$ of weight $k$ can be identified with
ordinary modular forms of weight $2k$ (rather than $k$).
This annoying factor of 2 in the weights
is unavoidable and is not just caused by a bad choice of conventions:
one construction in this paper starts with an ordinary modular
form for $SL_2(\Z)$ of weight $k$, and ends up with an ordinary modular form
of weight $2k$, and this factor of 2 is essentially caused by the doubling
of weights when lifting forms on $O_M$ to forms on $Pin_M$.
We now study the parabolic subgroups of $O_M$. The subgroup
fixing a nonzero null sublattice of $M$ is a maximal parabolic subgroup,
and this
null sublattice can have rank 1 or 2. If it has rank 1 we will call the
corresponding parabolic subgroup a Fourier group, and if it has
rank 2 we call the corresponding parabolic subgroup a Jacobi group.
The reason for this terminology is that the ``Fourier--Jacobi''
expansion
of an automorphic form with respect to a parabolic subgroup
is essentially either a Fourier series
expansion or an expansion in terms of Jacobi forms, depending on
whether the parabolic subgroup is a Fourier group or a Jacobi group.
What we call the Jacobi group is essentially a central extension
of the Jacobi group in [E-Z, p. 10]; see also [E-Z Theorem 1.4]
and [E-Z Theorem 6.1] for other appearances of this central extension.
Suppose that $U$ is a 2-dimensional primitive null sublattice
of $M$, and let $J$ be the corresponding Jacobi subgroup of
$GO_M$. (A sublattice $U$ of $M$ is called primitive if $U=M\cap U\otimes \R$.)
There is an obvious induced action of $J$ on the
lattices $U$ and $U^\perp/U=K$, so we get a homomorphism from
$J$ to $GO_K\times GL_U$. The connected component of the
kernel of this homomorphism is
a Heisenberg group whose structure we will now describe. (This is the
``same'' Heisenberg group that turns up regularly in the theory of
theta functions.) If $u\in U $ and $v\in u^\perp/u$ then
we define an automorphism $t_{u,v}$ of $M$ by
$$t_{u,v}(w)=w+(w,u)v-((w,v)+(w,u)(v,v)/2)u.$$
For fixed $u$ these form a group of automorphisms of $M$ fixing
$u$ and all elements of $u^\perp/u$. If $u$, $v$ is a positively oriented
basis of $U$ and $r\in R$ then we define $t_{r} $ by
$$t_r= t_{ru,v}.$$
This depends on $U$ but not on the choice of positively
oriented basis for $U$, and commutes with all automorphisms
$t_{u,v}$ for $u\in U$, $v\in U^\perp$.
The automorphisms $t_{u,v}$ for $u\in U$, $v\in U^\perp$
satisfy the relations
$$t_{u_1,v_1}t_{u_2,v_2}=t_{u_2,v_2}t_{u_1,v_1}t_{r}$$
where $r=(v_1,v_2)$ times the determinant of a linear transformation
taking a positively oriented basis of $U$ to $u_1,u_2$.
They generate a Heisenberg group of dimension $2s+1$
whose center is the group of elements
of the form $t_{r}$.
If $\Phi$ is an automorphic function and $J$ is a Jacobi group we define
the Jacobi expansion of $\Phi$ as follows. We let $t_{r}$
for $r\in \R$ be the elements of the center of the Jacobi group.
We define $\phi_m$ for $m\in \Z$ by
$$\phi_m(v) = \int_{r\in \R/\Z}\Phi(t_{r}(v))e^{2\pi i mr}dr.$$
This is well defined because $\Phi(t_{r}(v)=\Phi(v)$ for $r\in \Z$.
The Jacobi expansion of $\Phi$ is then
$$\Phi=\sum_{m\in \Z} \phi_m,$$
and the functions $\phi_m$ have the following properties.
\item{1} $\phi_m$ is a homogeneous function of weight $k$.
\item {2} $\phi_m$ is holomorphic on $P$.
\item {3} $\phi_m(t_{r}(v))=e^{2\pi i mr } \phi_m(v)$.
\item {4} $\phi_m$ is invariant (or possibly antiinvariant) under
the Jacobi group $J(\Z)^+$.
\item {5} $\phi_m$ is ``holomorphic at the cusps'' (at least if $\Phi$ is);
the meaning of this is described below in the section on Fourier
subgroups.
Functions with these properties are called {\bf Jacobi forms} of index
$m$ and weight $k$ and signature $s$. If the lattice $M$ is 5
dimensional (i.e., the signature is 1) then these are more or less the
same as the Jacobi forms of [E-Z] multiplied by an elementary function.
The signature $s$ is the signature of any of the 3 lattices $K$, $L$, or
$M$ associated with the Jacobi form.
If $\phi_m$ is a Jacobi form we can analytically continue it
to the space of all norm 0 vectors $v$ in $M\otimes \C$ such that
$\Im((v,u_1)/(v,u_2))>0$, where $u_1$ and $u_2$ is any oriented
base of $U$, by saying that $\phi_m(t_{r}(v))=e^{2\pi i mr } \phi_m(v)$
must hold for all complex values of $r$.
Now suppose that $u$ is a primitive norm 0 element
of $M$, and let $F$ be the corresponding Fourier group.
There is an induced action of $F$ on the lattice $L=u^\perp/u$,
which gives a homomorphism from $F$ to the group $GO_L$.
The connected component of the
kernel of this homomorphism is a unipotent abelian group
containing the elements $t_{u,v}$ for $v\in L$.
Suppose that $\Phi$ is either an automorphic form
or a Jacobi form of a 2-dimensional lattice containing $u$.
Then $\Phi(t_{u,v}(w))=\Phi(w)$ for $v\in L$.
We define $A_m$ for $m\in L'$ by
$$A_m(w) = \int_{v\in L\otimes\R/L}\Phi(t_{u,v}(w))e^{2\pi i mv}dv.$$
The Fourier expansion of $\Phi$ is then
$$\Phi=\sum_{m\in L'} A_m.$$
We will see shortly that the $A_m$'s are elementary factors times
exponential functions, so this is essentially just the usual Fourier
series expansion of $\Phi$.
We say that $\Phi$ is holomorphic at $F$ if the Fourier coefficients
$A_m$ are 0 unless $m$ lies in the closed positive cone of $L$.
(The vectors of nonpositive norm in $L\otimes \R$ form two
closed cones; the positive one is defined to be the one containing
a norm 0 vector $v$ such that $u,v$ is a positively oriented
basis of the 2-dimensional space they span in $M$.)
If $\phi$ is a Jacobi form corresponding to some Jacobi group
$J$, then we say that $\phi$ is holomorphic (at the cusps)
if the Fourier expansion of $\phi$ at every cusp of $J$ is holomorphic,
i.e., if the Fourier expansion of $\phi$ at every Fourier subgroup $F$
such that $F\cap J$ is parabolic is holomorphic. This condition on
$F$ just means that $F$ is the Fourier group of some
1-dimensional lattice contained in the 2-dimensional null lattice
of $J$.
If $F$ is a Fourier group of a norm zero vector $u$, we will construct
another model $H_u$ of the Hermitian space $H$, on which the action
of $F$ is easier to visualize. We put $L=u^\perp/u$ so that
$L$ is a Lorentzian lattice, and we write $L^1(R)$ for the
space (vectors of $M\otimes R$ which have inner product 1 with
$u$)/$Ru$,
so that $L^1$ is an affine space over $L$.
We define $H_u$ to be the vectors in $x+iy\in L^1(\C)$ such that $y$
is in the positive open cone of $L\otimes \R$.
If we write $P^1$ for the vectors of $P$ having inner product 1 with
$u$, then $H$ is naturally isomorphic to $P^1$ because each
fiber of $P$ over $H$ has a unique element in $P^1$. Also each
element of $P^1$ represents an element of $H_u$. This maps $P^1$ onto
$H_u$, because given any point $x+iy\in M\otimes \C$ representing
a point in $H_u$, we can add a multiple of $u$ to $x$ to make the norm
of $x$ equal to that of $y$, and can then add a multiple of $u$ to $y$
to make $x$ and $y$ orthogonal. The point $x+iy$ then lies in $P^1$.
Hence we have constructed isomorphisms from $H$ to $P^1$
and from $P^1$ to $H_u$, so $H_u$ can be identified with $H$.
We identify functions of degree $-k$ on $P$ with functions on
$P^1$ by restriction, and we identify functions on $P^1$ with
functions on $H_u$ by using the isomorphism from $P^1$ to $H_u$.
Hence functions of degree $-k$ on $P$ (in particular automorphic
forms of weight $k$) can be identified with functions on $H_u$. \bigskip
{\bf 3. Classical theory}
\bigskip
We will show that the definitions in the previous section
are equivalent to the usual definitions of Jacobi forms
(at least when $M$ has dimension 5) by working out a simple case
explicitly. We choose $K$ to be an even positive definite lattice
and we let $L=K\oplus II_{1,1}$, $M=L\oplus II_{1,1}$. We can write
vectors of $M$ in the form $(\kappa,\alpha,{-\delta},\gamma,\beta)$ with $\kappa\in K$,
$\alpha,\beta,\gamma,\delta \in \Z$,
and this vector has norm $\kappa^2/2+\alpha{\delta}-\gamma\beta$.
We put $u_1=(0,0,1,0,0)$, $u_2=(0,0,0,0,1)$. We define $J$ to
be the Jacobi group of $\langle u_1,u_2\rangle$, and
we let $F$ be the Fourier group of $u=u_2$.
We let $H_u$ be the Hermitian symmetric space defined in the previous
section. We can identify $H_u$ with a subspace of $L\otimes \C$
(because we have a canonical vector $(0,0,0,1,0)$ which has
inner product $-1$ with $u$.)
It is particularly easy to describe functions on $H$ which are
invariant under $F(\Z)$. If we consider the associated
function $\Phi_u$ on $H_u$, then we can expand $\Phi_u$ as a Fourier series
(as $f$ is invariant under translation by $L$), and the Fourier
coefficients have to be invariant under the natural action
of $F$ on $L'$.
As an example we work out the condition on $\Phi_u$ that corresponds to
the weight $k$ automorphic form
$f$ being invariant under the transformation taking
$(v,\gamma,\beta)$ to $(v,\beta,\gamma)$ ($v\in L\otimes\C$).
By definition, $\Phi_u(v)=f(v,1,v^2/2)$
and $f(v,\gamma,\beta)=f(v,\beta,\gamma)$, and $f(v,\gamma,\beta)=
\gamma^{-k}\Phi_u(v/\gamma)$ (for $v^2=2mn$).
From this it follows that
$$\Phi_u(-2v/(v,v))=((v,v)/2)^k\Phi_u(v).$$
Suppose now that $\phi$ is a Jacobi form of index $m$ and weight $k$. We
will show how to identify $\phi$ with a classical Jacobi form. The
conditions on $\phi$ are
$$\eqalign{
\phi(z,\alpha,{-\delta},\gamma,\beta)&=
r^k\phi(rz,r\alpha,-r{\delta},r\gamma,r\beta)\cr
&=e^{-2\pi i mr}\phi(z,\alpha,{-\delta}+r\gamma,\gamma,\beta-r\alpha)\cr
&=\phi(z,a\alpha+b\gamma,-d{\delta}-c\beta,c\alpha+d\gamma,a\beta+b{\delta})\cr
&=\phi(z+\alpha\lambda,\alpha,{-\delta}+(z,\lambda)+\alpha\lambda^2/2,\gamma,\beta)\cr
&=\phi(z+\gamma\mu,\alpha,{-\delta},\gamma,\beta+(z,\mu)+\gamma\mu^2/2)\cr
}$$
for $r\in \R$, $\lambda\in K$, $\mu\in K$, $z\in K\otimes\C$,
${ab\choose cd}\in SL_2(\Z)$, $\alpha,\beta,\gamma,\delta\in \R$.
We extend $\phi$ so that it is defined for all norm 0 vectors such
that $\Im(\alpha/\gamma)>0$ by insisting that $\phi$ should satisfy
$\phi(z,\alpha,{-\delta}+r\gamma,\gamma,\beta-r\alpha)=e^{2\pi i
mr}\phi(z,\alpha,{-\delta},\gamma,\beta)$ for all complex values of
$r$. If we define $\phi_u(z,\tau)$ for $\tau\in \C$,$\Im(\tau)>0$,
$z\in K\otimes\C$
by $$\phi_u(z,\tau)= \phi(z,\tau,0,1,z^2/2)$$ then we find that
$\phi_u$ has the following properties.
$$\eqalign{
\phi_u(z/(c\tau+d),(a\tau+b)/(c\tau+d))&=
(c \tau+d)^ke^{2\pi i m c(z^2/2)/(c\tau+d)}\phi_u(z,\tau)\cr
\phi_u(z+\lambda\tau+\mu,\tau)&=
e^{-2\pi i m(z\lambda+\tau
\lambda^2/2)}\phi_u(z,\tau)\quad (\lambda,\mu\in K).\cr
}$$
Conversely, if we are given $\phi_u$ with these properties and we
define $\phi$ by $$\phi(z,\alpha,{-\delta},\gamma,\beta)=
\phi_u(z/\gamma,\alpha/\gamma)\gamma^{-k}e^{-2\pi i m
{\delta}/\gamma}$$ then $\phi$ transforms like a Jacobi form. If $K$
is a one dimensional lattice spanned by a vector of norm 2, then the
relations for $\phi_u$ are equivalent to the relations in [E-Z, p. 1]
defining classical Jacobi forms (except for a misprint in their
equation (1), where the term $2\pi i mcz$ should be $2\pi i mcz^2$).
There is an extra factor of 2 in some of the norms in some of our
formulas compared to the ones in [E-Z]; this appears because we
normalize $K$ to be an even lattice generated by an element of norm 2,
while in [E-Z] $K$ is a lattice generated by an element of norm 1.
We now summarize some facts about Jacobi forms, which are
straightforward extensions of standard results about Siegel modular
forms and Jacobi forms of signature 1. We say that a Jacobi form of
signature $s$ has singular weight if its weight is $0$ or $s/2$. We
say that an automorphic form on $O_{M}^+(\R)$ has singular weight if
its weight is 0 or $s/2$. We define a theta function of weight
$k=s/2$ and index $m\in \Z$ to be a linear combination of functions of
the form $$\theta_{K+r}(z,\tau)=\sum_{\lambda\in
K+r}q^{\lambda^2/2}\zeta^{m\lambda}$$ where $K$ is some positive
definite rational lattice of dimension $s$ and $r\in K\otimes \Q$
($q=e^{2\pi i \tau}$, $\zeta^\lambda= e^{2\pi i (z,\lambda)}$). Any
theta function is a holomorphic Jacobi form of singular weight.
\proclaim Theorem {3.1}. Any
(nearly) holomorphic Jacobi form $\phi$ of positive index can be
written as a sum of products of theta functions and (nearly)
holomorphic modular forms (though these theta functions and modular
forms may have higher level than $\phi$).
For the case of Jacobi forms of signature 1 this is
theorem 5.1 of [E-Z]. The proof for higher signatures
is essentially the same.
\proclaim Corollary {3.2}. Any holomorphic Jacobi form of weight
0 is constant, and there are no nonconstant Jacobi forms of weight
less than the singular weight $s/2$. Any holomorphic Jacobi form of
singular weight $s/2$ is a sum of theta functions.
Proof. This follows from theorem 3.1 and the fact that
any theta function has weight $s/2$.
\proclaim Corollary {3.3}. Any holomorphic automorphic form
either has weight 0 in which case it is constant, or
has weight at least $s/2$. If it has singular weight $s/2$ then
all the Fourier coefficients corresponding to vectors of nonzero norm
vanish.
Proof. If $f$ can be written as a sum of Jacobi forms (which
is the only case we will use in this paper and is always true if
$s\ge 2$) then this follows from the previous corollary. This is
the analogue of the second proof in [F, appendix IV]. In general
the corollary can be proved by using the Laplacian operator, as in the first
proof given in [F, appendix IV].
In particular there is a gap between 0 and $s/2$, such that
there are no modular forms with weights in this gap. This phenomenon
does not occur for Siegel modular forms because it just happens that
all half integers less than the largest singular weight are
also singular weights. (In both cases the number of singular
weights is equal to the real rank of the corresponding Lie group.)
