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.

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.

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 ℘ 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.

1. Introduction.

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 η(τ)^{−s} = q^{−s/24} ∏_{n>0}(1 − q^{n})^{−s} = ∑_{n} c(n)q^{n} and ρ is a certain vector then

Φ(v) = e^{−2πi(ρ,v)} ∏_{r>0}(1 − e^{−2πi(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(τ). More precisely,

j(τ) = q^{−1} ∏_{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 + ⋯ 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(τ) − 1728. The usual product formula Δ(τ) = q ∏_{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

∏_{[σ]}(j(τ) − j(σ)) = q^{−H(−D)} ∏_{n>0}(1 − q^{n})^{c_{0}(n^{2})}

where σ 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 aσ^{2} + bσ + c = 0 (a, b, c ∈ Z) of some fixed discriminant b^{2}−4ac = D < 0 (except that when σ 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(τ)−1728 or j(τ) by (j(τ) − 1728)^{1/2} or j(τ)^{1/3}). The exponents c_{0}(n^{2}) are coefficients of the unique nearly holomorphic weight 1/2 modular form for Γ_{0}(4) whose power series ∑_{n∈Z} c_{0}(n)q^{n} is of the form q^{D} + O(q) and whose coefficients c_{0}(n) vanish unless n ≡ 0, 1 mod 4 (Kohnen’s “plus space” condition). The product on the left, as a function of j(τ), 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)

∏_{[τ_{1}],[τ_{2}]}(j(τ_{1}) − j(τ_{2}))^{4/w_{1}w_{2}} = ± ∏_{x∈Z,n,n′>0,x^{2}+4nn′=d_{1}d_{2}} n^{ϵ(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 ϵ(n′) = ±1 is defined in [G-Z]. It is also related to the denominator formula

j(σ) − j(τ) = p^{−1} ∏_{m>0,n∈Z}(1 − p^{m}q^{n})^{c(mn)}

of the monster Lie algebra (where p = e^{2πiσ}, q = e^{2πiτ}, and j(τ) − 744 = ∑_{n} c(n)q^{n} = q^{−1} + 196884q + ⋯). These 3 product formulas for ∏(j(σ) − j(τ)) cover the cases when both, one, or neither of σ and τ 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 ⊗ R, and if θ_{K}(τ) is the theta function of K, then the constant term of θ(τ)/Δ(τ)^{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(σ) − j(τ) of the monster Lie algebra given above. This function obviously transforms under a group of the form (SL_{2}(Z)×SL_{2}(Z)).(Z/2Z), 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

Φ(v) = ∑_{w∈W} ∑_{n>0} det(w)τ(n)e^{−2πin(w(ρ),v)} = e^{−2πi(ρ,v)} ∏_{r>0}(1 − e^{−2πi(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 Φ satisfies the functional equation

Φ(2v/(v, v)) = −((v, v)/2)^{12}Φ(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 ≤ 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(ρ + ∑_{ℓ≥0} ϕ|V_{ℓ}), where ϕ 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 ϕ 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 ∑_{ℓ≥0} ϕ|V_{ℓ} was an automorphic form. The two main extra complications we have to deal with when ϕ 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” ρ has to be chosen correctly.

Notation (in roughly alphabetical order).

+ If G is a subgroup of a real orthogonal group then G^{+} means the elements of G of positive spinor norm. ′ If L is a lattice then L′ means the dual of L. ⊥ If u is a vector (or sublattice) of a lattice then u^{⊥} means the orthogonal complement

of u.

¯ If λ is a vector in a lattice then λ̄ is the orthogonal projection of λ into some sublattice. α A coordinate of a vector in M. a An entry of a matrix \binom{ab}{cd} in SL_{2}(Z).

A(v) A Fourier coefficient of an automorphic form on O_{M}(R)^{+}. β A coordinate of a vector in M. b An entry of a matrix \binom{ab}{cd} in SL_{2}(Z).

B_{k} A Bernoulli number: ∑_{k∈Z} B_{k}t^{k}/k! = t/(e^{t} − 1). γ A coordinate of a vector in M.

Γ_{0}(N) \{\binom{ab}{cd} ∈ SL_{2}(Z)|c ≡ 0 mod N\}

Γ(z) Euler’s gamma function.

c An entry of a matrix \binom{ab}{cd} in SL_{2}(Z). 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. C The complex numbers. C The positive cone in a Lorentzian lattice. δ A coordinate of a vector in M. δ_{n}^{m} 1 if m = n, 0 otherwise. Δ The delta function, Δ(τ) = q ∏_{n>0}(1 − q^{n}). d The number of elements of a vector system, an entry of a matrix \binom{ab}{cd} in SL_{2}(Z). D The discriminant of a quadratic divisor or an imaginary quadratic irrational or an

imaginary quadratic field. e^{±z} means e^{z} if ℜ(z) < 0, e^{−z} if ℜ(z) > 0. e_{n} The Dynkin diagrams or lattices of e_{8}, e_{10}, and so on. E_{k} An Eisenstein series of weight k, equal to 1 − (2k/B_{k}) ∑_{n>0} σ_{k−1}(n)q^{n}. ζ e^{2πiz}. ζ^{y} = e^{2πi(y,z)}. f A function. F A Fourier group. See section 2.

F (τ) = ∑_{n>0} σ_{1}(2n − 1)q^{2n−1}. 2F_{1} The hypergeometric function.

g An element of the group G, or a function. G A group. GL A general linear group. GO A general orthogonal group. η(τ) = q^{1/24} ∏_{n>0}(1 − q^{n}).

h The height of a vector. See section 13. h(τ) h(τ) = h_{0}(4τ) + h_{1}(4τ) is a modular form of weight 1/2.

H, H_{u} Hermitian symmetric spaces of O_{n+2,2}(R). H(n) The Hurwitz class number of n. See section 13.

H_{N,j}(n) A generalization of the Hurwitz class number of n. See section 13. H_{N,j}(τ) A function with coefficients H_{N,j}(n). See section 13.

θ, θ_{K} Theta functions of lattices or cosets of lattices. See section 3.

I_{ν} A modified Bessel function. II_{m,n} The even unimodular Lorentzian lattice of dimension m + n and signature m − n.

ℑ The imaginary part of a complex number. j The elliptic modular function j(τ) = q^{−1} + 744 + 196884q + · · ·. J A Jacobi group. See section 2. J_{ℓ} A double coset of J(Z)^{+}. κ A vector of K. k The weight of an automorphic form or Jacobi form. K An even positive definite lattice of dimension s.

K_{μ} A modified Bessel function. λ An element of K. Λ The Leech lattice. See [C-S]. ℓ An integer, usually indexing Hecke operators. L An even Lorentzian lattice of dimension s + 2, sometimes equal to K ⊕ II_{1,1}. μ An element of K, or a real number. m The index of a Jacobi form or vector system. See section 3 or 6. M An even lattice of dimension s + 4 and signature s, sometimes equal to L ⊕ II_{1,1}.

M[U] The lattice generated by U and all vectors of M having integral inner product with

everything in U. ν A real number. n An integer, often indexing the coefficients of a modular form. N The level of a modular form. O An orthogonal group.

O(q^{n}) A sum of terms of order at most q^{n}.

p e^{2πiσ} p_{m}(n) The number of partitions of n into parts of m colors.

P A principle C^{∗} bundle over H. ℘^{(n)} ℘^{(n)}(z, τ) = \frac{d}{dz}^{n}℘(z, τ), where ℘ is the Weierstrass function. See section 7.

q e^{2πiτ} Q A quadratic form; Q(v) = (v, v)/2. Q The rational numbers. ρ A Weyl vector. See section 6. R A commutative ring, usually Z, Q, R, or C. ℜ The real part of a complex number. r An element of the ring R, or a vector of K. R The real numbers. σ A complex number with positive imaginary part.

σ_{k−1}(n) = ∑_{d|n} d^{k−1} if n > 0, −B_{k}/2k if n = 0.

s The signature of a lattice or Jacobi form. sl_{2}(R) The set of 2 by 2 real matrices of trace 0.

SL A special linear group. SO A special orthogonal group.

τ A complex number with positive imaginary part, or Ramanujan’s function τ(n). t_{u,v}, t_{r} Automorphisms of a lattice. See section 2.

T_{ℓ} A Hecke operator. T_{ℓ,ℓ} A Hecke operator.

u A norm zero vector in a lattice, often contained in U. U A 2-dimensional null lattice, usually a sublattice of M. U_{ℓ} A Hecke operator acting on Jacobi forms.

ϕ,ϕ_{u} Jacobi forms. See sections 2 and 3. Φ An automorphic form on O_{M}(R)^{+}. ψ A Jacobi form. See sections 2 and 3. Ψ A meromorphic modular form with a modular product expansion. v A vector in a vector system. See section 6. V A vector system. See section 6. V_{ℓ} A Hecke operator acting on Jacobi forms. See section 4. V_{ℓ,ℓ} A Hecke operator acting on Jacobi forms. See section 4. W A Weyl chamber. See section 12. x A real number or an element of K ⊗ R, often equal to ℜ(z). y A real number or an element of K ⊗ R, often equal to ℑ(z). z An element of the complexification of K, or a complex number, often equal to x + iy. Z The integers.

Terminology. Nearly holomorphic. Meromorphic with all poles at the cusps. Automorphic form. See section 2. 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. Fourier group. A certain parabolic subgroup of O_{M}. See section 2. Height. The height of a vector is the minimum inner product with a Weyl vector. See section 13. Index. See section 6 for the index of a vector system, and sections 2 and 3 for the index of a Jacobi form. Jacobi form. See sections 2,3. Jacobi group. A parabolic subgroup of O_{M}. See section 2. Koecher principle. An nearly holomorphic automorphic form on a simple group of rank greater than 1 is automatically holomorphic at the cusps. Modular product. An infinite product whose exponents are the coefficients of nearly holomorphic modular forms. See section 5. Primitive sublattice. A sublattice Z of M is primitive if Z contains any vector of M a nonzero multiple of which is in Z. Rational quadratic divisor. The zero set of a(y, y) + (b, y) + c where a ∈ Z, b ∈ L, c ∈ Z. See section 5. 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. Singular weight. Weight s/2 or 0. See section 3. Spezialschar. (“Special space.”) A space of automorphic forms whose Fourier coefficients satisfy certain relations. See section 9 and [M paper I]. 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. Theta function. A modular form or Jacobi form depending on a lattice. See section 3. Vector system. A multiset of vectors in a lattice with some of the properties of a root system. See section 6. Weyl chamber. A generalization of the Weyl chamber of a root system to vector systems. See section 6.

Weyl vector. A generalization of the Weyl vector of a root system to vector systems. See sections 6, 12.

2. Automorphic forms and Jacobi forms

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 ⊗ R preserving the quadratic form Q of M ⊗ 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) ∈ 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 ∈ 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 ⊗ R.

The group O_{M}(R)^{+} is the image of the pin group P in_{M}(R) induced by the natural homomorphism from P in_{M} to O_{M}, and similarly SO_{M}(R)^{+} is the image of the spin group Spin_{M}(R). Notice that the map from P in_{M} to O_{M} is an epimorphism of algebraic groups, but the map from P in_{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 P in_{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 ⊗ 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 ∈ M ⊗ 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 ⊗ 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 ⊗ C maps P and H to their complex conjugates P̄ and H̄. If we identify P̄ and H̄ 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 Φ on P is a nearly holomorphic automorphic form of weight k ∈ Z for O_{M}(Z)^{+} if it has the following properties.

1 Φ is holomorphic on P.

2 Φ is homogeneous of degree −k, i.e., Φ(vz) = z^{−k}Φ(v) for z ∈ C.

3 Φ is invariant under O_{M}(Z)^{+}, i.e., Φ(γv) = Φ(v) for γ ∈ O_{M}(Z)^{+}. More generally, we also allow Φ(γv) = det(γ)Φ(v) and call such forms antiinvariant under O_{M}(Z)^{+}. If Φ is also “holomorphic at the cusps” (see below) then we call Φ a holomorphic

automorphic form, or automorphic form for short. For s ≥ 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 Φ 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 P in_{M}(R)). The conditions on Φ 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 Φ 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 P in_{M}(R). This is because the weight k indexes representations of an S^{1} subgroup of O_{M}(R)^{+} or P in_{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} → SO_{M} corresponds to the double cover SL_{2} → P GL_{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 P in_{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 ∩ U ⊗ R.) There is an obvious induced action of J on the lattices U and U^{⊥}/U = K, so we get a homomorphism from J to GO_{K} × 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 ∈ U and v ∈ u^{⊥}/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^{⊥}/u. If u, v is a positively oriented basis of U and r ∈ 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 ∈ U, v ∈ U^{⊥}. The automorphisms t_{u,v} for u ∈ U, v ∈ U^{⊥} 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 Φ is an automorphic function and J is a Jacobi group we define the Jacobi expansion of Φ as follows. We let t_{r} for r ∈ R be the elements of the center of the Jacobi group. We define ϕ_{m} for m ∈ Z by

ϕ_{m}(v) = ∫_{r∈R/Z} Φ(t_{r}(v))e^{2πimr}dr.