Similarly the gap between weights 0 and $s/2$ of holomorphic Jacobi forms
of signature $s$ is not really noticeable
in [E-Z] because $s/2$ is equal to $1/2$.
\bigskip
{\bf 4. Hecke operators for Jacobi groups.}
\bigskip
Suppose that $M$ is an even lattice of dimension $s+4$ and signature
$s$ and that $U$ is a 2-dimensional primitive null sublattice
and $J$ the corresponding Jacobi group. We will assume that
we are in the simplest (``level 1'') case, so we assume that
the map from $M$ to $U'$ is onto, and we assume that
the discrete group we are working with is the full
group $J(\Z)^+$ (rather than some congruence subgroup).
Suppose that $Y$ is a 2-dimensional lattice containing $U$.
We define $M[Y]$ to be the lattice generated by $Y$ and the vectors
of $M$ that have integral inner product with all vectors of $Y$.
The fact that $M$ maps onto $U'$ implies $M$ can be written as
$K\oplus II_{2,2}$ where $U$ is contained in $II_{2,2}$, and
this implies that $M[Y]$ is isomorphic to $M$ under
some isomorphism mapping $Y$ to $U$.
(In the higher level case this is not always true, and we have to
restrict ourselves to lattices $Y$
having this property.)
We define $J_\l$ to be the set of all isomorphisms of positive spinor norm
from some lattice of the form $M[Y]$ with $[Y:U]=\l$
to $M$. This is
a union of double cosets of $J(\Z)^+=J_1$ because $J(\Z)^+$ acts on the
set of lattices $Y$ with $[Y:U]=\l$. Two elements $a$ and $b$ of
$J_\l$ are in the same right $J$-coset if and only if
$ab^{-1}$ is in $J_1$, which happens if and only if $a^{-1}$ and $b^{-1}$
both map $U$ to the same lattice $Y$. Hence the right cosets
of $J(\Z)^+$ in $J_\l$ correspond exactly to the lattices $Y$ with
$[Y:U]=\l$, and in particular there are only a finite number of
such right cosets.
If $\phi$ is a Jacobi form for $J(\Z)^+$ we define the Hecke operator
$V_\l$ by
$$(\phi|V_\l)(v)=(1/\l)\sum_{g\in J_1\backslash J_\l}\phi(gv).$$
This operator maps Jacobi forms of weight $k$ and index $m$ for $J_1$
to Jacobi forms of weight $k$ and index $m\l$. (The index gets
multiplied
by $\l$ because the elements of $J_\l$ act on $\Lambda^2(U)$ and
hence on the center of the nilradical of $J_1$ as multiplication by $\l$.)
We define the operator $V_{\l,\l}$ similarly except that we
restrict to the coset corresponding to lattices $Y$ such
that $Y/U=(\Z/\l\Z)^2$, so that
$$(\phi|V_{\l,\l})(v)=(1/\l^2)\phi(gv)$$
where $g$ maps $(1/\l)U$ to $U$. The operator $V_{\l,\l} $ maps Jacobi
forms of weight $k$ and index $m$ to Jacobi forms of weight $k$ and
index $m\l^2$. The operator $\l^{2-k}V_{\l,\l}$ is denoted by $U_\l$
in [E-Z].
We define an action of $GL_2$ on $M$ by
$${ab\choose cd}(z,\alpha,-\delta,\gamma,\beta) =
(z, (a\alpha+b\gamma)/\l,-d\delta-c\beta,(c\alpha+d\gamma)/\l, b\delta+a\beta)$$
where $\l=ad-bc$. A set of right coset representatives of
$J_1\backslash J_\l$ is then given by the usual set of matrices
${ab\choose 0d}$ with $0\le b0$ means that $x$
has positive inner product with this vector.
We define a {\bf modular product} to be
an infinite product of the form
$$\Phi(y)=e^{-2\pi i(\rho,y)}\prod_{x\in L,x>0}(1-e^{-2\pi i(x,y)})^{c(x)}$$
where $y\in L\otimes \C$, $\Im(y)\in C$, $c(x)$ is the coefficient of $q^{-(x,x)/2}$ of some
nearly holomorphic modular form $f_x$ of weight $-s/2$, and the modular forms
$f_{x_1}$ and $f_{x_2}$ are equal whenever $x_1-x_2$ lies in $NL$ for
some fixed integer $N$.
We define a {\bf rational quadratic divisor} to be
the set of points $y$ with $\Im(y)\in C$ such that $a(y,y)+(b,y)+c=0$ for
some $a\in \Z$, $b\in L$, $c\in \Z$ with $(b,b)-4ac>0$.
If $M=L\oplus II_{1,1}$, then the points $y$ is some rational
quadratic divisor are just the points $(y,1,y^2/2)\in M\otimes \C$
that are orthogonal to the norm $b^2-4ac$ vector $(b,-2a,-c)$ of $M$.
Hence rational quadratic divisors correspond to equivalence classes
of positive norm vectors of $M$, where two vectors are equivalent if
they are rational multiples of each other.
For example, if $L$ is a 1-dimensional lattice then
a rational quadratic divisor
is just an imaginary quadratic irrational number in the upper half plane.
\proclaim Theorem {5.1}. Any modular product $\Phi(y)$ converges
to a holomorphic function whenever $\Im(y)$ is in $C$
and has sufficiently large norm. This function can be
analytically continued to a multivalued meromorphic function
for all $y$ with $\Im(y)\in C$ all of whose singularities and zeros
lie on rational quadratic divisors.
We will see later that along any
rational quadratic divisor $a(y,y)+(b,y)+c=0$ the function $\Phi(y)$
is locally of the form $(a(y,y)+(b,y)+c)^s$ times a holomorphic
function for some complex number $s$ (except of course where the
rational quadratic divisor meets other singularities or zeros of $\Phi$).
The complex number $s$ is called the multiplicity of the zero
of $\Phi$ along this rational quadratic divisor. The function $\Phi$ is holomorphic
if and only if the multiplicity of every rational quadratic divisor is
a nonnegative integer.
If we allow the modular forms $f_r$ in the definition
of a modular product to have poles in the upper half plane,
then their coefficients $c(n)$ increase exponentially fast which implies
that the product defining $\Phi$ does not converge anywhere. On the other
hand,
if we insist that the modular forms $f_r$ should be holomorphic, then
their coefficients $c(n)$ have polynomial growth, which implies that
the infinite product for $\Phi$ converges whenever $\Im(y)\in C$, so that
$\Phi$ is holomorphic and nonzero in this region. An example of
this
case is
$f(\tau)=12\sum_{n\in \Z}q^{n^2}$ and
$\Phi(\tau)=q\prod_{n>0}(1-q^n)^{24}$.
\proclaim Theorem {5.2}. Suppose that $k$ is a positive integer and
$$\Phi(y)=\sum_{x\in L,x>0}A(x)e^{-2\pi i (x,v)}$$
where $A(x)= \sum_{d|x}d^{k-1}c_x(-(x,x)/2d^2)$ and $c_x(n)$
is the coefficient of $q^n$ of some nearly holomorphic modular form $f_x$
of weight $k-s/2$, such that $f_x$ depends only on $x\bmod N$
for some fixed integer $N$. Then the sum for $\Phi$ converges
whenever $\Im(y)\in C$ and $-(\Im(y),\Im(y))$ is sufficiently large,
and can be analytically continued to a meromorphic function
defined for all $y$ with $\Im(y)\in C$, all of whose
singularities are poles of order $k$ lying on rational quadratic divisors.
The proof of theorem 5.2 is similar to that of theorem 5.1
and slightly simpler, so we will omit it. If $k=0$ then the sum
in theorem 5.2 is, up to some elementary factors,
the logarithm of the product in theorem
5.1, so the main change in the proof is that we do not first
need to take logarithms. Lemmas 5.3 and 5.4 are sufficiently
general for the extension of the proof to theorem 5.2.
The idea of the proof of theorem 5.1
is that $\log(\Phi(y))$ is given
by a Fourier series whose coefficients depend on the
coefficients of modular forms. The singularities of any periodic
function are closely related to the asymptotic behavior of its
Fourier
coefficients, and we know the asymptotic behavior of the coefficients
of modular forms because of the Hardy-Ramanujan-Rademacher series.
Hence we can find all the singularities of $\log(\Phi(y))$, which gives
us all the singularities and zeros of $\Phi(y)$.
Before giving the proof of theorem 5.1
we prove two preliminary lemmas.
\proclaim Lemma {5.3}. Suppose that $f(\tau)=\sum_{n\in \Z}c(n)q^n$
is a nearly holomorphic modular form
which has half integral weight $k$ (which may be positive or negative
or zero). Suppose that its
expansion at the cusp $a/c$ ($c>0$, $ad-bc=1$) is
$$(c\tau+d)^{-k}f((a\tau+b)/(c\tau+d))=
\sum_{n\in \Q} c_{a/c}(n)e^{2\pi in\tau}.$$
Then for any
positive number $\epsilon$ we can find a finite sum
of terms of the series
$$\sum_{m>0}\sum_{c>0}\sum_{0\le a-1/2$, $ \Re(z)>0$, with $z=2\pi t\sqrt{(y,y)}$,
$\mu=s/2$.
($K_\mu$ is the modified Bessel function of the third kind; [E vol. 2, 7.2.2].)
We find that
$$\int_{x\in C}e^{-2\pi i(x,y)}f(\sqrt{(-x,x)})d^{s+2}x
=2\int_{t=0}^\infty f(t)K_{s/2}(2\pi t\sqrt{(y,y)})t^{s/2+1}(y,y)^{-s/4} dt.$$
We now substitute $f(t)=I_{1-k}(4\pi t\sqrt{m/2})t^{k-1}$ into this
and find that
$$I=4\pi2^{(1-k)/2}\int_{t=0}^\infty I_{1-k}(4\pi t\sqrt{m/2})K_{s/2}(2\pi t\sqrt{(y,y)}){(y,y)}^{-s/4} t^{s/2+k} dt.$$
By [E vol. 2, 7.14.2, formula 35]
$$ \eqalign{&2^{\rho+1}\alpha^{\nu+1-\rho}\Gamma(\nu+1)\int_0^\infty
K_\mu(\alpha t)I_{\nu}(\beta t)t^{-\rho}dt \cr
=&\beta^{\nu}\Gamma(\nu/2-\rho/2+\mu/2+1/2)\Gamma(\nu/2-\rho/2-\mu/2+1/2)\times
\cr
&\times{}_2F_1(\nu/2-\rho/2+\mu/2+1/2,\nu/2-\rho/2-\mu/2+1/2;
\nu+1;\beta^2/\alpha^2)\cr
}$$
whenever $\alpha>\beta$, $\Re
(\nu-\rho+1\pm\mu)>0$. (${}_2F_1$ is the hypergeometric function
[E volume 1, chapter II].)
If we set $\mu=s/2$, $\nu=1-k$, $\beta=4\pi \sqrt{m/2}$,
$\alpha=2\pi\sqrt{(y,y)}$,
$\rho= -k-s/2$ in this we find that
$$\eqalign{
&2^{1-k-s/2}(2\pi\sqrt{(y,y)})^{s/2+2}\Gamma(2-k)\int_0^\infty
K_{s/2}(2\pi t\sqrt{(y,y)})I_{1-k}(4\pi t\sqrt{m/2})t^{k+s/2}dt\cr
=&(4\pi\sqrt{m/2})^{1-k}\Gamma(1+s/2){}_2F_1(1+s/2,1;2-k;2m/(y,y))\cr
}$$
so that
$$\eqalign{
&2\pi\int_0^\infty K_{s/2}(2\pi t\sqrt{(y,y)})I_{1-k}(4\pi t\sqrt{m/2})t^{k+s/2}dt\cr
=&m^{(1-k)/2}{(y,y)}^{-1-s/4}2^{-(k+1)/2}{}_2F_1(1+s/2,1;2-k;2m/(y,y))\Gamma(1+s/2)/\Gamma(2-k)\cr
}$$
We find that
$$\eqalign{
I=& 2^{1-k}m^{(1-k)/2}(y,y)^{-1-s/2}{}_2F_1(1+s/2,1;2-k;2m/(y,y))\Gamma(1+s/2)/\Gamma(2-k)\cr
}$$
The hypergeometric function ${}_2F_1(a,b;c;z)$ can be analytically continued
whenever $z$ is not $0$, 1, or $\infty$, so the function $I$ can be analytically continued whenever $(y,y) $ is not 0 or $2m$.
(If $k$ is a positive integer at least 2 then
the hypergeometric function has a pole in $k$, but this cancels out with
the pole of $\Gamma(2-k)$, so $I$ is still a well defined analytic function.)
This proves lemma 5.4.
We can now prove theorem 5.1.
We can ignore the factor $e^{-2\pi i (\rho,y)}$, and can assume that
the product is taken only over those values of $x$ in
some coset $v+NL$ of $NL$, so that $c(x)$ is equal to the coefficient
$c(-(x,x)/2)$ of
$q^{-(x,x)}$ of some fixed nearly holomorphic modular form $f(\tau)$.
If we expand the coefficients using lemma 1 as a finite sum of Bessel
functions plus a remainder term, then the estimate $O(e^{\epsilon\sqrt{n}})$
for the remainder shows that the product using the remainder terms can
be made to converge whenever $\Im(y)$ has norm at least $\delta$
for any given positive $\delta$. It is also easy to check that
the infinite product over the vectors $x$ of negative or zero norm
converges whenever $\Im(y)\in C$, so we can ignore these terms
in the product. (In fact, the same is also true for the terms where
$r$ has norm less than any given constant.) Therefore it is
sufficient to prove theorem 5.1 for the infinite product
$$\prod_{x\in L+v,x\in C}(1-e^{-2\pi i(x,y)})^{I_{1-k}(4\pi
\sqrt{m(-x,x)/2})((-x,x)/2)^{(k-1)/2}}$$
(after replacing $L$ by $NL$).
The logarithm of this is
$$-\sum_{x\in L+v,x\in C}\sum_{n>0}e^{-2\pi i n(x,y)}I_{1-k}(4\pi
\sqrt{m(-x,x)/2})((-x,x)/2)^{(k-1)/2}/n.$$
The sum of all terms with $n$ large converges whenever the norm of $\Im(y)$
is at least $\delta$ for any given positive constant $\delta$,
so it is sufficient to prove that the sum of the
terms for any fixed $n$ can be analytically continued
with at most logarithmic singularities along rational quadratic divisors.
If we replace $L$ by $nL$ we find that we have to show that the
function
$$\sum_{x\in L+v,x\in C}e^{-2\pi i (x,y)}I_{1-k}(4\pi \sqrt{m(-x,x)/2})((-x,x)/2)^{(k-1)/2}$$
has only logarithmic singularities along rational quadratic divisors.
This is a finite linear combination of sums of the form
$$\sum_{x\in L,x\in C}e^{-2\pi i (x,y+r)}I_{1-k}(4\pi
\sqrt{m(-x,x)/2})((-x,x)/2)^{(1-k)/2}$$
for some larger lattice $L$ and some rational vectors $r\in L\otimes \Q$.
If we apply the Poisson summation formula to the integral in lemma 5.4
we can evaluate this sum explicitly, and
by lemma 5.4 all its singularities lie on quadratic divisors of the form
$(y+r+v,y+r+v)=2m$
for some vectors $v$ in the dual of $L$. (More precisely,
the singularities are of the form form $-m^{(k-1)/2}\log(1-2m/(y+r+v,y+r+v))$.)
If we exponentiate this we find that all singularities and zeros
of the product in theorem 1 lie on rational quadratic divisors because $b$
is rational and $r+v\in L\otimes \Q$, which
proves theorem 5.1.
\bigskip
{\bf 6. Vector systems and the Macdonald identities.}
\bigskip
In this section we show that certain infinite products
parameterized by vectors of a lattice are Jacobi forms.
The Macdonald identities for root systems are more or less
a special case of this result.
We first define vector systems in a lattice, which are a
generalization
of indecomposable root systems. Suppose that $K$ is a positive definite
integral lattice,
and that we are given nonnegative
integers $c(v)$ for $v\in K$ which are zero for
all but a finite number of vectors of $K$. We say that
the function $c$ is a {\bf vector system} if it has the following 2
properties.
\item {1.} $c(v)=c(-v)$.
\item {2.} The function taking $\lambda$ to $\sum_{v\in K}c(v)(\lambda,v)^2$
is constant on the sphere of norm 1 vectors $\lambda\in K\otimes\R$.