This is well defined because Φ(t_{r}(v) = Φ(v) for r ∈ Z. The Jacobi expansion of Φ is then

Φ = ∑_{m∈Z} ϕ_{m},

and the functions ϕ_{m} have the following properties.

1 ϕ_{m} is a homogeneous function of weight k.

2 ϕ_{m} is holomorphic on P.

3 ϕ_{m}(t_{r}(v)) = e^{2πimr}ϕ_{m}(v).

4 ϕ_{m} is invariant (or possibly antiinvariant) under the Jacobi group J(Z)^{+}.

5 ϕ_{m} is “holomorphic at the cusps” (at least if Φ is); the meaning of this is described below in the section on Fourier subgroups. Functions with these properties are called 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 ϕ_{m} is a Jacobi form we can analytically continue it to the space of all norm 0 vectors v in M ⊗ C such that ℑ((v, u_{1})/(v, u_{2})) > 0, where u_{1} and u_{2} is any oriented base of U, by saying that ϕ_{m}(t_{r}(v)) = e^{2πimr}ϕ_{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^{⊥}/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 ∈ L. Suppose that Φ is either an automorphic form or a Jacobi form of a 2-dimensional lattice containing u. Then Φ(t_{u,v}(w)) = Φ(w) for v ∈ L. We define A_{m} for m ∈ L′ by

A_{m}(w) = ∫_{v∈L⊗R/L} Φ(t_{u,v}(w))e^{2πimv}dv.

The Fourier expansion of Φ is then

Φ = ∑_{m∈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 Φ. We say that Φ 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 ⊗ 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 ϕ is a Jacobi form corresponding to some Jacobi group J, then we say that ϕ is holomorphic (at the cusps) if the Fourier expansion of ϕ at every cusp of J is holomorphic, i.e., if the Fourier expansion of ϕ at every Fourier subgroup F such that F ∩ 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^{⊥}/u so that L is a Lorentzian lattice, and we write L^{1}(R) for the space (vectors of M ⊗ 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 ∈ L^{1}(C) such that y is in the positive open cone of L ⊗ 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 ∈ M ⊗ 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}.

3. Classical theory

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 ⊕ II_{1,1}, M = L ⊕ II_{1,1}. We can write vectors of M in the form (κ, α, −δ, γ, β) with κ ∈ K, α, β, γ, δ ∈ Z, and this vector has norm κ^{2}/2 + αδ − γβ. 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 ⟨u_{1}, u_{2}⟩, 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 ⊗ 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 Φ_{u} on H_{u}, then we can expand Φ_{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 Φ_{u} that corresponds to the weight k automorphic form f being invariant under the transformation taking (v, γ, β) to (v, β, γ) (v ∈ L ⊗ C). By definition, Φ_{u}(v) = f(v, 1, v^{2}/2) and f(v, γ, β) = f(v, β, γ), and f(v, γ, β) = γ^{−k}Φ_{u}(v/γ) (for v^{2} = 2mn). From this it follows that

Φ_{u}(−2v/(v, v)) = ((v, v)/2)^{k}Φ_{u}(v).

Suppose now that ϕ is a Jacobi form of index m and weight k. We will show how to identify ϕ with a classical Jacobi form. The conditions on ϕ are

ϕ(z, α, −δ, γ, β) = r^{k}ϕ(rz, rα, −rδ, rγ, rβ)
= e^{−2πimr}ϕ(z, α, −δ + rγ, γ, β − rα)
= ϕ(z, aα + bγ, −dδ − cβ, cα + dγ, aβ + bδ)
= ϕ(z + αλ, α, −δ + (z, λ) + αλ^{2}/2, γ, β)
= ϕ(z + γμ, α, −δ, γ, β + (z, μ) + γμ^{2}/2)

for r ∈ R, λ ∈ K, μ ∈ K, z ∈ K ⊗ C, \binom{ab}{cd} ∈ SL_{2}(Z), α, β, γ, δ ∈ R.

We extend ϕ so that it is defined for all norm 0 vectors such that ℑ(α/γ) > 0 by insisting that ϕ should satisfy ϕ(z, α, −δ + rγ, γ, β − rα) = e^{2πimr}ϕ(z, α, −δ, γ, β) for all complex values of r. If we define ϕ_{u}(z, τ) for τ ∈ C,ℑ(τ) > 0, z ∈ K ⊗ C by

ϕ_{u}(z, τ) = ϕ(z, τ, 0, 1, z^{2}/2)

then we find that ϕ_{u} has the following properties.

\begin{aligned}ϕ_{u}(z/(cτ + d), (aτ + b)/(cτ + d)) & = (cτ + d)^{k}e^{2πimc(z^{2}/2)/(cτ+d)}ϕ_{u}(z, τ) \\ ϕ_{u}(z + λτ + μ, τ) & = e^{−2πim(zλ+τλ^{2}/2)}ϕ_{u}(z, τ) (λ, μ ∈ K). \end{aligned}

Conversely, if we are given ϕ_{u} with these properties and we define ϕ by

ϕ(z, α, −δ, γ, β) = ϕ_{u}(z/γ, α/γ)γ^{−k}e^{−2πimδ/γ}

then ϕ transforms like a Jacobi form. If K is a one dimensional lattice spanned by a vector of norm 2, then the relations for ϕ_{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 2πimcz should be 2πimcz^{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 ∈ Z to be a linear combination of functions of the form

θ_{K+r}(z, τ) = ∑_{λ∈K+r} q^{λ^{2}/2}ζ^{mλ}

where K is some positive definite rational lattice of dimension s and r ∈ K ⊗Q (q = e^{2πiτ}, ζ^{λ} = e^{2πi(z,λ)}). Any theta function is a holomorphic Jacobi form of singular weight.

Theorem 3.1. Any (nearly) holomorphic Jacobi form ϕ 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 ϕ).

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.

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.

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 ≥ 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.

4. Hecke operators for Jacobi groups.

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 ⊕ 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 ϕ is a Jacobi form for J(Z)^{+} we define the Hecke operator V_{ℓ} by

(ϕ|V_{ℓ})(v) = (1/ℓ) ∑_{g∈J_{1}\J_{ℓ}} ϕ(gv).