We will write $V$ for the ``multiset'' of vectors in
a vector system, so we think of $V$ as containing $c(v)$ copies
of each vector $v\in K$, and we write $\sum_{v\in V}f(v)$
instead of $\sum_{v\in K} c(v)f(v)$ (and similarly for products over
$V$).
The second axiom for a vector system says that the directions of the vectors
in it are evenly distributed over the unit sphere in some weak sense.
We say the vector system $V$ is trivial if it only contains
vectors of zero norm. A decomposable root system is not usually
a vector system.
\proclaim Lemma {6.1}. If $G$ is any group acting on the lattice $K$
that acts
irreducibly on $K\otimes \R$ and contains $-1$ then any orbit $V$ of $G$, or any finite
union of orbits
of $G$, is a vector system. In particular if the automorphism group of $K$
acts irreducibly on $K\otimes \R$ then the set $V$ of vectors
of any fixed norm is a vector system, and any finite multiset of vectors of $V$
invariant under the automorphism group of $K$ is a vector system.
Proof. If $\sum_{v\in V}(\lambda,v)^2$ were not constant
on the unit sphere, then the points at which it took its
maximum value would span a proper subspace of $K\otimes \R$
invariant under the $G$,
contradicting the fact that $G$
acts irreducibly on $K\otimes \R$. This proves lemma 6.1.
We define the {\bf index} $m$ of a vector system by
$$m = \sum_{v\in V} {(v,v)\over 2\dim(K)}.$$
\proclaim Lemma {6.2}. If $V$ is a vector system and $\lambda$ and $\mu$
are any vectors of $K$ then
$$\eqalign{
\sum_{v\in V}(v,\lambda)(v,\mu)& = 2m(\lambda,\mu)\cr
\sum_{v\in V}v(v,\lambda)& = 2m\lambda.\cr
}$$
Proof. If $\lambda=\mu$ the first line follows from axiom 2 by integrating
$\lambda$ over the unit sphere. The case for arbitrary $\lambda$ and $\mu$
follows from the case for $\lambda=\mu$ by polarization.
The second identity follows from the first because both sides
are vectors having the same inner product with all vectors $\mu$.
This proves lemma 6.2.
\proclaim Lemma {6.3}. The index $m$ of a vector system $V$ is a nonnegative
integer, and is 0 if and only if the vector system is trivial.
Proof. The only nontrivial fact to prove is that $m$ is integral.
Suppose that $n$ is the highest common factor of
all the integers $(\lambda,\mu)$ for $\lambda\in K$, $\mu\in K$. The sum on the
left of lemma 6.2 is divisible by $2n^2$ for any $\lambda,\mu\in K$
(the factor of 2 comes from the fact that if $v\in V$ then $-v\in V$),
so if we let $\lambda$ and $\mu$ run through all vectors of $K$ we see
from lemma 6.2 that $m$ is divisible by $2n^2/2n=n$ and is therefore
integral. This proves lemma 6.3.
The index $m$ is closely related to the dual Coxeter number
of a root system, and can be thought of as measuring the
``average amount of norm per dimension'' of the vector system.
If the vector system is
an indecomposable root system with roots of maximal length 2,
then its index is equal to the dual Coxeter number.
The hyperplanes orthogonal to
the vectors of a vector system $V$
divide $K\otimes \R$ into cones that we call
the {\bf Weyl chambers} of the vector system. (Warning:
unlike the case of root systems, the Weyl chambers need not
be all the same shape.) If we choose a fixed Weyl chamber $W$
then we can define the positive and negative vectors of the vector system
by saying that $v$ is positive or negative ($v>0$ or $v<0$) if
$v$ has positive or negative inner product with some vector
in the interior of the Weyl chamber. This does not depend on which
vector in the Weyl chamber we choose, and every vector of the
vector system is either positive or negative.
We define the {\bf Weyl vector} $\rho=\rho_W$ of $W$ by
$$\rho = {1\over 2} \sum_{v\in V, v>0}v.$$
\proclaim Lemma {6.4}. If $\lambda$ is in the dual of $K$, then
$2(\rho,\lambda)\equiv m(\lambda,\lambda)\bmod 2$, and in particular
$m(\lambda,\lambda)$ is integral.
(Notice that the Weyl vectors for
different Weyl chambers differ by elements of $K$, so that
$2(\rho,\lambda)$ is well defined mod 2 independently of the choice of
Weyl chamber.)
Proof.
$$\eqalign{
(2\rho,\lambda)&=\sum_{v>0}(v,\lambda)\cr
&\equiv \sum_{v>0}(v,\lambda)^2\bmod 2\cr
&= m(\lambda,\lambda),\cr
}$$
which proves lemma 6.4.
For example, if $K$ is an even unimodular lattice, then this
lemma shows that $\rho\in K$ because it has integral inner
product with every element of the dual of $K$. This does not imply that
the Weyl vector of the root system of $K$ lies in $K$ because the
root system of $K$ is not always a vector system.
Finally we define $d$ to be the number of vectors in
$V$ (counted with multiplicities), and we define the weight $k$ to be
half the number of zero vectors in $V$ (so $k=c(0)/2$).
For example, if $V$ is the weights of some representation of
a simple finite dimensional Lie algebra, then $d$ is the dimension
of this representation.
If $V$ is a vector system in $K$
we define the (untwisted) {\bf affine vector system} of $V$
to be the multiset of vectors $(v,n)\in K\oplus \Z$ with
$v\in V$. We say that $(v,n)$ is positive if either $n>0$
or $n=0, v>0$.
We select a Weyl chamber $W$ with its corresponding
Weyl vector $\rho$ and positive vectors, and define
$$\psi(z,\tau)= q^{d/24}\zeta^{-\rho}\prod_{v\in V,n\in \Z,(v,n)>0}
(1-q^n\zeta^v)$$
where $q^a=e^{2\pi i a\tau}$, $\zeta^v=e^{2\pi i (z,v)}$.
When $V$ is the vector system of a finite dimensional or affine
Kac-Moody algebra this is essentially
the denominator of the Weyl-Kac character formula.
The main aim of this section is to prove the following
generalization of the Macdonald identities for untwisted
affine root systems.
\proclaim Theorem {6.5}.
The function $\psi$ is a nearly holomorphic Jacobi form of weight
$k$ and index $m$.
More precisely,
$$\eqalign{
\psi(z,\tau+1) &= e^{2\pi i d/24}\psi(z,\tau)\cr
\psi(z/\tau,-1/\tau)
&= (-i)^{d/2-k}(\tau/i)^k e^{2\pi i m (z,z)/2\tau}\psi(z,\tau)\cr
\psi(z+\mu,\tau)&=(-1)^{2(\rho,\mu)}\psi(z,\tau)\cr
\psi(z+\lambda\tau,\tau)&=
(-1)^{2(\rho,\lambda)}q^{-m(\lambda,\lambda)/2}\zeta^{-m\lambda}\psi(z,\tau)\cr
}$$
for any $\lambda,\mu \in K'$. The function $\psi$ can be written
as a finite sum of theta functions times nearly holomorphic modular forms.
For example, if $V$ is an indecomposable root system of rank
$n$ together with $c(0)$ copies of the zero vector,
then the product is just the product occurring in the
Macdonald identity of the untwisted affine root system of $V$.
Moreover this product is a holomorphic Jacobi form
of singular weight, so can be written as a finite sum of theta
functions. This turns out to be a sum over the (finite)
Weyl group of theta functions, and this sum can be written as
a sum over the affine Weyl group. Hence we recover the
usual Macdonald identities.
I do not know of any cases other than the Macdonald identities
where the sum of theta functions times modular forms has been
worked out explicitly.
Proof of theorem 6.5.
We start with the two easy transformations. It is obvious that
$$\psi(z,\tau+1)= e^{2\pi i d/24} \psi(z,\tau)$$
because $q^{d/24}$ is the only factor which is changed
by adding 1 to $\tau$.
The only factor of $\psi$ that changes under adding $\mu$ to $z$
is $\zeta^{-\rho}$ which gets multiplied by
$(-1)^{2(\mu,\rho)}$,
so
$$\psi(z+\mu) = (-1)^{2(\mu,\rho)}\psi(z).$$
For the transformation of adding $\lambda\tau$ to $z$ we first assume that
$\lambda$ is in the Weyl chamber and calculate as follows.
$$\eqalign{
\psi(z+\lambda\tau,\tau)
=&q^{d/24} e^{-2\pi i (\rho,z+\lambda\tau)}\prod_{r,n}(1-q^{n+(\lambda,v)}\zeta^v)\cr
=&\psi(z,\tau)e^{-2\pi i (\rho,\lambda)\tau}
\prod_{v\in V,00}(1-\zeta^v).$$
Then $\psi_0(0,\tau)=\eta(\tau)^d$, so that
$$\psi_0(0,-1/\tau)={(\tau/i)}^{d/2}\psi_0(0,\tau).$$
Also
$$\phi(z,\tau)=
{\psi_0(z/\tau,-1/\tau)\prod_{v>0}(1-e^{2\pi i (v,z/\tau)})
\over \psi_0(z,\tau)\prod_{v>0}(1-e^{2\pi i (v,z)})e^{2\pi i z^2/2\tau}}
$$
so if we take the limit as $z$ tends to 0 and use the fact that
$\phi(z,\tau)$ does not depend on $z$ we find that
$$\eqalign{
\phi(z,\tau)&= ({\tau/i})^{d/2}/\prod_{v>0}\tau\cr
&= i^{k-d/2}({\tau/i})^{k}.\cr
}$$
From the definition of $\phi$, this is equivalent to
the final transformation law for
$\psi$. This proves theorem 6.5.
The results in this section can easily be extended to cover the
analogues of twisted affine root systems. We will briefly sketch
how to do
this in the remainder of this section. A pure
affine vector system of level $N$ is defined to be
the multiset of vectors of the form $(v,Nn+(v,\lambda))\in K\oplus\Z$
as $v$ runs through the vectors of some vector system
and $n$ runs through all integers, and $\lambda$ is some fixed vector of
$K'$. We define an affine vector system of level dividing $N$ to be
a union of pure affine vector systems of level dividing $N$.
For each affine vector system of level dividing $N$
we can define a function $\psi(z,\tau)$
as an infinite product over half the vectors in the affine vector
system as above. The function $\psi$ is then a nearly holomorphic
Jacobi form for the congruence
subgroup $\Gamma_0(N)= \{{ab\choose cd}\in SL_2(\Z)|c\equiv 0 \bmod N\}$
of $SL_2(\Z)$ (and can therefore be written
as a sum of theta functions times nearly holomorphic modular functions).
We can prove this in the same way as above: the product
for each pure affine vector system of level $N$ is a Jacobi form for
the conjugate $\{{ab\choose cd}|ad-bc=1, a, nb,c/n, d\in \Z\}$
of $SL_2(\Z)$, so the product for
the union of the pure affine vector systems of level dividing $N$
is a Jacobi form
for the intersection of these conjugates, which contains $\Gamma_0(N)$.
For affine root systems we always have $1\le N\le 4$. \bigskip
{\bf 7. The Weierstrass $\wp$ function.}
\bigskip
In this section we prove some identities involving
the Weierstrass $\wp$ function that we will use in section 9.
The results of this section and section 9 are not used elsewhere in this paper.
We recall that the Weierstrass $\wp$ function is defined
for $\Im(\tau)>0$, $z\in \C$ by
$$\wp(z,\tau) = {1\over z^2}+
\sum_{(m,n)\ne (0,0)} \left({1\over (z-m\tau-n)^{2}}-{1\over(m\tau+n)^{2}}\right)$$
and satisfies the functional equations
$$\eqalign{
\wp(z+\lambda\tau+\mu,\tau)&= \wp(z,\tau)\cr
\wp\left({z\over c\tau+d},{a\tau+b\over c\tau+d}\right)
&= (c\tau+d)^2\wp(z,\tau).\cr
}$$
In other words $\wp$ is a meromorphic Jacobi form of weight 2 and
index 0 and signature 1 (see [E-Z p. 2]).
We also recall the formulas
for the Eisenstein series $E_2$
$$\eqalign{ E_2(\tau)&= {3\over
\pi^2}\sum_m\left(\sum_n'{1\over (m\tau+n)^2}\right)\cr
&=1-24\sum_{n>0}\sigma_1(n)q^n\cr }$$
where $\sum_n'$ means we omit
$n=0$ if $m=0$. This function satisfies the functional
equation $E_2((a\tau+b)/(c\tau+d))=(c\tau+d)^2E_2(\tau)+12c(c\tau+d)/2\pi i $
for ${ab\choose cd}\in SL_2(\Z)$.
The Eisenstein series $E_k(\tau)$ for $k$ even and $k\ge 4$
is equal to $1-(2k/B_k)\sum_{n>0}\sigma_{k-1}(n)q^n$ and
satisfies the functional equation $E_k((a\tau+b)/(c\tau+d))=
(c\tau+d)^kE_k(\tau)$ for ${ab\choose cd}\in SL_2(\Z)$.
By differentiating the partial fraction
decomposition
$$
{1\over z}+\sum_{n\ne 0}\left({1\over z-n}+{1\over n}\right)
=\pi \cot(\pi z)
=-\pi i -2\pi i \sum_{n>0}e^{2\pi i nz}
$$
(valid for $\Im(z)>0$) we find that
$$\sum_{n\in \Z}{1\over (z+n)^2} = (2\pi i )^2 \sum_{n>0}ne^{\pm 2\pi i nz}$$
(valid for $z$ not real), where
$e^{\pm x}$ means $e^x$ if $|e^x|<1$ and $e^{-x}$ if $|e^x|>1$.
From this we see
that
$$\wp(z,\tau)= (2\pi i )^2\sum_{m\in \Z}\sum_{n>0}ne^{\pm2\pi i
n(z+m\tau)}-{\pi^2\over 3}E_2(\tau)$$
whenever $\Im(z+m\tau)$ is nonzero
for all integers $m$.
By differentiating repeatedly with respect to $z$ we find that
for positive even integers $k$
the derivatives $\wp^{(k-2)}(z,\tau)={d\over dz}^{k-2}\wp(z,\tau)$
satisfy
$$\eqalign{
\wp^{(k-2)}(z+\lambda\tau+\mu,\tau)&= \wp^{(k-2)}(z,\tau)\cr
\wp^{(k-2)}\left({z\over c\tau+d},{a\tau+b\over c\tau+d}\right)
&= (c\tau+d)^{k}\wp^{(k-2)}(z,\tau)\cr
\wp^{(k-2)}(z,\tau)&=
(2\pi i )^{k}\sum_{m\in \Z}\sum_{n>0}n^{k-1}e^{\pm2\pi i n(z+m\tau)}
-\delta_k^2E_2(\tau)\pi^2/3\cr
}$$
(where $\delta_m^n$ is 1 if $m=n$ and 0 otherwise).
Suppose that $K$ is a positive definite even lattice of dimension
$s$ and that $c(r)$ is an integer defined for $r\in K$ such that
$c(r)=c(-r)$ and $c(r)=0$ for all but a finite number of $r\in K$.
Choose a vector $\rho$ not orthogonal to any $r$ with $c(r)\ne 0$,
and say that the pair $(r,n)\in K\oplus \Z$
is positive if $n>0$ or $n=0$ and $(r,\rho)>0$.
\proclaim Theorem {7.1}. Suppose that $k$ is an even positive integer
and suppose that if $k=2$ then $\sum_{r\in K}c(r)=0$. Then the function
$$\psi(z,\tau)=
-c(0)B_k/2k+\sum_{(r,n)>0} \sum_{a|(r,n)}a^{k-1}c(r/a)q^n\zeta^r$$
can be extended to a meromorphic function defined for
$\Im(\tau)>0$, $z\in K\otimes \C$ which satisfies the functional equations
$$\eqalign{
\psi\left({z\over c\tau+d},{a\tau+b\over c\tau+d}\right)
&= (c\tau+d)^k\psi(z,\tau)\cr
\psi(z+\lambda\tau+\mu,\tau)&=\psi(z,\tau)\cr
}$$
for $\lambda,\mu\in K'$, ${ab\choose cd}\in SL_2(\Z)$.
In other words, $\psi$ is a meromorphic Jacobi form of weight $k$
and index 0.
Proof. The function $\psi$ is equal to
$$ -c(0){B_k\over 2k}E_k(\tau)
+(2\pi i )^{-k}\sum_{r>0}c(r)\left(\wp^{(k-2)}((r,z),\tau)
+ \delta_k^2E_2(\tau)\pi^2/3\right)$$
so theorem 7.1 follows from the functional equations of the derivatives of
the Weierstrass $\wp$ function and the Eisenstein series $E_k(\tau)$.
(If $k=2$ then the assumption $\sum_{r\in K}c(r)=0$ implies that
the coefficient of $E_2(\tau)$ is 0.)
\bigskip
{\bf 8. Generators for $O_M(\Z)^+$.}
\bigskip
In this section we prove a technical result which says that the
orthogonal group $O_M(\Z)^+$ is generated by a Fourier and a Jacobi subgroup.
We will use this result to construct automorphic forms for
$O_M(\Z)^+$ by showing that the form transforms correctly under
both the Fourier and the Jacobi groups, and hence under the whole of
$O_M(\Z)^+$.
We let $M$ be the lattice $II_{s+2,2}$ (with $8|s$).
We let $u$ be a primitive norm 0 vector of $M$ and let $U$
be a 2-dimensional primitive null sublattice containing $u$.
We let $F$ and $J$ be the Fourier and Jacobi groups of $u$ and $U$.
The main theorem of this section is
\proclaim Theorem {8.1}. $O_M(\Z)^+$ is generated by $F(\Z)^+$ and
$J(\Z)^+$.
We first prove two lemmas.
\proclaim Lemma {8.2}. If $L=II_{s+1,1}$, $v\in L\otimes \R$ and
$v\notin L$ then there is some $\lambda\in L$ with
$1/2\le (v-\lambda)^2\le 3/2$.
Proof. Choose a primitive norm 0 vector $u_1\in L$ with
$(u_1,v)$ not an integer, which we can do because the norm 0
vectors of $L$ generate $L$. Choose another norm 0 vector
$u_2\in L$ with $(u_1,u_2)=-1$. We can find an integer
$m$ so that $0<|(v-mu_2,u_1)|\le 1/2$. But then
$(v-mu_2-nu_1)^2=A+Bn$ for some fixed $A$ and $B$ with
$0<|B|\le 1$, so we can choose $n$ so that $1/2\le A+nB\le 3/2$.
This proves lemma 8.2.
\proclaim Lemma {8.3}. If $u$ is a primitive norm 0 vector in $M$ and $G$ is the group
generated by the reflection of norm 2 vectors $r$ with $(r,u)=-1$ then
$G$ acts transitively on primitive norm 0 vectors of $M$.
Proof. Suppose $u_1$ is a primitive norm 0 vector of $M$.
If $(u_1,u)\ge 0$ then it is easy to find an element $g\in G$
such that $(g(u_1),g)<0$, so we can assume that $(u_1,u)<0$.
We will prove lemma 8.3 by induction on $-(u_1,u)$.
If $-(u_1,u)=1$ then the reflection of the root
$u_1-u$ maps $u_1$ to $u$. If $-(u_1,u)>1$ then
we choose coordinates $M=L\oplus II_{1,1}$ for $M$
so that $u=(0,0,1)$. If $u_1=(v,m,n) $ then $v/m$ is not in $L$,
otherwise $m$ would divide $v$ and hence $n$ as $(v,v)=2mn$,
which is impossible as $u$ is primitive and $m=-(u_1,u)>1$.
Therefore by lemma 8.2 we can
choose a vector $\lambda\in L$ with $1/2\le(\lambda-v/m)^2\le 3/2$.
Then a simple calculation shows that if $g$ is the reflection of
the vector $(\lambda,1,\lambda^2/2-1)$ then
$00$
to meromorphic automorphic forms $\Phi$ of weight $k$ for $O_{M}(\Z)^+$
all of whose singularities are poles of order $k$
along rational quadratic divisors.
The Hecke operator $V_0$ is defined by
$$(\phi|V_0)(z,\tau) = -c(0,0)B_{k}/2k+\sum_{(r,n)>0}
\sum_{d|(r,n)} d^{k-1}c(r,0)q^n\zeta^r)$$
which is equal to $(-c(0,0)B_k/2k)E_k(\tau)$ if $\phi$ is
holomorphic.
The map from $\phi$ to $\Phi$
is an isomorphism from the space of holomorphic Jacobi forms $\phi$ of level
1 and weight $k$ to the ``Spezialschar'' of all holomorphic automorphic forms
$\Phi= \sum_{r,m,n}A(r,m,n)\zeta^rp^mq^n$
of weight $k$ whose coefficients satisfy
$$A(r,m,n) = \sum_{d|(r,m,n)}d^{k-1} A(r/d, 1, mn/d^2).$$
Proof. When $\phi$ is holomorphic this is a straightforward
generalization of [E-Z, theorem 6.2] due to Gritsenko [G]
and the proof in [E-Z] works with
minor changes. (See also [M, paper I].) The first coefficient of the
Fourier-Jacobi expansion of any automorphic form is a Jacobi form of
level 1, and the coefficients of any form in the Spezialschar are
obviously determined by those of the first Fourier-Jacobi coefficient,
so the main thing to check is that $\Phi$ is an automorphic form. It
transforms like an automorphic form under the Jacobi group because all
its coefficients do. It is also invariant under the automorphism of
$L$ taking $(\lambda,m,n)$ to $(\lambda,n,m)$ because the formula for
$A(\lambda,m,n)$ is symmetric in $m$ and $n$. However this
automorphism together with the Jacobi group generates the whole of
$O_M(\Z)^+$ by theorem 8.1, so $\sum_{m\ge 0} p^m(\phi|V_m)(z,\tau)$ is an
automorphic form.
When $\phi$ has poles at the cusps the proof is similar except that we
need the following extra arguments. Firstly, the series for $\Phi$
does not converge everywhere, so we need to use theorem 5.2 to
show that the series for $\Phi$ can be analytically continued. Secondly,
$\phi|V_0$ is no longer a modular form, so
we need to use theorem 7.1 to show that $\Phi|V_0$ is a nearly holomorphic
Jacobi form. If $k=2$ then the condition $\sum_{r\in K}c(r)=0$ is
satisfied by lemma 9.2, because it is the constant coefficient
of the nearly holomorphic weight 2 form $\phi(0,\tau)$. This proves theorem
9.3.
Example 1. (Gritsenko [G].) If we let $f$ be the constant form 1 of weight 0,
we find a singular automorphic form for $O_M(\Z)^+$ of weight
$s/2$ whose
Fourier coefficients $A(\lambda, m,n)$ are given by
$A(\lambda,m,n)= \sigma_{s/2-1}(d)$ if $\lambda^2=2mn$ and
the highest common factor of $\lambda$, $m$, and $n$ is $d$
(where $\sigma_{s/2-1}(0)$ is defined to be $-B_{s/2}/s$).
In particular singular automorphic forms with nonzero constant terms
exist for all the groups $O_{II_{8n+2,2}}(\Z)^+$.
When $\phi$ is not holomorphic the meromorphic automorphic form $\Phi$
will have poles along rational quadratic divisors. We can remove these
poles by multiplying $\Phi$ by some of the functions produced in
section 10, where we construct holomorphic automorphic forms with
zeros along any given rational quadratic divisor.
Example 2. Let $\Phi$ be the meromorphic automorphic form
for $O_{II_{26,2}}(\Z)^+$ constructed from the
nearly holomorphic modular form $j'$ as in theorem 9.3. Then
$\Phi$ has weight 14 and
the singularities of $\Phi$ are poles of order 2
along all rational quadratic divisors of discriminant 2.
In section 10 we will find a holomorphic automorphic form
$\Phi_1$ which has weight 12 and has a zero of order 1 along every
rational quadratic divisor of discriminant 2. Hence
$\Phi\Phi_1^2$ is a holomorphic automorphic form of weight
38 with the same zeros as $\Phi$. \bigskip
{\bf 10. The zero weight case}
\bigskip
This section is the heart of the paper where we put everything
together to construct some automorphic forms on $O_M(\Z)^+$ as
infinite products. We let $L$ be the even unimodular Lorentzian
lattice $II_{s+1,1}$, and we let $M=L\oplus II_{1,1}$. We choose a
negative norm vector in $L\otimes \R$ and write $r>0$ to mean that $r$
has positive inner product with this negative norm vector. We will
prove
\proclaim Theorem {10.1}. Suppose that $f(\tau)=\sum_nc(n)q^n$
is a nearly holomorphic modular form
of weight $-s/2$ for $SL_2(\Z)$ with integer coefficients,
with $24|c(0)$ if $s=0$.
There is a unique vector $\rho\in L$
such that
$$\Phi(v)=e^{-2\pi i (\rho, v)}\prod_{r>0}(1-e^{-2\pi i (r,v)})^{c(-(r,r)/2)}$$
is a meromorphic automorphic form of weight $c(0)/2$ for
$O_M(\Z)^+$.
All the zeros and poles of $\Phi$ lie on rational quadratic divisors,
and the multiplicity of the zero of $\Phi$ at the rational quadratic
divisor of the primitive positive norm vector $r\in M$ (see section 5) is
$$\sum_{n>0}c(-n^2(r,r)/2).$$
In particular if this number is always nonnegative then
$\Phi$ is holomorphic.
Proof. We write $L=K\oplus II_{1,1}$ where $K$ is the lattice $E_8^{s/8}$.
We let $\phi(\tau,z) $ be the nearly holomorphic Jacobi form
$f(\tau)\theta_K(z,\tau)$.
We define a vector system on $E_8^{s/8}$ to be the multiset of vectors
$v\in K$ with multiplicities
$c(v)=c(-(v,v)/2)$.
This is a vector system by lemma 6.1, as
$O_K(\Z)$ acts irreducibly on $K$ when $K$ is $E_8^{s/8}$
so the set of vectors of any fixed norm is a vector system.
We define the corresponding affine vector system $V$
to be the multiset of vectors $(v,n)\in K\oplus\Z$ with multiplicities
$c((v,n))=c(-(v,v)/2)$.
By theorem 6.5 the function
$\psi(z,\tau)$ associated to $V$ satisfies the following functional equations
$$\eqalign{
\psi(z,\tau+1) &=\psi(z,\tau)\cr
\psi(z/\tau,-1/\tau)
&= \tau^k e^{2\pi i m (z,z)/2\tau}\psi(z,\tau)\cr
\psi(z+\mu,\tau)&=\psi(z,\tau)\cr
\psi(z+\lambda\tau,\tau)
&= q^{-m(\lambda,\lambda)/2}\zeta^{-m\lambda} \psi(z,\tau)\cr
}$$
for any $\lambda,\mu \in K$. (This follows because
the integer $k=s/2$ is divisible by 4 and $d$ is divisible by 24,
and $\rho\in K$ by the remark after lemma 6.4.)
In particular $\psi$ is a Jacobi form of
weight $k$ and index $m$ for $J(\Z)^+$, and therefore
$$p^m\psi(z,\tau)$$
transforms like an automorphic form of weight $k$ under all elements
of $J(\Z)^+$.
On the other hand $\phi|V_\l$ is a Jacobi form of weight 0 and index
$\l$, so that
$$\exp(\sum_{\l>0}p^\l(\phi|V_\l)(z,\tau))$$
transforms like an automorphic form of weight $0$ for
all elements in the Jacobi group $J(\Z)^+$
whenever the product converges.
If we multiply these two expressions together and use theorem 4.2 we find that
$$\Phi(z,\tau,\sigma)=p^m\psi(z,\tau)\exp(\sum_{\l>0}p^\l(\phi|V_\l)(z,\tau))$$
transforms like an automorphic form of weight $k=c(0)/2$ for
all elements in the Jacobi group $J(\Z)^+$
whenever the product converges.
By theorem 5.1 $\Phi$ can be analytically
continued as a nonzero multivalued function
to all vectors with imaginary parts in the positive cone,
except for some singularities or zeros along rational quadratic divisors.
Next we check that $\Phi$ is invariant or antiinvariant under the
Fourier group $F(\Z)^+$. It is obviously invariant under the unipotent
radical of $F(\Z)^+$ (which is isomorphic to $L$), so we have to check
invariance under the group $O_L(\Z)^+$, which we will do by
considering the Fourier expansion of $\Phi$. We first check that it
is invariant under the element $g_1$ taking $(z, \alpha, -\delta,
\gamma, \beta)$ to $(z, -\delta,\alpha, \gamma, \beta)$. Under this
transformation the factor $e^{-2\pi i (\rho, v)}$ of $\psi$ is multiplied by
$$e^{2\pi i (\rho-g_1(\rho),v)},$$ and the factor $\prod(1-e^{-2\pi i
(r,v)})^{c(-(r,r)/2)}$ of $\psi$ is multiplied by a factor of
$\prod_{r>0,g_1(r)<0}-e^{2\pi i c(-(r,r)/2)(r,v)}$.
Hence to prove
(anti)invariance under $g_1$ we have to show that
$$\rho-g_1(\rho)-\sum_{r>0,g_1(r)<0}c(-(r,r)/2)r=0. $$
Before proceeding further with the proof of invariance of $\Phi$ under
$g_1$ we need to calculate the Weyl vector $\rho$ and check some of its
properties, which we do in 10.2 to 10.7.
\proclaim Lemma {10.2}. If $f(\tau),g(\tau)$ are nearly holomorphic modular
functions for $SL_2(\Z)$ (possibly transforming according to some
nontrivial character of $SL_2(\Z)$) then the constant term of
the $q$ expansion of $f(\tau)g'(\tau)$
vanishes.
Proof. The $SL_2(\Z)$ invariant differential form
$f(\tau)g'(\tau)d\tau$ has only one pole on the compactification
of the upper half plane modulo $SL_2(\Z)$ (which is at the cusp
$i\infty$) and therefore its residue there must vanish. But its
residue
is just the constant term of $f(\tau)g'(\tau)$. This proves lemma 10.2.
\proclaim Lemma {10.3}. Suppose that $\theta(\tau)$ is a nearly
holomorphic modular
form of weight $s/2$ and $f(\tau)$ is a nearly holomorphic modular
form of weight $-s/2$ (both of level 1). Then the constant term of
the $q$ expansion of
$$s\theta(\tau)f(\tau)E_2(\tau)/24-\theta'(\tau)f(\tau)$$
is zero.
Proof.
This follows by applying lemma 10.2 to the modular functions
$f(\tau)\eta(\tau)^s$ and $\theta(\tau)\eta(\tau)^{-s}$,
since $\eta'(\tau)/\eta(\tau) =E_2(\tau)/24$. (Alternatively
we can observe that $\theta'(\tau)-s\theta(\tau)E_2(\tau)/24$
is a nearly holomorphic
modular form of weight $s/2+2$, so the expression in lemma 10.3
is a nearly holomorphic
modular form of weight 2 with only one pole, whose residue must be
0.) This proves theorem 10.3.
%We can suppose that $f$ is of the form $E_4^a\Delta^b$
%for some integers $a$ and $b$ with $a\ge 0$, $8a+24b=-s$.
%Then the constant term of the function in lemma 10.3 is equal to the constant term of
%$$\eqalign{
%&\theta(\tau)f(\tau)E_2(\tau)/24+\theta(\tau)f'(\tau)/s\cr
%=&\theta(\tau)f(\tau)(\Delta'(\tau)/24\Delta(\tau)
%+f'(\tau)/sf(\tau))\cr
%=&(1/24s)\theta(\tau)f(\tau)((-8a-24b)\Delta'(\tau)/\Delta(\tau)
%+24aE_4'(\tau)/E_4(\tau)+24b\Delta'(\tau)/\Delta(\tau))\cr
%=&(1/24s)\theta(\tau)f(\tau)8aj'(\tau)/j(\tau).\cr
%}$$
%
%If $a=0$ then lemma 10.3 follows immediately. If $a>0$ then
%$f(\tau) $ vanishes at the zeros of $j(\tau)$. This implies that
%$\theta(\tau)f(\tau)$ is a nearly holomorphic modular function
%vanishing at the zeros of $j$, so that $\theta(\tau)f(\tau)/j(\tau)$
%is a nearly holomorphic modular function
%of level 1, and lemma 10.3 now follows from lemma 10.2.