This operator maps Jacobi forms of weight k and index m for J_{1} to Jacobi forms of weight k and index mℓ. (The index gets multiplied by ℓ because the elements of J_{ℓ} act on Λ^{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/ℓZ)^{2}, so that

(ϕ|V_{ℓ,ℓ})(v) = (1/ℓ^{2})ϕ(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 mℓ^{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

\binom{ab}{cd}(z, α, −δ, γ, β) = (z, (aα + bγ)/ℓ, −dδ − cβ, (cα + dγ)/ℓ, bδ + aβ)

where ℓ = ad − bc. A set of right coset representatives of J_{1}\J_{ℓ} is then given by the usual set of matrices \binom{ab}{0d} with 0 ≤ b < d, ad = ℓ. Using this set of representatives we can calculate the relations between the Hecke operators and the Fourier expansion of f|T_{ℓ} in the same way as for ordinary modular forms, and we get the following results.

Theorem 4.1.

1 If ℓ and ℓ′ are coprime then V_{ℓ}V_{ℓ′} = V_{ℓℓ′}.

2 V_{ℓ,ℓ}V_{ℓ′,ℓ′} = V_{ℓℓ′,ℓℓ′}.

3 If p is prime then V_{p}V_{p^{n}} = V_{p^{n+1}} + pV_{p,p}V_{p^{n−1}}. (More generally, V_{ℓ}V_{ℓ′} =

∑_{d|(ℓ,ℓ′)} dV_{d,d}V_{ℓℓ′/d^{2}}.) 4 The operators of the form V_{ℓ} and V_{ℓ,ℓ} for ℓ ≥ 1 all commute with each other.

Proof. For signature 1 see [E-Z Theorem 4.2]; the proof in the general case is similar. (There is a misprint in equation (10) of [E-Z, Theorem 4.2]; the term V_{ℓ} ∘ U_{ℓ′} should be V_{ℓ} ∘ V_{ℓ′}.)

If we choose a norm 0 vector u and let ϕ_{u} be the classical Jacobi form associated to ϕ and u as in the previous section then we can calculate the effect of V_{ℓ} on the Fourier coefficients of ϕ_{u} as follows.

Theorem 4.2. If ϕ_{u}(z, τ) = ∑ c(r, n)q^{n}ζ^{r} then

ϕ_{u}|V_{ℓ,ℓ} = ℓ^{k−2} ∑_{r,n} c(r/ℓ, n)q^{n}ζ^{r}

and

ϕ_{u}|V_{ℓ} = ∑_{r,n} q^{n}ζ^{r} ∑_{a|(r,ℓ,n)} a^{k−1}c(r/a, nℓ/a^{2}).

Proof. This is a straightforward calculation using the standard set of coset representatives. Details for the case of signature 1 are given in the proof of [E-Z, Theorem 4.2], and the proof in the general case is similar.

5. Analytic continuation.

In this section we prove that certain infinite sums and products on L ⊗ C can be analytically continued to multivalued functions whose singularities and branch points (and zeros in the case of products) are known explicitly.

We let L be an (s + 2)-dimensional Lorentzian lattice (of signature s), and we let C be one of the two cones of negative norm vectors in L ⊗ R. We choose some vector in −C that is not orthogonal to any nonzero vector of L, and the expression x > 0 means that x has positive inner product with this vector.

We define a modular product to be an infinite product of the form

Φ(y) = e^{−2πi(ρ,y)} ∏_{x∈L,x>0}(1 − e^{−2πi(x,y)})^{c(x)}

where y ∈ L ⊗ C, ℑ(y) ∈ 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 rational quadratic divisor to be the set of points y with ℑ(y) ∈ C such that a(y, y) + (b, y) + c = 0 for some a ∈ Z, b ∈ L, c ∈ Z with (b, b) − 4ac > 0. If M = L ⊕ II_{1,1}, then the points y is some rational quadratic divisor are just the points (y, 1, y^{2}/2) ∈ M ⊗ 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.

Theorem 5.1. Any modular product Φ(y) converges to a holomorphic function whenever ℑ(y) is in C and has sufficiently large norm. This function can be analytically continued to a multivalued meromorphic function for all y with ℑ(y) ∈ 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 Φ(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 Φ). The complex number s is called the multiplicity of the zero of Φ along this rational quadratic divisor. The function Φ 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 Φ 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 Φ converges whenever ℑ(y) ∈ C, so that Φ is holomorphic and nonzero in this region. An example of this case is f(τ) = 12 ∑_{n∈Z} q^{n^{2}} and Φ(τ) = q ∏_{n>0}(1 − q^{n})^{24}.

Theorem 5.2. Suppose that k is a positive integer and

Φ(y) = ∑_{x∈L,x>0} A(x)e^{−2πi(x,v)}

where A(x) = ∑_{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 mod N for some fixed integer N. Then the sum for Φ converges whenever ℑ(y) ∈ C and −(ℑ(y), ℑ(y)) is sufficiently large, and can be analytically continued to a meromorphic function defined for all y with ℑ(y) ∈ 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(Φ(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(Φ(y)), which gives us all the singularities and zeros of Φ(y). Before giving the proof of theorem 5.1 we prove two preliminary lemmas.

Lemma 5.3. Suppose that f(τ) = ∑_{n∈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

(cτ + d)^{−k}f((aτ + b)/(cτ + d)) = ∑_{n∈Q} c_{a/c}(n)e^{2πinτ} .

Then for any positive number ϵ we can find a finite sum of terms of the series

∑_{m>0} ∑_{c>0} ∑_{0≤a<c,0≤d<c,c|(ad−1)} 2πc_{a/c}(−m)e^{2πi(an−md)/c}I_{1−k}(4π\sqrt{mn}/c)(m/n)^{(1−k)/2}/c

which differs from c(n) by at most O(e^{ϵ\sqrt{n}}). (I_{1−k}(z) is the modified Bessel function of the first kind; see [E 7.2.2].)

Proof. This is the Hardy-Ramanujan-Rademacher series for the coefficients of nearly holomorphic modular forms. The case when the form f has negative weight and level 1 is proved in [R] (and in this case the series converges absolutely to c(n)). The case when f has level greater than 1 is an easy generalization of the case when f has level 1. If f has weight 0, then Rademacher showed that the series still converges to c(n) provided they are added in the right order and n ≠ 0, which again implies lemma 5.3. If f has positive weight then a similar argument shows that the series we get is an asymptotic series for c(n) with the stated error term. (In this case the series does not usually converge to c(n).) This proves lemma 5.3.

Lemma 5.4. If m is positive, k is real and s ≥ −1 then the integral

I = 2π ∫_{x∈C} e^{−2πi(x,y)}I_{1−k}(4π\sqrt{m(−x, x)/2})((−x, x)/2)^{(k−1)/2}d^{s+2}x

converges if ℑ(y) ∈ C and −(ℑ(y), ℑ(y)) is sufficiently large, and can be analytically continued to a multivalued holomorphic function in the region of all y ∈ L ⊗ C with (y, y) ≠ 0 and (y, y) ≠ 2m.

In other words, the integral can be extended to a function which is (multivalued) holomorphic for ℑ(y) ∈ C apart from a singularity along a rational quadratic divisor.

Proof. We first evaluate the integral

∫_{x∈C} e^{−2πi(x,y)}f(\sqrt{(−x, x)})d^{s+2}x

where y ∈ iC, (y, y) is sufficiently large, and f is any continuous function defined for non negative real numbers which does not increase more than exponentially fast at infinity.

This integral is equal to

\frac{2π^{(s+1)/2}}{Γ((s + 1)/2)} ∫_{x=0}^{∞} ∫_{r=0}^{x} e^{−2πx\sqrt{(y,y)}}f(\sqrt{x^{2} − r^{2}})r^{s}drdx.

(The factor in front is the area of a sphere of radius 1 in s + 1-dimensional space.) If we put t^{2} = x^{2} − r^{2} and then change x to tx, we find that this integral is equal to

\begin{aligned}& \frac{2π^{(s+1)/2}}{Γ((s + 1)/2)} ∫_{t=0}^{∞} ∫_{x=t}^{∞} e^{−2πx\sqrt{(y,y)}}f(t)(x^{2} − t^{2})^{(s−1)/2}tdxdt \\ = & \frac{2π^{(s+1)/2}}{Γ((s + 1)/2)} ∫_{t=0}^{∞} ∫_{x=1}^{∞} e^{−2πxt\sqrt{(y,y)}}f(t)(x^{2} − 1)^{(s−1)/2}t^{s+1}dxdt. \end{aligned}

The integral over x can carried out explicitly using Gubler’s formula ([E vol. 2, 7.3.4, formula 15])

Γ(μ + 1/2)K_{μ}(z) = π^{1/2}(z/2)^{μ} ∫_{1}^{∞} e^{−zx}(x^{2} − 1)^{μ−1/2}dx

which is valid for ℜ(μ) > −1/2, ℜ(z) > 0, with z = 2πt\sqrt{(y, y)}, μ = s/2. (K_{μ} is the modified Bessel function of the third kind; [E vol. 2, 7.2.2].) We find that

∫_{x∈C} e^{−2πi(x,y)}f(\sqrt{(−x, x)})d^{s+2}x = 2 ∫_{t=0}^{∞} f(t)K_{s/2}(2πt\sqrt{(y, y)})t^{s/2+1}(y, y)^{−s/4}dt.

We now substitute f(t) = I_{1−k}(4πt\sqrt{m/2})t^{k−1} into this and find that

I = 4π2^{(1−k)/2} ∫_{t=0}^{∞} I_{1−k}(4πt\sqrt{m/2})K_{s/2}(2πt\sqrt{(y, y)})(y, y)^{−s/4}t^{s/2+k}dt.

By [E vol. 2, 7.14.2, formula 35]

2^{ρ+1}α^{ν+1−ρ}Γ(ν + 1) ∫_{0}^{∞} K_{μ}(αt)I_{ν}(βt)t^{−ρ}dt
=β^{ν}Γ(ν/2 − ρ/2 + μ/2 + 1/2)Γ(ν/2 − ρ/2 − μ/2 + 1/2)×
×_{2}F_{1}(ν/2 − ρ/2 + μ/2 + 1/2, ν/2 − ρ/2 − μ/2 + 1/2; ν + 1; β^{2}/α^{2})

whenever α > β, ℜ(ν − ρ + 1 ± μ) > 0. (_{2}F_{1} is the hypergeometric function [E volume 1, chapter II].) If we set μ = s/2, ν = 1 − k, β = 4π\sqrt{m/2}, α = 2π\sqrt{(y, y)}, ρ = −k − s/2 in this we find that

\begin{matrix}2^{1−k−s/2}(2π\sqrt{(y, y)})^{s/2+2}Γ(2 − k) ∫_{0}^{∞} K_{s/2}(2πt\sqrt{(y, y)})I_{1−k}(4πt\sqrt{m/2})t^{k+s/2}dt \\ = & (4π\sqrt{m/2})^{1−k}Γ(1 + s/2)_{2}F_{1}(1 + s/2, 1; 2 − k; 2m/(y, y)) \end{matrix}

so that

\begin{matrix}2π ∫_{0}^{∞} K_{s/2}(2πt\sqrt{(y, y)})I_{1−k}(4πt\sqrt{m/2})t^{k+s/2}dt \\ = & m^{(1−k)/2}(y, y)^{−1−s/4}2^{−(k+1)/2}_{2}F_{1}(1 + s/2, 1; 2 − k; 2m/(y, y))Γ(1 + s/2)/Γ(2 − k) \end{matrix}

We find that

I =2^{1−k}m^{(1−k)/2}(y, y)^{−1−s/2}_{2}F_{1}(1 + s/2, 1; 2 − k; 2m/(y, y))Γ(1 + s/2)/Γ(2 − k)

The hypergeometric function _{2}F_{1}(a, b; c; z) can be analytically continued whenever z is not 0, 1, or ∞, 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 Γ(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^{−2πi(ρ,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(τ). If we expand the coefficients using lemma 1 as a finite sum of Bessel functions plus a remainder term, then the estimate O(e^{ϵ\sqrt{n}}) for the remainder shows that the product using the remainder terms can be made to converge whenever ℑ(y) has norm at least δ for any given positive δ. It is also easy to check that the infinite product over the vectors x of negative or zero norm converges whenever ℑ(y) ∈ 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

∏_{x∈L+v,x∈C}(1 − e^{−2πi(x,y)})^{I_{1−k}(4π\sqrt{m(−x,x)/2})((−x,x)/2)^{(k−1)/2}}

(after replacing L by NL).

The logarithm of this is

− ∑_{x∈L+v,x∈C} ∑_{n>0} e^{−2πin(x,y)}I_{1−k}(4π\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 ℑ(y) is at least δ for any given positive constant δ, 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

∑_{x∈L+v,x∈C} e^{−2πi(x,y)}I_{1−k}(4π\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

∑_{x∈L,x∈C} e^{−2πi(x,y+r)}I_{1−k}(4π\sqrt{m(−x, x)/2})((−x, x)/2)^{(1−k)/2}

for some larger lattice L and some rational vectors r ∈ L ⊗ 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 ∈ L ⊗ Q, which proves theorem 5.1.