\proclaim Theorem {10.4}. The Weyl vector $\rho$ is equal to
$$\left(\sum_{(r,v)>0}c(-r^2/2)r/2,\quad m,\quad d/24\right)$$ where $m$ is the constant
coefficient of $\theta_K(\tau)f(\tau)E_2(\tau)/24$, and $d$ is the
constant coefficient of $\theta_K(\tau)f(\tau)$.
Proof. The Weyl vector is $(\rho_K, m,d/24)$ where $\rho_K,m$, and $d$ are
as in section 6. The formulas for $\rho_K$ and $d$ then follow
immediately from the definitions in section 6. The integer $m$ in section 6 is
equal to the constant term of $\theta'_K(\tau)f(\tau)/s$,
so we have to show that the constant term of
$$\theta'_K(\tau)f(\tau)/s-\theta_K(\tau)f(\tau)E_2(\tau)/24$$
vanishes. But this follows from lemma 10.3.
This proves theorem 10.4.
Now we check that the Weyl vector $\rho$ lies in $L$ (and not just
$L\otimes \Q$).
\proclaim Lemma {10.5}. For any nonzero integer $n$ the constant term of
$\Delta(\tau)^n$ is divisible by 24.
Proof. $\Delta'(\tau)/\Delta(\tau)=1-24\sum_{m>0}\sigma_1(m)q^m$ is
congruent to 1 mod 24,
so $$(\Delta(\tau)^n)'=n\Delta(\tau)^{n-1}\Delta'(\tau)\equiv
n\Delta(\tau)^n\bmod 24n.$$
As the left hand side has zero constant coefficient, so does the right
hand side mod $24n$, which proves lemma 10.5 as $n\ne 0$.
\proclaim Lemma {10.6}. If $f$ is a nearly holomorphic modular form
of level 1 and negative weight then the constant term of $f$ is divisible
by 24.
Proof. We can write $f$ as an integral linear combination of functions
of the form $\Delta^mE_4^n$ with $m<0$, and lemma 10.6 then follows
from lemma 10.5 and the fact that $E_4\equiv 1 \bmod 24$.
\proclaim Corollary {10.7}. The Weyl vector $\rho_W$ lies in $L$.
Proof. Suppose that $K$ is the lattice $E_8^{3n}$.
We have to check that $m$ and $d/24$ are integers
and that $\rho_K=\sum_{(r,v)>0}c(-r^2/2)r/2$ lies in $K$.
We know that $\rho_K\in K$ by the remark after lemma 6.4.
The constant
term $d$ of $f(\tau)\theta_K(\tau)$ is divisible by 24 by lemma 10.6
and the fact that $\theta_K(\tau)= E_4(\tau)^{s/8}\equiv 1 \bmod 24$,
so $d/24$ is integral.
Also, $E_2\equiv 1\bmod 24$, so $m\equiv d/24\bmod 1$ is
an integer.
This proves corollary 10.7.
We now continue with the proof that $\Phi$ is invariant under $g_1$,
which was interrupted by the calculation of $\rho$.
We know that $\rho=(\rho_K, m,d/24)$ by theorem 10.4, so that
$g_1(\rho)=(\rho_K,d/24,m)$. The vector $(\kappa, a,b)$ is positive
if $a>0$, or $a=0, b>0$, or $a=b=0,
\kappa>0$, so that $(\kappa,a,b)>0$ and
$g_1(\kappa,a,b)=(\kappa,b,a)<0$ if and only if $a>0,b<0$.
Hence
$$\eqalign{
\sum_{r>0,g_1(r)<0}c(-(r,r)/2)r
&=\sum_\kappa\sum_{a>0}\sum_{b<0}c(ab-(\kappa,\kappa)/2)(\kappa,a,b)\cr
&=\sum_\kappa\sum_{n>0}\sum_{a|n}c(-n-(\kappa,\kappa)/2)(0,a,-a)\cr
&=\sum_{\kappa}\sum_{n>0}c(-n-(\kappa,\kappa)/2)(0,\sigma_1(n),-\sigma_1(n))\cr
&=(0,x,-x)\cr
}$$
where $x$ is the constant term of
$\theta_K(\tau)f(\tau)\sum_{n>0}\sigma_1(n)q^n$. By theorem 10.4
$x=m-d/24$ (as $E_2(\tau)=1-24\sum_{n>0}\sigma_1(n)q^n$). By comparing this with the expression for $\rho$ we see that we
have proved that
$$\rho-g_1(\rho)-\sum_{r>0,g_1(r)<0}c(-(r,r)/2)r=0. $$
This completes the proof that $\Phi$ is (anti)invariant under $g_1$.
Next we see that it transforms like an automorphic form under the
element $g_2$ taking $(z, \alpha, -\delta, \gamma, \beta)$ to $(z,
\alpha, -\delta, \beta, \gamma)$, because this is conjugate to $g_1$
under an element of the Jacobi group. This element $g_2$ acts as
$v\rightarrow 2v/(v,v)$ on $L\otimes \C$, and in particular commutes
with of $O_L(\Z)^+$. If $g_3$ is any element of $O_L(\Z)^+$ then
$\Phi(g_3(v))=e^{2\pi i (\lambda,v)}\Phi(v)$ for some $\lambda$
depending on $g_3$ because of the expression for $\Phi$ as an infinite
product.
We now see that $$\eqalign { &((v,v)/2)^k e^{2\pi i
(\lambda, v)}\Phi(v)\cr =&((g_3(v),g_3(v))/2)^k \Phi(g_3(v))\cr
=&\Phi(g_2(g_3(v)))\cr =&\Phi(g_3(g_2(v)))\cr =&e^{2\pi i (\lambda,
g_2(v))}\Phi(g_2(v))\cr =&e^{2\pi i
(\lambda,2v/(v,v))}((v,v)/2)^k\Phi(v)\cr }$$ so that $$e^{2\pi i
(\lambda,v)} =e^{2\pi i (\lambda, 2v/(v,v))}$$ for all $v$, which
implies that $\lambda=0$ and hence that $\Phi$ is invariant or
antiinvariant under $O_L(\Z)^+$.
We have shown that $\Phi$ transforms correctly under
both $F(\Z)^+$ and $J(\Z)^+$, so by theorem 8.1 $\Phi$ transforms
like an automorphic form under the whole of $O_M(\Z)^+$.
Next we have to find the singularities and zeros of $\Phi$,
which we know lie on rational quadratic divisors by theorem 5.1.
If $v\in M$ is a primitive positive norm vector corresponding
to a rational quadratic divisor, then there is some
primitive norm zero vector $u$ of $U$ orthogonal to $v$,
because $U$ is 2-dimensional. As $J(\Z)^+$ acts transitively on
such vectors $u$ we can assume that $u$ is the standard
choice $(0,0,0,0,1)$. This rational quadratic divisor
is then just the linear divisor of points
orthogonal to some positive norm vector of $L$. But such a divisor
intersects the region where the infinite product for $\Phi$
converges, (except where one of the factors is zero or singular)
so the only singularities or zeros along such a divisor must
be where one of the factors in the infinite product for
$f$ has a zero or singularity.
But if $r$ is a primitive positive norm vector of
$L$, then the order of the zero of $\Phi$ along the divisor of $r$
is just $$\sum_{n>0} c(-(nr,nr)/2),$$
coming from the factors
$$\prod_{n>0}(1-e^{-2\pi i(nr,v)})^{c(-(nr,nr)/2)}$$
in the infinite product for $\Phi$.
This completes the proof of theorem 10.1.
Example 1. If we take $L$ to be $II_{1,1} $ and $f$ to
be $j(\tau)-744$ we recover the denominator formula
for the monster Lie algebra (which can of course be proved easily
without using theorem 10.1).
Example 2. Suppose we take $L$ to be $II_{25,1}$ and $f$ to be
$1/\Delta(\tau)$. Then we find that
$\Phi$ is an antiinvariant automorphic form of weight 12
for $O_{II_{26,2}}(\Z)^+$ whose zeros are the rational quadratic divisors
corresponding to vectors of norm 2. Any antiinvariant automorphic form
must vanish at these zeros, and so must be divisible by $\Phi$.
By the Koecher boundedness principle the quotient is an invariant
automorphic form. Hence multiplication by $\Phi$ is an isomorphism
from invariant automorphic forms of weight $k$ to antiinvariant automorphic forms of weight $k+12$. In particular any antiinvariant automorphic form of
weight less that 24 must be a multiple of $\Phi$ because
the only invariant forms of weight less than 12 are constant. (From theorem
9.3 we know that there is an invariant form of weight 12, so there is
a nontrivial antiinvariant form of weight 24.) The form $\Phi$
is also the denominator function of the fake monster Lie algebra.
As it has singular weight, all its nonzero Fourier coefficients correspond
to vectors of norm 0. The multiplicities of norm 0 vectors
are always easy to work out explicitly, so we find that
$$\Phi(v)
=e^{-2\pi i (\rho,v)}\prod_{r>0}(1-e^{-2\pi i (r,v)})^{p_{24}(1-r^2/2)}
=\sum_{w\in W}\sum_{n>0}\det(w)\tau(n) e^{-2\pi i n(w(\rho),v)},$$
which gives a new proof of the denominator formula of the fake monster Lie
algebra.
This is the only case when theorem 10.1 produces a holomorphic
automorphic form of singular weight. When the weight is not singular,
the Fourier coefficients are much harder to describe explicitly. \bigskip
{\bf 11. The negative weight case.}
\bigskip
In sections 9 and 10 we have shown how to construct holomorphic
automorphic forms from Jacobi forms of positive or zero weight. In this section
we show that Jacobi forms of negative weight do not seem to
give new examples of automorphic forms, at least not in any obvious way.
Suppose that $f=\sum c(n)q^n$
is a nearly holomorphic modular form of weight $k<-s/2$.
We can try to apply the construction of sections 9 or 10 to $f$ to produce
some function $\Phi$ which might be similar to an automorphic form.
The first problem with this function is that it has polylogarithm
singularities. We can turn these into poles by applying a high power
of the Laplace operator, and then we get a meromorphic
automorphic function. Unfortunately we will see in this section that
this meromorphic automorphic is not new; it is the function
associated to the modular form $(d/d\tau)^{1-k}f(\tau)$.
We would expect the Fourier coefficients $A(v)$ ($v\in L$)
of $\Phi$ to look something
like
$$A(v)=\sum_{d|v}d^{k-1}c(-(v,v)/2d^2).$$
(We will not worry about what the coefficient of $0$ is or whether this
does somehow define an automorphic function, since the point of this section
is that even if these problems can be solved we still do not seem to
get new automorphic forms.)
If we apply the $(1-k)$'th power of the Laplacian to this we get
a function whose Fourier coefficients are
$$\sum_{d|v}d^{k-1}(-(v,v)/2)^{1-k}c(-(v,v)/2d^2).$$
On the other hand, if we apply the operator ${d\over d\tau}^{1-k}$
to $f$ we get a nearly holomorphic modular form of positive weight
$2-k$ with Fourier coefficients $n^{1-k}c(n)$, because the
$(1-k)$'th derivative of a meromorphic modular form of weight $k\le 0$
is a meromorphic modular form of weight $2-k$. The automorphic
form associated to ${d\over d\tau}^{1-k}f$ in theorem 9.3 has Fourier coefficients
$$\sum_{d|v}d^{(2-k)-1}(-(v,v)/2d^2)^{1-k}c(-(v,v)/2d^2).$$
These are equal to the Fourier coefficients above. So the good news is
that if we apply the $(1-k)$'th power of the Laplacian to $\Phi$
we do seem to get a meromorphic automorphic form (assuming we can
define $\Phi$), but the bad news is that this is not a new
automorphic form.
We can of course still apply
powers of the Laplacian to $\Phi$ if $f$ has weight $\ge -s/2$,
and this gives a few examples where some power of the Laplacian
applied to some meromorphic automorphic form is a meromorphic automorphic form.
It may be possible to apply some of the Laplacian to
$\Phi$ to produce a function with logarithmic singularities
and then exponentiate this to get a function which can be written
as an infinite product. One problem with this is that
if we apply an arbitrary power of the Laplacian to
something that transforms like an automorphic form, the result
usually does not transform like an automorphic form. \bigskip
{\bf 12. Invariant modular products}
\bigskip
In this section we will define Weyl vectors and Weyl chambers of
modular products. These are sometimes the Weyl vectors and Weyl
chambers of hyperbolic reflection groups, and even when they are not
they still have many of the properties of hyperbolic reflection
groups. Conversely, we can often find automorphic forms associated to
hyperbolic reflection groups which have the same Weyl vectors and Weyl
chambers. We will also give some applications to even unimodular lattices.
Suppose that
$$\Phi(y)=e^{-2\pi i (\rho,y)}\prod_{x>0}(1-e^{-2\pi i(x,y)})^{c(x)}$$
is a modular product for some Lorentzian lattice
$L$ which defines a holomorphic automorphic form. In particular
$\Phi$ is invariant up to sign
for some finite index subgroup
$G$ of $O_L(\Z)^+$.
The hyperplanes orthogonal to the positive norm vectors $x$ with $c(x)\ne 0$
divide up the cone $C$ into closed chambers that we will call
the Weyl chambers of $\Phi$. If $W$ is a Weyl chamber of $\Phi$,
we define the Weyl vector $\rho_W$ of $W$ by
$$\Phi(y)=\pm e^{-2\pi i (\rho_W,y)}\prod_{(x,-W)>0}(1-e^{-2\pi i(x,y)})^{c(x)}$$
(where $(x,-W)>0$ means that $(x,w)>0$ for any $w$ in the interior of $-W$).
We list some of the properties of Weyl vectors of $\Phi$.
\item{1} If $g\in G$ then $\rho_{g(W)}=g(\rho_W)$.
\item{2} If $W_1$ and $W_2$ are Weyl chambers then
$$\rho_{W_1}= \rho_{W_2}+\sum_{(x,-W_1)<0,(x,-W_2)>0}c(x)x.$$
\item{3} Any Weyl vector has coefficient
$\pm 1$ in the Fourier expansion of $\Phi$.
In particular any Weyl vector lies in the closure of $C$ and
has norm at most 0, because this is true of any vector corresponding
to a nonzero coefficient of a holomorphic automorphic form.
\item{4} Any Weyl vector of maximal norm with Weyl chamber
$W$ has positive inner product with
all the positive real roots of $W$. In particular $\rho_W$ lies
in the interior of $W$ if $\rho_W$ has negative norm, and on the
boundary if it has zero norm. In either case, $W$ is the only Weyl
chamber containing $\rho_W$. (If $\rho_W$ does not have maximal norm
I do not know whether or not it necessarily lies in $W$.)
\item{5} If $W_1$ and $W_2$ are 2 adjacent chambers
separated only by the hyperplane $x^\perp$ with $(x,W_1)>0$, and $x$ is a root
of $G$, then $(\rho_{W_1},x) =\sum_{r\in \Q,r>0}c(rx)(rx,rx)/2$.
This follows by applying property 1 with $g$ equal to reflection
in $x^\perp$. In particular if $c(rx)$ is 1 when $r=1$ and 0 otherwise
then $(\rho_W,x)=-(x,x)/2$, so that $\rho_W$ does indeed behave like
a Weyl vector with respect to the simple root $x$.
\item{6} Any two Weyl vectors differ by a vector of $L$.
Weyl chambers behave differently depending on whether the Weyl vector $W$
has negative or zero norm. When its norm is negative, the Weyl chamber
has only a finite number of sides, a finite automorphism group,
and its image in hyperbolic space has finite volume. (The finite
volume property follows because $W/{\rm Aut}(W)$ is a subset of
a fundamental domain of $G$ which has finite volume.) When the Weyl
vector
has zero norm, the Weyl chamber may have an infinite number of sides,
an infinite automorphism group, and infinite volume. The automorphism group
then has a free abelian subgroup of finite index, and
the quotient of the Weyl chamber by this free abelian subgroup
has finite volume. It is quite
common for both of these cases to occur for the same function $\Phi$.
In any case we obtain a canonical decomposition of
hyperbolic space into Weyl chambers each of which
has finite volume modulo the action of a free abelian subgroup.
To avoid confusion we will also list a few properties that Weyl
vectors
and Weyl chambers do not always have. Weyl chambers are not always
acted on transitively by some group. Weyl chambers may have quite different
shapes, and their Weyl vectors may have different lengths. Reflection
in the hyperplane separating two Weyl chambers is
not always an automorphism of $L$.