6. Vector systems and the Macdonald identities.

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 ∈ K which are zero for all but a finite number of vectors of K. We say that the function c is a vector system if it has the following 2 properties.

1. c(v) = c(−v).

2. The function taking λ to ∑_{v∈K} c(v)(λ, v)^{2} is constant on the sphere of norm 1 vectors λ ∈ K ⊗ 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 ∈ K, and we write ∑_{v∈V} f(v) instead of ∑_{v∈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.

Lemma 6.1. If G is any group acting on the lattice K that acts irreducibly on K ⊗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 ⊗ 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 ∑_{v∈V} (λ, 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 ⊗ R invariant under the G, contradicting the fact that G acts irreducibly on K ⊗ R. This proves lemma 6.1.

We define the index m of a vector system by

m = ∑_{v∈V} \frac{(v, v)}{2 dim(K)}.

Lemma 6.2. If V is a vector system and λ and μ are any vectors of K then

\begin{aligned}∑_{v∈V}(v, λ)(v, μ) & = 2m(λ, μ) \\ ∑_{v∈V} v(v, λ) & = 2mλ. \end{aligned}

Proof. If λ = μ the first line follows from axiom 2 by integrating λ over the unit sphere. The case for arbitrary λ and μ follows from the case for λ = μ by polarization. The second identity follows from the first because both sides are vectors having the same inner product with all vectors μ. This proves lemma 6.2.

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 (λ, μ) for λ ∈ K, μ ∈ K. The sum on the left of lemma 6.2 is divisible by 2n^{2} for any λ, μ ∈ K (the factor of 2 comes from the fact that if v ∈ V then −v ∈ V), so if we let λ and μ 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 ⊗ R into cones that we call the 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 Weyl vector ρ = ρ_{W} of W by

ρ = \frac{1}{2} ∑_{v∈V,v>0} v.

Lemma 6.4. If λ is in the dual of K, then 2(ρ, λ) ≡ m(λ, λ) mod 2, and in particular m(λ, λ) is integral. (Notice that the Weyl vectors for different Weyl chambers differ by elements of K, so that 2(ρ, λ) is well defined mod 2 independently of the choice of Weyl chamber.)

Proof.

(2ρ, λ) = ∑_{v>0}(v, λ)

≡ ∑_{v>0}(v, λ)^{2} mod 2

= m(λ, λ),

which proves lemma 6.4.

For example, if K is an even unimodular lattice, then this lemma shows that ρ ∈ 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) affine vector system of V to be the multiset of vectors (v, n) ∈ K ⊕ Z with v ∈ 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 ρ and positive vectors, and define

ψ(z, τ) = q^{d/24}ζ^{−ρ} ∏_{v∈V,n∈Z,(v,n)>0}(1 − q^{n}ζ^{v})

where q^{a} = e^{2πiaτ}, ζ^{v} = e^{2πi(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.

Theorem 6.5. The function ψ is a nearly holomorphic Jacobi form of weight k and index m. More precisely,

\begin{aligned}ψ(z, τ + 1) & = e^{2πid/24}ψ(z, τ) \\ ψ(z/τ, −1/τ) & = (−i)^{d/2−k}(τ/i)^{k}e^{2πim(z,z)/2τ} ψ(z, τ) \\ ψ(z + μ, τ) & = (−1)^{2(ρ,μ)}ψ(z, τ) \\ ψ(z + λτ, τ) & = (−1)^{2(ρ,λ)}q^{−m(λ,λ)/2}ζ^{−mλ}ψ(z, τ) \end{aligned}

for any λ, μ ∈ K′. The function ψ 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

ψ(z, τ + 1) = e^{2πid/24}ψ(z, τ)

because q^{d/24} is the only factor which is changed by adding 1 to τ. The only factor of ψ that changes under adding μ to z is ζ^{−ρ} which gets multiplied by (−1)^{2(μ,ρ)}, so

ψ(z + μ) = (−1)^{2(μ,ρ)}ψ(z).

For the transformation of adding λτ to z we first assume that λ is in the Weyl chamber and calculate as follows.

\begin{matrix}ψ(z + λτ, τ) = & q^{d/24}e^{−2πi(ρ,z+λτ)} ∏_{r,n}(1 − q^{n+(λ,v)}ζ^{v}) \\ = & ψ(z, τ)e^{−2πi(ρ,λ)τ} ∏_{v∈V,0<n≤−(λ,v)}(−ζ^{v}q^{n+(λ,v)}) \\ = & ψ(z, τ)e^{−2πi(ρ,λ)τ} ∏_{v<0}(−1)^{(λ,v)}ζ^{−v(λ,v)}q^{−(λ,v)((λ,v)+1)/2} \\ = & ψ(z, τ) ∏_{v<0}(−1)^{(λ,v)}ζ^{−v(λ,v)}q^{−(λ,v)(λ,v)/2} \\ = & ψ(z, τ)(−1)^{2(ρ,λ)}ζ^{−mλ}q^{−m(λ,λ)/2} \end{matrix}

where in the last step we use lemma 6.2 twice. If this is true for two values of λ then it is also true for their sum and difference. The vectors λ in the Weyl chamber generate the whole of K′, so this transformation law holds for all λ ∈ K′.

Next we consider the function

ϕ(z, τ) = ψ(z/τ, −1/τ)e^{−2πimz^{2}/2τ} /ψ(z, τ).

This function has no zeros or poles because the zeros of ψ(z, τ) (which are the divisors of points z with (v, z) in the lattice generated by 1 and τ) are the same as the zeros of ψ(z/τ, −1/τ) (and have the same multiplicities). The transformations we have just proved for ψ imply that ϕ(z, τ) = ϕ(z + λτ + μ, τ) for all λ, μ ∈ K. Hence for any fixed τ, ϕ(z, τ) is a holomorphic abelian function of z and is therefore constant. We will now work out ϕ(z, τ) by taking the limit of ϕ(z, τ) as z tends to 0. (We cannot set z = 0 because the numerator and the denominator of ϕ(z, τ) usually both vanish at z = 0.)

We define ψ_{0}(z, τ) by

ψ_{0}(z, τ) = ψ(z, τ)/ ∏_{v>0}(1 − ζ^{v}).

Then ψ_{0}(0, τ) = η(τ)^{d}, so that

ψ_{0}(0, −1/τ) = (τ/i)^{d/2}ψ_{0}(0, τ).

Also

ϕ(z, τ) = \frac{ψ_{0}(z/τ, −1/τ) ∏_{v>0}(1 − e^{2πi(v,z/τ)})}{ψ_{0}(z, τ) ∏_{v>0}(1 − e^{2πi(v,z)})e^{2πiz^{2}/2τ}}

so if we take the limit as z tends to 0 and use the fact that ϕ(z, τ) does not depend on z we find that

\begin{aligned}ϕ(z, τ) & = (τ/i)^{d/2}/ ∏_{v>0} τ \\ & = i^{k−d/2}(τ/i)^{k}. \end{aligned}

From the definition of ϕ, this is equivalent to the final transformation law for ψ. 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, λ)) ∈ K ⊕ Z as v runs through the vectors of some vector system and n runs through all integers, and λ 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 ψ(z, τ) as an infinite product over half the vectors in the affine vector system as above. The function ψ is then a nearly holomorphic Jacobi form for the congruence subgroup Γ_{0}(N) = \{\binom{ab}{cd} ∈ SL_{2}(Z)|c ≡ 0 mod 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 \{\binom{ab}{cd}|ad − bc = 1, a, nb, c/n, d ∈ 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 Γ_{0}(N). For affine root systems we always have 1 ≤ N ≤ 4.

7. The Weierstrass ℘ function.

In this section we prove some identities involving the Weierstrass ℘ 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 ℘ function is defined for ℑ(τ) > 0, z ∈ C by

℘(z, τ) = \frac{1}{z^{2}} + ∑_{(m,n)≠(0,0)} (\frac{1}{(z − mτ − n)^{2}} − \frac{1}{(mτ + n)^{2}} )

and satisfies the functional equations

\begin{aligned}℘(z + λτ + μ, τ) & = ℘(z, τ) \\ ℘ (\frac{z}{cτ + d}, \frac{aτ + b}{cτ + d}) & = (cτ + d)^{2}℘(z, τ). \end{aligned}

In other words ℘ 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}

\begin{aligned}E_{2}(τ) & = \frac{3}{π^{2}} ∑_{m} (∑_{n}^{′} \frac{1}{(mτ + n)^{2}} ) \\ & = 1 − 24 ∑_{n>0} σ_{1}(n)q^{n} \end{aligned}

where ∑′_{n} means we omit n = 0 if m = 0. This function satisfies the functional equation E_{2}((aτ +b)/(cτ +d)) = (cτ +d)^{2}E_{2}(τ)+12c(cτ +d)/2πi for \binom{ab}{cd} ∈ SL_{2}(Z). The Eisenstein series E_{k}(τ) for k even and k ≥ 4 is equal to 1 − (2k/B_{k}) ∑_{n>0} σ_{k−1}(n)q^{n} and satisfies the functional equation E_{k}((aτ + b)/(cτ + d)) = (cτ + d)^{k}E_{k}(τ) for \binom{ab}{cd} ∈ SL_{2}(Z).

By differentiating the partial fraction decomposition

\frac{1}{z} + ∑_{n≠0} (\frac{1}{z − n} + \frac{1}{n}) = π cot(πz) = −πi − 2πi ∑_{n>0} e^{2πinz}

(valid for ℑ(z) > 0) we find that

∑_{n∈Z} \frac{1}{(z + n)^{2}} = (2πi)^{2} ∑_{n>0} ne^{±2πinz}

(valid for z not real), where e^{±x} means e^{x} if |e^{x}| < 1 and e^{−x} if |e^{x}| > 1. From this we see that

℘(z, τ) = (2πi)^{2} ∑_{m∈Z} ∑_{n>0} ne^{±2πin(z+mτ)} − \frac{π^{2}}{3} E_{2}(τ)

whenever ℑ(z + mτ) is nonzero for all integers m.

By differentiating repeatedly with respect to z we find that for positive even integers k the derivatives ℘^{(k−2)}(z, τ) = \frac{d}{dz}^{k−2}℘(z, τ) satisfy

\begin{aligned}℘^{(k−2)}(z + λτ + μ, τ) & = ℘^{(k−2)}(z, τ) \\ ℘^{(k−2)} (\frac{z}{cτ + d}, \frac{aτ + b}{cτ + d}) & = (cτ + d)^{k}℘^{(k−2)}(z, τ) \\ ℘^{(k−2)}(z, τ) & = (2πi)^{k} ∑_{m∈Z} ∑_{n>0} n^{k−1}e^{±2πin(z+mτ)} − δ_{k}^{2}E_{2}(τ)π^{2}/3 \end{aligned}

(where δ_{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 ∈ K such that c(r) = c(−r) and c(r) = 0 for all but a finite number of r ∈ K. Choose a vector ρ not orthogonal to any r with c(r) ≠ 0, and say that the pair (r, n) ∈ K ⊕ Z is positive if n > 0 or n = 0 and (r, ρ) > 0.

Theorem 7.1. Suppose that k is an even positive integer and suppose that if k = 2 then ∑_{r∈K} c(r) = 0. Then the function

ψ(z, τ) = −c(0)B_{k}/2k + ∑_{(r,n)>0} ∑_{a|(r,n)} a^{k−1}c(r/a)q^{n}ζ^{r}

can be extended to a meromorphic function defined for ℑ(τ) > 0, z ∈ K ⊗C which satisfies the functional equations

\begin{aligned}ψ (\frac{z}{cτ + d}, \frac{aτ + b}{cτ + d}) & = (cτ + d)^{k}ψ(z, τ) \\ ψ(z + λτ + μ, τ) & = ψ(z, τ) \end{aligned}

for λ, μ ∈ K′, \binom{ab}{cd} ∈ SL_{2}(Z). In other words, ψ is a meromorphic Jacobi form of weight k and index 0.

Proof. The function ψ is equal to

−c(0)\frac{B_{k}}{2k} E_{k}(τ) + (2πi)^{−k} ∑_{r>0} c(r) (℘^{(k−2)}((r, z), τ) + δ_{k}^{2}E_{2}(τ)π^{2}/3)

so theorem 7.1 follows from the functional equations of the derivatives of the Weierstrass ℘ function and the Eisenstein series E_{k}(τ). (If k = 2 then the assumption ∑_{r∈K} c(r) = 0 implies that the coefficient of E_{2}(τ) is 0.)

8. Generators for O_{M}(Z)^{+}.

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

Theorem 8.1. O_{M}(Z)^{+} is generated by F (Z)^{+} and J(Z)^{+}.

We first prove two lemmas.

Lemma 8.2. If L = II_{s+1,1}, v ∈ L ⊗ R and v ∉ L then there is some λ ∈ L with 1/2 ≤ (v − λ)^{2} ≤ 3/2.