We will apply these considerations about Weyl vectors to
the Lorentzian lattice $II_{s+1,1}$. In the special case $s=24$,
our results immediately imply
several well known results about Niemeier lattices (for example,
the existence and uniqueness of the Leech lattice, Conway's result
that the Leech lattice is the Dynkin diagram of the reflection group
of $II_{25,1}$, and the fact that the number of roots of a Niemeier
lattice is divisible by 24).
We let $\Phi(y)$ be the automorphic form
$$\Phi(y)=e^{-2\pi i (\rho_W,y)}\prod_{x>0}(1-e^{-2\pi i (x,y)})^{c(x)}$$
where $c(x) $ is the coefficient of $q^{-(x,x)/2}$ in
some nearly holomorphic modular form $f(\tau)=\sum_nc(n)q^n$ of level 1 and
weight $-s$.
We let $K$ be the even
$s$-dimensional lattice corresponding to $u$, and we identify
$II_{s+1,1}$ with $K\oplus II_{1,1}$, so that vectors of
$II_{s+1,1}$ can be written in the form $(v,m,n)$ with
$v\in K$, $m,n\in \Z$, and $(v,m,n)^2=v^2-2mn$. We choose the vector
$u$ to be $(0,0,1)$. We choose a vector $v\in K$ not orthogonal
to any vectors of $K$ of small norm, and we let $W$ be the Weyl
chamber of $II_{s+1,1}$ containing $u$ and $(v,0,0)$.
\proclaim Theorem {12.1}. If $24|s$ and $s>0$ then the constant term of
$\theta_K(\tau)/\Delta(\tau)^{s/24}$
is divisible by 24.
Proof. We take $f$ to be $\Delta^{-s/24}$.
By theorem 10.4 the constant term of
$\theta_K(\tau)/\Delta(\tau)^{s/24}$
is equal to $d=24(\rho,u_1)$ where $\rho$ is the Weyl vector of
the Weyl chamber containing $u$ and $u_1=(0,-1,0)$.
By corollary 10.7, $\rho\in L$ so the inner product $(\rho,u)$
is an integer, and this proves theorem 12.1.
For example, when $s=24$ and
$f(\tau)=\Delta(\tau)^{-1}=q^{-1}+24+\ldots$, this theorem is just the
well known fact that the number of norm 2 vectors of any Niemeier
lattice $K$ is divisible by 24.
The automorphic form $\Phi(y)$ is a cusp form if and only if
there are no Weyl vectors of zero norm, which is true unless
$f(\tau)$ has weight $-s/2$ and there is an extremal lattice
of dimension $s$. (An extremal even unimodular lattice in dimension
$s$ is one with no nonzero vectors of norm at most $2[s/24]$.)
According to [C-S, Chapter 7, section 7], there is
exactly one extremal
lattice
in 24 dimensions (the Leech lattice), at least 2 in dimension 48,
and none in dimensions larger than about 41000.
For Niemeier lattices it is well known that there are either no
roots (the Leech lattice) or the roots span the vector space of the
lattice (any other Niemeier lattice). For
even unimodular lattices $K$ of dimension divisible by 24
this has the following
generalization:
\proclaim Theorem 12.2. An even unimodular lattice $K$
of dimension $s$ divisible by 24 is either extremal (no nonzero vectors
of norm at most $s/12$), or the
vectors of norm at most $s/12$ span the vector space $K\otimes \R$.
Proof. Consider the Weyl chamber $W$ containing a norm 0
vector $u$ corresponding to $K$. If $K$ is not extremal, then
$u$ is not a Weyl vector, so the intersection of the Weyl chamber with
a small neighborhood of $u$ has finite volume. This implies that
the Weyl chamber has a cusp at $u$, which implies that the vectors of
norm at most $s/12$ in $K$ span $K$ as a vector space.
We can also use the ideas of this section to give an amusing proof
of the existence and uniqueness of the Leech lattice (i.e.,
a 24 dimensional even unimodular lattice with no norm 2 vectors).
We do this by considering
the automorphic function $\Phi$ for the group $O_{L}(\Z)^+$
in theorem 10.1 for $L=II_{25,1}$ and $f(\tau)=1/\Delta(\tau)=q^{-1}+24
+324q+\ldots$. This form has weight $24/2=12$ which is singular,
so all its nonzero Fourier coefficients correspond to norm 0
vectors of $L$. In particular the Weyl vector $\rho$ has norm zero,
so by the remarks above it corresponds to an extremal lattice,
i.e. a 24-dimensional unimodular lattice with no roots.
This proves the existence of the Leech lattice. To prove uniqueness,
we observe that all faces of any Weyl chamber are orthogonal to norm 2
vectors which are roots, so $O_L(\Z)^+$ acts transitively on the Weyl chambers.
Any norm 0 vector of an extremal lattice is the Weyl vector of
some Weyl chamber, so there can be only one orbit of such norm
zero vectors, so the Leech lattice is unique.
This also proves Conway's result [C-S, chapter 27]
that the Leech lattice is essentially the Dynkin diagram of
$II_{25,1}$, which in turn easily implies that the Leech lattice has covering
radius $\sqrt 2$. \bigskip
{\bf 13. Heights of vectors.}
\bigskip
Suppose that we have fixed an automorphic form $\Phi$ for $O_M(\Z)^+$
which is a modular product,
which defines a system of Weyl chambers and Weyl vectors as in section 12.
We define
the {\bf height} of a vector $\lambda$ in the positive cone of
$L=II_{s+1,1}$ to be the inner product $-(\rho,\lambda)$ where $\rho$ is
the Weyl vector of any Weyl chamber containing $\lambda$. The height
is a continuous positive function on the positive cone which is linear
in the interior of any Weyl chamber, and can be extended to norm 0
vectors of $L$. We have already found a formula for the height of
$v$ when $v$ has norm 0 in theorem 10.4.
In this section we will find formulas for the height
when $\lambda$ has norm $-2$ or $-2p$ for $p$ a prime. We do this by
looking at the restriction of $f$ to multiples $\tau\lambda$ of
$\lambda$. This is a modular form in $\tau$ for the group
$\Gamma_0(p)=\{{ab\choose cd}\in SL_2(\Z)|c\equiv 0\bmod p\}$ whose zeros in the upper half plane are known
explicitly and whose zero at the cusps has an order related to the
height of $\lambda$.
We let $v$ be a vector of norm $-2N$, where for the moment $N$ is
any positive integer. We write $c(n)$ for the coefficients
of $f(\tau)=\sum_nc(n)q^n$, where $f$ is the nearly holomorphic
modular form of weight $-s/2$ from which $\Phi$ is constructed
as in section 10.
We define an isomorphism from $(L\oplus II_{1,1})\otimes \R$ to
$v^\perp\otimes \R\oplus sl_2(\R)$ which takes $(\lambda,m,n)$
to
$$\bar\lambda\oplus \pmatrix{-(\lambda,v)&2n\cr 2mN&(\lambda,v)\cr}$$
where $\bar\lambda$ is the projection of $\lambda$ into $v^\perp$
and $sl_2(\R)$ is the set of real $2\times 2$ matrices of trace 0.
We let the group $SL_2(\R)$ act on $sl_2(\R)$ by conjugation,
and on $v^\perp$ by the trivial action, which induces an action
of $SL_2(\R)$ on $(L\oplus II_{1,1})\otimes \R$. This action
is given by
$$\eqalign{
&{ab\choose cd} (\lambda,m,n) = \cr
&\left(\lambda+(bdm-b{c\over N}(\lambda,v)-a{c\over N}n)v,\quad
d^2m-{c\over N}d(\lambda,v)-{c^2\over N}n,\quad
ab(\lambda,v)+a^2n-b^2mN \right).\cr
}$$
In particular, if ${ab\choose cd}\in \Gamma_0(N)$ (so that $N|c$) then
this maps $L\oplus II_{1,1}$ into $L\oplus II_{1,1}$.
This defines an action of $\Gamma_0(N)$ on $L\oplus II_{1,1}$,
and hence an action on the Hermitian symmetric space $H$.
We embed the upper half plane into $H$ by mapping $\tau$ to $\tau v$,
which is represented by the point $(\tau v, 1, -\tau^2N)\in M\otimes
\C$. The action of $\Gamma_0(N)$ on $H$ restricts
to the usual action ${ab\choose cd}(\tau)=(a\tau+b)/(c\tau+d)$
on the upper half plane. In particular if we restrict
an automorphic form of weight $k$ on $H$ to multiples of $v$ we get
a modular form of weight $2k$ for $\Gamma_0(N)$.
The restriction of $\Phi$ to multiples of
$v$ will often be identically 0.
We define $\Phi_0$ to
be $\Phi$ divided by all the factors in the product defining $\Phi$
which are identically
zero on multiples of $v$, so that
$$\Phi_0(y)=e^{-2\pi i(\rho,y)}\prod_{x\in L,x>0,(x,v)\ne 0}(1-e^{-2\pi i(x,y)})^{c(x)}.$$
We define $H_{N,j}(-D) $ to be the number of complex numbers
$\tau$ in a fundamental domain of $\Gamma_0(N)$ satisfying
a nonzero quadratic equation of the form $a\tau^2+b\tau+c=0$ with
$a\b,c\in \Z$, $N|a$, $b\equiv j\bmod 2N$, and $b^2-4ac=D<0$ if $D<0$, and
define $H_{N,j}(0)$ to be $-|SL_2(\Z)/\Gamma_0(N)|/12$ if $j\equiv 0\bmod
2N $ and 0 otherwise.
(Points on the boundary
of the fundamental domain have to be counted with fractional
multiplicity in the usual way.) We define the
function $\H_{N,j}(\tau) $ by
$$\H_{N,j}(\tau)=\sum_n H_{N,j}(n)q^{n}.$$
We put $H_N(n)=\sum_{j\bmod 2N}H_{N,j}(n)$ and
$\H_N(n)=\sum_{j\bmod 2N}\H_{N,j}(n)$.
For example, the first few functions are
$$\eqalign{
\H_1(\tau) &= -1/12 +(1/3)q^{3}+(1/2)q^4 + q^{7}+ q^8+q^{11}+(4/3)q^{12}+\cdots\cr
\H_2(\tau) &= -1/4 +(1/2)q^{4} +2q^{7}+q^{8}+2q^{12}+\cdots\cr
\H_3(\tau) &= -1/3 +(1/3)q^{3}+2q^{8}+2q^{11}+(4/3)q^{12}+\cdots \cr
}$$
\proclaim Theorem 13.1. Suppose $v$ is a vector of norm $-2N<0$
and let $\Phi_0(v\tau)$ the function $\Phi_0$ defined above,
restricted to the multiples $v\tau$ of $v$ for $\tau\in \C$, $\Im(\tau)>0$.
Then $\Phi_0(v\tau)$ is a modular form for $\Gamma_0(N)$ of weight $k_0$
equal to the constant term of $f(\tau)\theta_{v^\perp}(\tau)$.
If $\tau$ is a root of an equation $a\tau^2+b\tau+c=0$ with $a/N,b,c\in \Z$,
$(a/N,b,c)=1$, $b^2-4ac=D<0$, then the order of the zero of
$\Phi_0(\tau v)$ is
$$\sum_{d>0}\sum_{(\lambda,v)=db}c(d^2D/4N-\bar\lambda^2/2).$$
The sum of the orders of the zeros
of $\Phi_0(\tau v)$ at the cusps of a fundamental domain of $\Gamma_0(N)$
is the constant term of
$$-\sum_{b\bmod 2N} f(\tau)\theta_{v^\perp,b}\H_{N,b}(\tau/4N).$$
Under the Fricke involution $\tau\rightarrow -1/N\tau$
$\Phi_0(\tau v)$ transforms as
$$\Phi_0(-v/N\tau)= \pm(\sqrt N\tau)^{k_0}\Phi_0(\tau v).$$
Proof.
Under the group $\Gamma_0(N)$, direct calculation shows that
the function $\Phi$ transforms as
$$\Phi\left({-2N\bar y+(ay+bv,cy+dv)v\over(cy+dv,cy+dv)}\right)
=\pm\left({(cy+dv,cy+dv)\over -2N}\right)^k\Phi(y).$$
From this we find that the function $\Phi_0(\tau v)$
transforms as
$$
\Phi_0(v(a\tau+b)/(c\tau+d))=\pm(c\tau+d)^{2k+2\sum_{x>0,(x,v)=0}c(-(x,x)/2)}\Phi_0(v\tau)
$$
for ${ab\choose cd}\in \Gamma_0(N)$,
so that $\Phi_0(\tau v) $ is a nonzero
modular form for $\Gamma_0(N)$ of weight equal to
$${2k+2\sum_{x>0,(x,v)=0}c(-(x,x)/2)}=\sum_{(x,v)=0}c(-(x,x)/2)$$
which is the constant term of $f(\tau)\theta_{v^\perp}(\tau)$.
This follows because $\Phi_0(\tau v)$ is essentially
the first nonvanishing coefficient in the Taylor series
expansion of $\Phi$ orthogonal to $\C v$ and is obtained
by differentiating $\Phi$ $c(-(x,x)/2)$ times for each
positive norm vector $x$ orthogonal to $v$, and each differentiation
contributes 2 to the weight of $\Phi_0(\tau v)$.
We have to calculate the zeros of $\Phi_0(\tau v)$. The zeros of
$\Phi_0(y)$ are the divisors of positive norm vectors $(\lambda,m,n)$ of
$M$, each with multiplicity $c(-\lambda^2/2+mn)$. We have to remember
only to count the zeros from one of the vectors $(\lambda,m,n)$ and
$-(\lambda,m,n)$, and also remember to count the zeros from positive
multiples of $(\lambda,m,n)$. This gives a contribution of
$c(-\lambda^2/2+2mn)$ to the zero of $\Phi_0(\tau v) $ at $\tau$ whenever
$((v\tau,1,v^2\tau^2/2),(\lambda,m,n))=0$, or in other words when
$mN\tau^2+(v,\lambda)\tau-n=0$. In particular $\tau$ must be an
imaginary quadratic irrational. Suppose that $a\tau^2+b\tau+c=0$
with $a/N,b,c\in \Z$, $(a/N,b,c)=1$, $b^2-4ac=D$. Then we must have
$m=da$, $(v,\lambda)=db$, $-n=dc$ for some integer $d$.
But then $d^2D/4N = (v,\lambda)^2/4N+mn=\bar\lambda^2/2+(mn-\lambda^2/2)$
where $\bar \lambda=\lambda-v(\lambda,v)/(v,v)$ is the projection of
$\lambda$ into $v^\perp$. Hence the multiplicity of the zero of
$\Phi_0(\tau v)$ at $\tau$ is
$$\sum_{d>0}\sum_{\lambda,(\lambda,v)=bd}c(d^2D/4N-\bar\lambda^2/2).$$
As $\Phi_0(\tau v)$ is a modular form for $\Gamma_0(N)$, the total number of
zeros in a fundamental domain is equal to $|SL_2(\Z)/\Gamma_0(N)|/12$
times its weight, which is the constant term of $-\H_{N,b}$ times the constant term of $f(\tau)\theta_{v^\perp}(\tau)$.
Hence the number of zeros at all the cusps is this number minus
the number of complex zeros of $\Phi_0(\tau v)$ in a fundamental domain.
We have just worked out the multiplicity of
a zero at any complex number $\tau$,
so we can work out the total number of complex zeros in
a fundamental domain, and we find that the number
of zeros at the cusps is as stated in 13.1.
Finally the transformation formula for $\Phi_0(\tau v)$ under
the Fricke involution follows from
the formula $\Phi(2y/(y,y))=\pm ((y,y)/2)^k\Phi(y)$.
This proves theorem 13.1.
\proclaim Corollary 13.2. Suppose that $v$ is a vector of norm $-2N\le 0$.
\item {} If $N=0$ and $v$ is primitive then the height of
$v$ is the constant term of
$$\theta_{v^\perp/v}(\tau)f(\tau)E_2(\tau)/24.$$
\item{}If $N=1$ then the height of $v$ is the constant term of
$$-\theta_{v^\perp{}'}(\tau)f(\tau)\H(\tau).$$
\item{}If $N$ is prime then the height of $v$ is the constant term of
$$-{1\over 2}\sum_{b\bmod 2N}\theta_{v^\perp{},b}(\tau)f(\tau)\H_{N,b}(\tau).$$
Proof. The case $N=0$ follows from theorem 10.4. If $N=1$ then the
height is just the order of the zero of $\Phi_0$ at the cusp $i\infty$
and as $SL_2(\Z)$ has only one cusp the corollary then follows from
theorem 13.1. If $N$ is prime then $\Gamma_0(N)$ has 2 cusps represented
by $0$ and $i\infty$. These two cusps are exchanged by the Fricke
involution, so the $\Phi_0(\tau v)$ has zeros of the same order
at both cusps.
Therefore the order of the zero at $i\infty$ is half the sum of
the orders at all cusps. This case of the corollary then follows from
theorem 13.1.
The fact that the height is always an integer can be used to find some
congruences between the coefficients of theta functions of lattices.
As an example we will work out these congruences for the theta
functions of some unimodular lattices explicitly. The
isomorphism classes of 25 dimensional unimodular lattices can be
identified with the orbits of norm $-4$ vectors $v$ in $II_{25,1}$,
where the lattice $v^\perp$ is isomorphic to the lattice of vectors of
even norm in the corresponding 25-dimensional unimodular lattice. We
put $N=2$ and $f(\tau)=1/\Delta(\tau)$ and find that the constant term
of
$$-{1\over 2} \sum_{b\bmod 4} \theta_{v^\perp,b}\H_{2,b}(\tau)f(\tau)$$ is the
height of $v$ and therefore is an integer. The first few coefficients
of $\H_{2,b}$ are given by $\H_{2,0}(\tau/8)=-1/4+q+\cdots$,
$\H_{2,1}(\tau/8)=\H_{2,3}(\tau/8)= q^{7/8}+\cdots$, and
$\H_{2,2}(\tau/8)=
(1/2)q^{1/2}+\cdots$. The coefficients of the theta functions
are given by $\theta_{v^\perp,0}(\tau)+\theta_{v^\perp,2}(\tau)=
\theta_K(\tau)$, and $\theta_{v^\perp,1}(\tau)=\theta_{v^\perp,3}(\tau)=
aq^{1/8}+\cdots$, where $a=1$ if $K$ is the sum of a one dimensional lattice
and a Niemeier lattice, and $a=0$ otherwise.
Putting everything together, we find that
$$8{\rm height}(v)= 20+r_2-2r_1-8a$$
where $r_n$ is the number of vectors of norm $n$ in $K$.
In particular
$$r_2\equiv 2r_1+4\bmod 8.$$
A consequence of this is that any 25 dimensional even unimodular
lattice has minimum norm at most 2.
More generally we find that if $K$ is a unimodular
lattice of dimension $s+1\equiv 1\bmod 24$ with $s>0$ then the constant term of
$$(\H_{2,0}(\tau/8)+\H_{2,2}(\tau/8))\theta_K(\tau)/\Delta(\tau)^{s/24}$$
is divisible by 2. As the Fricke involution
acts on a fundamental domain of $\Gamma_0(2)$
fixed point freely except at the images of the points
$\tau=i$ and $\tau=\sqrt 2i$ the coefficients of
the series $\H_{2,0}$ and $\H_{2,2}$ are usually even;
more precisely $\H_{2,0}(\tau)\equiv -1/4 +\sum_{n>0}q^{8n^2}\bmod 2$
and $\H_{2,2}(\tau)\equiv (1/2)\sum_{n>0}q^{4n^2}\bmod 2$.
This implies that the constant term of
$$(2\theta(\tau)+\theta(\tau/2))\theta_K(\tau)/\Delta(\tau)^{s/24}$$
is divisible by 8.
We can get similar congruences for unimodular lattices whose dimension
is not $1\bmod 24$ by adding on copies of the 1-dimensional unimodular lattice
until the dimension is $1\bmod 24$. These congruences can probably
also be deduced by constructing automorphic forms
for the groups $O_{K\oplus I_{2,2}}(\Z)^+$. \bigskip
{\bf 14. Product formulas for modular forms.}
\bigskip
In this section we will prove theorem 14.1 below. This immediately
implies the product formula for the modular polynomial stated in the
introduction, because the product
$\prod_{[\sigma]} (j(\tau)-j(\sigma))$ obviously satisfies
the conditions in theorem 14.1, and has zeros corresponding
to a function $f_0(\tau)$ of the form $q^D+O(q)$.
Recall that $H(n) $ is the Hurwitz class number for the discriminant
$-n$ if $n>0$,
and $H(0)=-1/12$. (So $\sum H(n)q^n=-1/12 + q^3/3 +q^4/2 +q^7 +q^8
+q^{11} +(4/3)q^{12}+\cdots$. )
\proclaim Theorem {14.1}.
Suppose that $f_0(\tau)=\sum c_0(n)q^n$ is a nearly holomorphic modular form of weight $ 1/2$ for
$\Gamma_0(4)$
with integer coefficients
whose coefficients $c_0(n)$ vanish unless $n$ is 0 or 1 mod 4.
We put
$$\Psi(\tau)=q^{-h}\prod_{n>0}(1-q^n)^{c_0(n^2)}$$
where $h$ is the constant term of $f_0(\tau)\sum H(n)q^n$.
Then $\Psi(\tau)$ is a
meromorphic modular form for some character of $SL_2(Z)$,
of integral weight, leading coefficient 1,
whose coefficients are integers, and all of whose zeros
and poles are either cusps or imaginary quadratic irrationals.
This correspondence gives an isomorphism between the additive group of
functions
satisfying the conditions on $f_0$ and the multiplicative group
of functions satisfying the conditions on $\Psi$.
Under this isomorphism,
the weight of the modular form $\Psi$ is $c_0(0)$, and the multiplicity
of the zero of $\Psi$ at a quadratic irrational $\tau$ of discriminant $D<0$
is $\sum_{d>0}c_0(Dd^2)$. (The discriminant of $\tau$
is $D=b^2-4ac$, where $a,b,$ and $c$ are integers with no common factor
such that $a\tau^2+b\tau+c=0$.)
Example 1. Under this isomorphism
$f_0(\tau)=12\theta(\tau)=12+24q+24q^4+24q^9+\cdots$
corresponds to $\Psi(\tau)=\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}$
of weight $c_0(0)=12$;
this is the usual product formula for the $\Delta$ function.
Example 2.
Put
$$\eqalign{
F(\tau)&=\sum_{n>0,n \,{\rm odd}} \sigma_1(n)q^n = q+4q^3+6q^5\cdots\cr
\theta(\tau)&=\sum_{n\in Z} q^{n^2} = 1+2q+2q^4+\cdots\cr
f_0(\tau)&=F(\tau)\theta(\tau)(\theta(\tau)^4-2F(\tau))(\theta(\tau)^4-16F(\tau))E_6(4\tau)/\Delta(4\tau)+56\theta(\tau)\cr
&=q^{-3}-248q+26752q^4-\cdots\cr
&= \sum_{n} c_0(n)q^n.\cr
}$$
Then the corresponding infinite product $\Psi(\tau)$ has
weight $c_0(0)=0$ and has zeros of order $c_0(-3)=1$
at the point of discriminant $-3$
(which are the conjugates of $(1+i\sqrt 3)/2$)
and nowhere else, so it must be $j(\tau)^{1/3}$.
The Fourier series of $3f_0(\tau)$ is
$$3f_0(\tau)=
3q^{-3}-744q + 80256q^4 - 257985q^5 + 5121792q^8 -
12288744q^9+\cdots$$
so
$$\eqalign{
j(\tau) &=q^{-1}+744+196884q+21493760q^2+\cdots\cr
&= q^{-1}\prod_{n>0}(1-q^n)^{3c_0(n^2)}\cr
&= q^{-1}(1-q)^{-744}(1-q^2)^{80256}(1-q^3)^{-12288744}\cdots\cr
}$$
Example 3. The Eisenstein series $E_4$, $E_6$, $E_8$, $E_{10}$, and
$E_{14}$ all satisfy the condition on $\Psi$ and so can be written as
infinite products corresponding to some functions $f_0(\tau)$. An
explicit formula for $E_4$ follows easily from the infinite product
expansions of $j$ and $\Delta$ because $j=E_4^3/\Delta$, and an
explicit formula for $E_6$ is given in example 2 of section 15. The
other cases follow from $E_8=E_4^2$, $E_{10}=E_4E_6$, and $E_{14} =
E_4^2E_6$. The remaining Eisenstein series cannot be written as
modular products.
Example 4. There are exactly 13 integers $n$ for
which $j(\tau)-n$ satisfies the conditions on $\Psi(\tau)$
and hence can be written as a modular product; these
are the values of $j(\tau)$ at values of
imaginary quadratic $\tau$ for which $j(\tau)$ is integral
which are well known to be
$
j((1+i\sqrt{3})/2)=0 $, $
j(i)=2^6.3^3$, $
j((1+i\sqrt{7})/2)=-3^35^3 $, $
j(i\sqrt{2})=2^65^3$, $
j((1+i\sqrt{11})/2)=-2^{15} $, $
j(i\sqrt{3})=2^{4}3^35^3$, $
j(2i)=2^33^311^3$, $
j((1+i\sqrt{19})/2)=-2^{15}3^3 $, $
j((1+i\sqrt{27})/2)=-2^{15}3.5^3 $, $
j(i\sqrt{7})=3^3.5^3.17^3$, $
j((1+i\sqrt{43})/2)=-2^{18}3^35^3 $, $
j((1+i\sqrt{67})/2) =-2^{15}3^35^311^3$, $
j((1+i\sqrt{163})/2)=-2^{18}3^35^323^329^3$.
There are therefore exactly 14 modular forms of weight 12
with integer coefficients for $SL_2(\Z)$
which are modular products: the forms $\Delta(\tau)(j(\tau)-n)$
and the form $\Delta(\tau)$.
We will prove theorem 14.1 as follows. We find a spanning set
for the set of modular forms $f_0$ in theorem 14.1 and check
for each of them that the conclusion of theorem 14.1 is true
using the automorphic forms on $II_{s+2,2}$ constructed in
section 10. Then we check that all the modular forms
$\Psi$ satisfying the conclusion of theorem 14.1 can be written
as a product of the modular products constructed from
the forms $f_0$.
\proclaim Lemma {14.2}. Every sequence of integers
$c_0(n)$ for $n\le 0$, $n\equiv 0,1\bmod 4$
which are almost all zero
is the set of coefficients of nonpositive degree
for a unique modular form $f_0$ satisfying the conditions of theorem
14.1.
Proof. This is similar to the proof of theorem 5.4 of [E-Z] with a few sign
changes.
If $f_0(\tau)=\sum_nc_0(n)q^n$ is a modular form satisfying the conditions
of theorem 14.1 with $c_0(n)=0$ for $n\le 0$ then we define
$h_0(\tau)=\sum_{n} c_0(4n)q^n$ and $h_1(\tau)=\sum_n
c_0(4n+1)q^{n+1/4}$.
If we use the fact that $f_0(\tau)=h_0(4\tau)+h_1(4\tau)$ satisfies the
relation $f_0(\sigma/(4\sigma+1))=\sqrt{(4\sigma+1)}f_0(4\sigma)$
and put $\tau=4\sigma+1$ we find
that $h_0$ and $h_1$ satisfy the relation
$$h_0(-1/\tau)-ih_1(-1/\tau)= \sqrt{\tau}(h_0(\tau)-ih_1(\tau)).$$
If we let $\tau$ be imaginary and take real and imaginary parts of
this we find that $h_0$ and $h_1$ satisfy the relations
$$\eqalign{
h_0(\tau+1) &=h_0(\tau)\cr
h_1(\tau+1) &=ih_1(\tau)\cr
h_0(-1/\tau)&=(1/2-i/2)\sqrt{\tau}(h_0(\tau)+h_1(\tau))\cr
h_1(-1/\tau)&=(1/2-i/2)\sqrt{\tau}(-h_0(\tau)+h_1(\tau)).\cr
}$$
This implies that $h_0$ and $h_1$ modular forms of weight
$1/2$ which have zeros of order at least $1/4$ at all cusps, so
$h_0$ and $h_1$ are both zero
because $h_0/\eta$ and $h_1/\eta$ are
holomorphic modular functions vanishing at all cusps.
This proves that the form $f_0$ in lemma 14.2 is unique if it exists.
To prove the existence of $f_0$ we must exhibit
such a form $f_0$ whose Laurent series starts off $q^n=\cdots$
for every $n\le 0$ with $n\equiv 0,1 \bmod 4$. It is sufficient to
do this for $n=0$ or $-3$, because we can then get all values
of $n$ by multiplying by powers of $j(4\tau)$. For $n=0$ we
can use the function $\sum_{n\in \Z}q^{n^2}$,
and for $n=-3$ we can use the function
$F(\tau)\theta(\tau)(\theta(\tau)^4-2F(\tau))(\theta(\tau)^4-16F(\tau))E_6(4\tau)
/\Delta(4\tau)+56\theta(\tau)$ where $F(\tau)$ is defined in
example 2 above.
This proves lemma 14.2.
\proclaim Lemma {14.3}. There is a norm $-2$ vector $v\in II_{25,1}$
such that $\theta_{{v^\perp}'}(\tau) = 1+2q^{1/4}+O(q)$.
Proof: We take $v$ to be the image of a norm $-2$ vector of
$II_{1,1}$ in $II_{25,1}=II_{1,1}\oplus\hbox{(Leech lattice)}$.
\proclaim Lemma {14.4}. There is a norm $-2$ vector $v\in II_{25,1}$ such that
$\theta_{{v^\perp}'}(\tau) = 1+6q+O(q^{5/4})=1+O(q)$.
Proof. If we take $\rho$ to be a primitive norm 0 vector
of $II_{25,1}$ corresponding to the Leech lattice,
then $\rho$ is a Weyl vector for the reflection group of the
Leech lattice and the simple roots are the norm 2 vectors $r$
with $(r,v)=-1$ and they correspond to vectors of the Leech lattice.
If we take $r_1$ and $r_2$ to be simple roots having inner product
$-1$ (corresponding to two vectors of the Leech lattice at distance
$\sqrt 6$) then the vector $v=\rho +r_1+r_2$ is a norm $-2$
vector of the Leech lattice. It is easy to check that it
is in the Weyl chamber of the reflection group of $II_{25,1}$,
and that the simple roots of $v^\perp$ are just $r_1$ and $r_2$
so that $v^\perp$ has root system $a_2$ and therefore has 6
norm 2 vectors. It is also easy to check that there are
no norm 0 vectors of $II_{25,1}$ having inner product 1 with $v$,
which implies that there are no vectors of norm $1/2$ in
${v^\perp}'$. This proves lemma 14.4.
\proclaim Lemma {14.5}. The $\Z$-module of functions $f_0$ satisfying the conditions
of theorem 14.1 is spanned by functions of the form
$\theta_{{v^\perp}'}(4\tau)f(4\tau)$, where $v$ is a norm -2 vector of
a lattice $II_{s+1,1}$ and $f=\sum_nc(n)q^n$ is a nearly holomorphic
modular form of level 1 and weight $-s/2$.
Proof: By using lemmas 14.1 and 14.4 we can find functions which are
linear combinations of the functions mentioned in lemma 14.5,
and whose Fourier series
start off $q^{-4n}+\ldots$ or $2q^{-4n-3}+\ldots$ for any
nonnegative integer $n$. By lemma 14.2 these functions span the module
of functions $f_0$ satisfying the conditions of theorem 14.1.
We can now prove theorem 14.1.
Suppose that $v$ is a norm $-2$ vector of $II_{25,1} $ and $f$
is a nearly holomorphic modular form of weight $-12$,
and
$$\Phi_0(y)=e^{-2\pi i(\rho,y)}\prod_{x\in L,x>0,(x,v)\ne 0}(1-e^{-2\pi i(x,y)})^{c(x)}$$
is the function defined in section 13.
By theorem 13.1 $\Psi(\tau)=\Phi_0(\tau v)$ is
a modular form for $SL_2(\Z)$ of weight equal to
${2k+2\sum_{x>0,(x,v)=0}c(-(x,x)/2)}=\sum_{(x,v)=0}c(-(x,x)/2)$ which
is the constant term of $f(\tau)\theta_{v'}(\tau)$. By theorem 13.1 again
the zeros of $\Psi$ are as stated in theorem 14.1.
We can find the infinite product decomposition of $\Phi_0(\tau v)$ from that of
$\Phi_0(y)$ by restriction, and we see that
$$\Phi_0(\tau v)=q^h\prod_{n>0}(1-q^n)^{c_0(n^2)}$$
where $h$ is the height of $v$ and $c_0(n)$ is the coefficient of
$q^n$ in $f(4\tau)\theta_{{v^\perp}'}(4\tau)=f_0(\tau)$.
By theorem 13.1 the height $h$ is given by the formula stated
in theorem 14.1.
Finally we have to check that the map from functions $f_0$ to
functions $\Psi$ in theorem 14.1 is an isomorphism. If the image
of $f_0$ is 1, then $\Psi$ has no zeros and weight 0, so
the coefficients $c_0(n)$ are 0 if $n\le 0$, so $f_0$ is 0
by lemma 14.2, which proves that the map is injective.
If $\Psi$ is any function satisfying the conditions
of theorem 14.1, then again by lemma 14.2 we can find
a function $f_0$ such that the corresponding infinite
product has the same complex zeros and poles as $\Psi$.
By taking the quotient we can assume that $\Psi$ has no zeros
or poles except at cusps. But this implies that $\Psi$
must be a positive or negative power of $\eta(\tau)^2$
(as $\Psi$ has integral weight), and we obtain these
functions $\Psi$ by taking $f_0$ to be an integral
multiple of $\theta(\tau)=1+2q+2q^4+\cdots$. This proves
theorem 14.1.
Remark: in the case when we take $v$ to be a norm $-2$ vector in the
lattice $II_{1,1}$ we can work out the Weyl vector explicitly.
By identifying its height with the integer $h$ in theorem
14.1 we recover the classical Hurwitz formula
$$\sum_{t\in \Z}H(4m-t^2)=\sum_{d|m}\max(d,m/d)$$
for positive integers $m$. \bigskip
{\bf 15. Generalized Kac-Moody algebras.}
\bigskip
Many automorphic forms for $O_M(\Z)^+$ that are modular products,
especially those of singular weight, are the denominator functions of
generalized Kac-Moody algebras. We give a few examples of this.
Example 1. If $f(\tau)$ is the Hauptmodul of an element
of the monster simple group then $f(\sigma)-f(\tau)$
is an automorphic function on $O_{2,2}(\R)$ with respect
to some discrete subgroup, and can be written as an
infinite product whose exponents can be described explicitly.
See [B] for details. More generally, if $A$ and $B$
are elements of $SL_2(\Z)$ then $f(A\sigma)-f(B\tau)$
can often be written explicitly as an infinite product,
and these expressions are often the denominator formulas
for generalized Kac-Moody algebras.
Example 2. The product formula
$$\eqalign{
E_6(\tau)&=1-504\sum_{n>0}\sigma_5(n)q^n\cr
&= 1-504q-16632q^2-122976q^3-\cdots\cr
& =\prod_{n>0}(1-q^n)^{a(n^2)}\cr
&= (1-q)^{504}(1-q^2)^{143388}(1-q^3)^{51180024}\cdots\cr
}$$
where
$$\eqalign{
&\sum_na(n)q^n\cr
=& q^{-4} + 6 + 504q + 143388q^4 + 565760q^5 + 18473000q^8 + 51180024q^9 + O(q^{12})\cr
=& (j(4\tau)-852)\theta(\tau)-2F(\tau)\theta(\tau)(\theta(\tau)^4-2F(\tau))(\theta(\tau)^4-16F(\tau))E_6(4\tau)/\Delta(4\tau)\cr
}$$
(see section 14) is the denominator formula for a generalized Kac-Moody algebra
of rank 1 whose simple roots are all multiples of
some root $\alpha$ of norm $-2$, the simple roots are $n\alpha$ ($\alpha>0$)
with multiplicity $504\sigma_3(\alpha)$, and the multiplicity
of the roots $n\alpha$ is $a(n^2)$.
(Kok Seng Chua pointed out that the formula in the published paper
is wrong: 876
should be changed to 852.)
The positive subalgebra
of this generalized Kac-Moody algebra is a free Lie algebra,
so we can also state this result by saying that the
free graded Lie algebra with $504\sigma_5(n)$
generators of each
positive degree $n$ has a degree $n$ piece of dimension $a(n^2)$.
There are similar examples corresponding to the infinite products
for the Eisenstein series $E_{10}$ and $E_{14}$.
Example 3. In [B] there is an example of
an infinite product formula for every element of
$2^{24}.O_{\Lambda}(\Z)$ (where $\Lambda$ is the Leech lattice)
given by taking the trace of this element on the cohomology
of the fake monster Lie algebra.
This is probably always an automorphic form of singular weight for some group
$O_M(\Z)^+$, although I have not checked this for all cases.
This automorphic form is often the denominator function for
some generalized Kac-Moody algebra or superalgebra.
This gives several examples of automorphic forms of singular
weight on groups $O_M(\Z)^+$ with level greater than 1.
Example 4. There seems to be a superalgebra of rank 10 associated
with the $E_8$ lattice in the same way that the fake monster
Lie algebra is associated with the Leech lattice.
In fact, there seem to be 2 closely related superalgebras,
one with zero Weyl vector and one with Weyl vector
equal to the Weyl vector of the reflection group of
$O_{I_{9,1}}(\Z)$ generated by the reflections of norm 1 vectors.
I have a construction for these superalgebras but have not yet
checked all the details. The denominator formula for
one of these superalgebras is proved in [B]. The superalgebra
is probably acted on by a group $2^8.W(E_8)$ where $W(E_8)$
is the Weyl group if the $E_8$ lattice. There are presumably
twisted versions of the denominator formula for this
superalgebra associated to conjugacy classes in $2^8.W(E_8)$.
These twisted denominator functions are probably automorphic
forms of singular weights, and this can probably be proved case
by case using the methods of this paper.
Example 5.
The product formula
$$\sum_{m,n\in \Z}(-1)^{m+n}p^{m^2}q^{n^2}r^{mn}
= \prod_{a+b+c>0}\left({1-p^aq^cr^b\over1+p^aq^cr^b}\right)^{f(ac-b^2)}$$
where $f(n)$ is defined by $\sum f(n)q^n = 1/(\sum_n (-1)^nq^{n^2}) =
1+2q+4q^2+8q^3+14q^4+\ldots$.
is the denominator formula for a generalized Kac-Moody
superalgebra of rank 3. This superalgebra is graded by $\Z^3$,
and the subspace of degree $(a,b,c)$ had dimension 3 if
$(a,b,c)=(0,0,0)$, and $f(ac-b^2)|f(ac-b^2)$ otherwise. (The symbol $m|n$
for the dimension of a superspace means that it is the sum of
an ordinary part of dimension $m$ and a super part of dimension
$n$.) This product formula can be proved using the ideas of this paper,
except that we need to use Jacobi forms of level 2 rather than level
1. The left hand side is essentially Siegel's theta function of
genus 2, and so is an automorphic form for $Sp_2(\Z)$.
As $Sp_2(\R)$ is locally isomorphic to $O_{3,2}(\R)$,
it is also an automorphic form for the group $O_M(\Z)^+$
where $M$ is the even lattice of determinant 2,
dimension 5 and signature 1.
\bigskip
{\bf 16. Hyperbolic reflection groups}
\bigskip
There is often an automorphic form with a modular product expansion
associated with the hyperbolic reflection group of a Lorentzian
lattice, especially when the reflection group of the lattice has finite
index in the automorphism group. We will give several examples of this.
Example 1. We let $L$ be the 10-dimensional even Lorentzian lattice
$II_{9,1}$, whose reflection group has Dynkin diagram $e_{10}$. We
let $f$ be the automorphic form of weight 252 corresponding to the weight $-4$
modular form $E_4(\tau)^2/\Delta(\tau)=q^{-1} +504+\cdots$. All the
real vectors of the corresponding vector system are norm 2 roots and
have multiplicity 1, so they are exactly the roots of the reflection
group of $L$. The Weyl chambers of $f$ are all conjugate and any Weyl
vector has norm $-1240$, and these are the same as the Weyl chambers
and Weyl vectors of the reflection group of $L$. Unfortunately the
norm 0 vectors all have multiplicity $504$ which is much larger than
the multiplicity $8$ of the norm 0 vectors of the Kac-Moody algebra
$e_{10}$, so this seems to give no useful information about the root
multiplicities of this Kac-Moody algebra. The function $f$ has a
simple zero at all rational quadratic divisors of norm 2 and no other
zeros, so it divides any antiinvariant automorphic form for the group
$O_{II_{10,2}(\Z)^+}$, and therefore gives an isomorphism from invariant
automorphic forms of weight $k$ to antiinvariant forms of weight
$k+252$. In particular any antiinvariant form of weight less than 256
is a multiple of $f$.
Example 2. We let $L$ be the 18-dimensional even Lorentzian lattice.
We let $f$ be the form associated to the weight $-8$ form
$E_4(\tau)/\Delta(\tau) = q^{-1}+ 256 +\cdots$. As in the previous
example, we find that the Weyl chambers and Weyl vectors (norm $-620$)
of $f$ are the same as those of the reflection group of $L$, and
multiplication by $f$ is an isomorphism from invariant forms of weight
$k$ to antiinvariant ones of weight $k+128$.
Example 3. We let $L$ be the even sublattice of index 2 in
$I_{21,1}$, which is the orthogonal complement of a $d_4$
lattice in $II_{25,1}$. The reflection group of $L$
has finite index in the full automorphism group and is generated by
the reflections of vectors of norms 2 and 4. If we take an automorphic form
$\Phi$ for $O_{II_{26,2}}(\Z)^+$ and divided it by
the factors vanishing on $L$, we get a function $\Phi_0$
which restricts to an automorphic form for $O_{L\oplus II_{1,1}}(\Z)^+$.
If we take $\Phi$ to be the automorphic form associated
to $1/\Delta(\tau)$ then the positive norm vectors of $L$ of nonzero
multiplicity
are the norm 2 vectors with multiplicity 1, and half the norm 4 roots with
multiplicity 8. This gives an automorphic form whose positive norm
vectors of nonzero multiplicity are multiples of the roots
of $L$. This example cannot correspond to any generalized Kac-Moody algebra
because all the positive norm roots of a generalized Kac-Moody algebra
have multiplicity 1.
Example 4. More generally if we take any subdiagram of the Dynkin
diagram $\Lambda$ of $II_{25,1}$ whose components are all
of the form $d_4$, $d_n$ for $n\ge 6$, $e_6$, $e_7$, or $e_8$
and we let $L$ be the orthogonal complement in $II_{25,1}$ of
the lattice of this subdiagram, then we get an example similar to
the previous example of an automorphic form whose
positive norm vectors of nonzero multiplicity
are closely related to the roots of some reflection group
of finite index in the full automorphism group of $L$.
Examples 1,2, and 3 correspond to the subdiagrams
$e_8^2$, $e_8$, and $d_4$.
\bigskip
{\bf 17. Open problems.}
\item {1.} Can the methods for constructing automorphic
forms as infinite products be used for semisimple groups other
than $O_{s+2,2}(\R)$?
\item{2.} Extend the methods of this paper to level greater than 1.
Can any nearly holomorphic Jacobi form of nonnegative weight
be used to construct a meromorphic automorphic form? In particular,
can the explicit calculations used in section 10 to prove the
existence of a Weyl vector be replaced by a more general argument?
Can any automorphic form (with rational integral Fourier coefficients)
all of whose zeros are rational
quadratic divisors be written as a modular product?
\item {3.} Are there a finite or infinite number of automorphic
forms of singular weight that can be written as modular products?
Are there any such forms on $O_{s+2,2}(\R)$ for $s>24$? The forms of
singular weight which are modular products are particularly interesting
because they often correspond to generalized Kac-Moody algebras.
If a form has singular weight then its Weyl vectors must be of norm
0, and the lattices corresponding to them have no vectors of small norm.
\item {4.} Find some interesting cases of the generalized Macdonald
identities of chapter 6 such that the sum of theta functions times
modular forms can be written down explicitly.
\item{5.} What are the eigenvalues of the Hecke operators
acting on the weight 12 form for $O_{II_{26,2}}(\Z)^+$
(which is an eigenform of the Hecke operators)? Does it correspond to
some Galois representation?
\item{6.} Can the automorphic forms that are modular products be
understood in terms of representation theory or Langlands philosophy?
(I do not even know how to understand the product formula
for $\Delta(\tau)$ in terms of representation theory.) One problem
with this is that the automorphic forms which are modular products
are often not eigenforms of Hecke operators; for example there
are 14 modular forms of level 1 and weight 12 which are modular products,
only one of which ($\Delta$) is an eigenform.
\item{7.} Many automorphic forms that are modular products
can be interpreted as the denominator formulas of generalized Kac-Moody
algebras. Is it possible to construct these generalized Kac-Moody algebras
explicitly (other than by generators and relations)?
For example, the fake monster Lie algebra is the
Lie algebra of chiral strings on a 26-dimensional torus.
\item{8.} Given an automorphic form for $O_M(\Z)^+$ with
a modular product we obtain a decomposition of the cone $C$
into Weyl chambers, each of which either has finite volume, or
has finite volume modulo the action of a free abelian group.
In the only cases I know of where this decomposition has been worked
out explicitly (mostly reflection groups or the examples associated
with $II_{25,1}$ in chapter 14)
the automorphism group acts transitively on the Weyl
chambers, but this is certainly not usually the case. What happens in
general? I would guess that there are usually a
large number of Weyl chambers, each one of which has
a fairly simple structure and not too many sides, but I have
no real evidence for this. The structure of any given Weyl chamber
can probably be worked out using some sort of extension
of Vinberg's algorithm for the case of reflection groups.
\item{9.} Is there any connection between the 3
product formulas for $j(\tau)-j(\sigma)$? (The three formulas
are the Gross-Zagier formula when both of $\sigma$, $\tau$
are imaginary quadratic integers, the formula of this paper when
one of them is, and the product formula for the monster Lie algebra
when neither of them are.)
\item{10.} Extend theorem 14.1 to higher levels. Can
any modular form (of any level) with integral coefficients,
such that all its zeros are cusps or imaginary quadratic irrationals,
be written as a modular product? What happens if the
coefficients are not required to be rational?
Is there a proof of theorem 14.1 which only uses modular
forms on $SL_2$ and not automorphic forms on larger groups?
In theorem 14.1 the coefficients of negative powers of $q$
in $f_0$ are related to the orders of the zeros of $\Phi_0$,
and the coefficients of $q^{n^2}$ appear in the infinite product expansion of
$\Phi_0$.
Is there any interpretation of the coefficients of $q^n$ for positive
values of $n$ that are not squares?
Does the correspondence in theorem 14.1 commute with some
action of a Hecke algebra? (The Hecke operators would have to act
multiplicatively rather than additively on meromorphic modular forms.)
The Shimura-Kohnen isomorphism [Ko] is an isomorphism
from the space of weight $k/2$ forms of level 4 whose Fourier
coefficients $c(n)$ vanish unless $n\equiv 0, (-1)^{(k-1)/2}\bmod 4$,
to the space of modular forms of level 1 and weight $k-1$ ($k$ odd).
In this isomorphism the coefficients $c(n^2)$ are closely related
to the eigenvalues of the Hecke operators. Is there any connection
with the isomorphism of theorem 14.1, which has the same condition
on the coefficients $c(n)$ and for which the coefficients $c(n^2)$
are also particularly important? (Notice that the Shimura-Kohnen
isomorphism is additive, while the one in theorem 14.1
is multiplicative.)
\item{11.} The extension of theorem 14.1 to higher levels
appears to give
many modular functions whose zeros are the imaginary quadratic
irrationals that are used to define Heegner divisors of modular
curves. R. Taylor suggested that this could be used to produce
linear relations between the Heegner points on the Jacobian.
\item{12.} Suppose $L$ is a Lorentzian lattice
whose reflection group has finite index in its automorphism group.
Is there always an automorphic form with a modular product expansion
such that the positive vectors of nonzero multiplicity
are more or less the same as the roots of $L$ (possibly multiplied
by nonzero rational numbers)?
\bigskip
{\bf Acknowledgements.} I thank the
N.S.F. for providing support and the referee for several useful
comments, in particular for simplifying the original proofs of
lemmas 10.2 and 10.3.
\bigskip
{\bf References.}
\bigskip
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\end