Proof. Choose a primitive norm 0 vector u_{1} ∈ 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} ∈ L with (u_{1}, u_{2}) = −1. We can find an integer m so that 0 < |(v − mu_{2}, u_{1})| ≤ 1/2. But then (v − mu_{2} − nu_{1})^{2} = A + Bn for some fixed A and B with 0 < |B| ≤ 1, so we can choose n so that 1/2 ≤ A + nB ≤ 3/2. This proves lemma 8.2.

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) ≥ 0 then it is easy to find an element g ∈ 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 ⊕ 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 λ ∈ L with 1/2 ≤ (λ − v/m)^{2} ≤ 3/2. Then a simple calculation shows that if g is the reflection of the vector (λ, 1, λ^{2}/2 − 1) then 0 < m/4 ≤ −(g(u_{1}), u) ≤ 3m/4 < m = −(u_{1}, u), so by induction u_{1} is conjugate to u under the group G. This proves lemma 8.3.

Now we prove theorem 8.1. We let G be the group generated by F (Z)^{+} and J(Z)^{+}. We choose coordinates M = K ⊕ II_{1,1} ⊕ II_{1,1} so that u = (0, 0, 0, 0, 1) and U is the set of vectors of the form (0, 0, ∗, 0, ∗). The reflection of (0, 1, −1, 0, 0) is in F (Z)^{+}, and is conjugate under J(Z)^{+} to the reflection of (0, 0, 0, 1, −1) so that G contains the reflection of at least one root having inner product −1 with u. However the normal abelian subgroup of F (Z)^{+} acts simply transitively on the set of such roots, so G contains the reflections of all roots r with (r, u) = −1. By lemma 8.3 G therefore acts transitively on the primitive norm 0 vectors of M. As it contains the stabilizer F (Z)^{+} of one such vector in O_{M}(Z)^{+} it must therefore be the whole of O_{M}(Z)^{+}. This proves theorem 8.1.

9. The positive weight case.

In this section we construct some examples of meromorphic automorphic forms whose poles are known explicitly, which can be used to construct holomorphic automorphic forms by multiplying them by other automorphic forms with zeros which cancel out the poles. The results of this section are not used elsewhere in this paper except in a few examples.

We recall that K is an even unimodular lattice of dimension s, and we let θ_{K} be the theta function of K of index 1, defined by

θ_{K}(z, τ) = ∑_{λ∈K} q^{(λ,λ)/2}ζ^{λ}.

This is a holomorphic Jacobi form of index 1 and singular weight s/2.

Theorem 9.1. The linear map taking f(τ) to f(τ)θ(z, τ) is an isomorphism from nearly holomorphic modular forms of weight k for SL_{2}(Z) to nearly holomorphic Jacobi forms for J(Z)^{+} of weight k + s/2 and index 1.

We omit the proof of this as it is similar to the proof of theorem 3.5 in [E-Z], but simpler because the lattice L is even and unimodular so we only need one generator θ_{K} for the space of Jacobi forms of index 1 as a module over the ring of modular forms. This theorem is also proved in Gritsenko [G].

Lemma 9.2. If f(τ) is a nearly holomorphic modular form for SL_{2}(Z) of weight 2 then it is the derivative of a modular function, and in particular has zero constant term.

Proof. The total number of zeros of f in a fundamental domain is 1/6 as f has weight 2 (where poles are counted as zeros of negative order), so f must have zeros of order at least 2 at conjugates of (1 + \sqrt{3}i)/2 and a zero at i. Hence f is divisible by j′(τ). But f/j′(τ) is then a nearly holomorphic modular function and therefore a polynomial in j(τ), which easily implies that f is the derivative of a polynomial in j(τ). This proves lemma 9.2.

Theorem 9.3. The map taking

ϕ(z, τ) = ∑_{r,n} c(r, n)q^{n}ζ^{r}

to Φ(z, τ, σ) = ∑_{m≥0} p^{m}(ϕ|V_{m})(z, τ) (where p = e^{2πiσ}, q = e^{2πiτ}, and ζ = e^{2πiz}) takes nearly holomorphic Jacobi forms ϕ of level 1 and weight k > 0 to meromorphic automorphic forms Φ 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

(ϕ|V_{0})(z, τ) = −c(0, 0)B_{k}/2k + ∑_{(r,n)>0} ∑_{d|(r,n)} d^{k−1}c(r, 0)q^{n}ζ^{r})

which is equal to (−c(0, 0)B_{k}/2k)E_{k}(τ) if ϕ is holomorphic. The map from ϕ to Φ is an isomorphism from the space of holomorphic Jacobi forms ϕ of level 1 and weight k to the “Spezialschar” of all holomorphic automorphic forms Φ = ∑_{r,m,n} A(r, m, n)ζ^{r}p^{m}q^{n} of weight k whose coefficients satisfy

A(r, m, n) = ∑_{d|(r,m,n)} d^{k−1}A(r/d, 1, mn/d^{2}).

Proof. When ϕ 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 Φ 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 (λ, m, n) to (λ, n, m) because the formula for A(λ, 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 ∑_{m≥0} p^{m}(ϕ|V_{m})(z, τ) is an automorphic form.

When ϕ has poles at the cusps the proof is similar except that we need the following extra arguments. Firstly, the series for Φ does not converge everywhere, so we need to use theorem 5.2 to show that the series for Φ can be analytically continued. Secondly, ϕ|V_{0} is no longer a modular form, so we need to use theorem 7.1 to show that Φ|V_{0} is a nearly holomorphic Jacobi form. If k = 2 then the condition ∑_{r∈K} c(r) = 0 is satisfied by lemma 9.2, because it is the constant coefficient of the nearly holomorphic weight 2 form ϕ(0, τ). 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(λ, m, n) are given by A(λ, m, n) = σ_{s/2−1}(d) if λ^{2} = 2mn and the highest common factor of λ, m, and n is d (where σ_{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 ϕ is not holomorphic the meromorphic automorphic form Φ will have poles along rational quadratic divisors. We can remove these poles by multiplying Φ 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 Φ 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 Φ has weight 14 and the singularities of Φ are poles of order 2 along all rational quadratic divisors of discriminant 2. In section 10 we will find a holomorphic automorphic form Φ_{1} which has weight 12 and has a zero of order 1 along every rational quadratic divisor of discriminant 2. Hence ΦΦ_{1}^{2} is a holomorphic automorphic form of weight 38 with the same zeros as Φ.

10. The zero weight case

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 ⊕ II_{1,1}. We choose a negative norm vector in L ⊗ R and write r > 0 to mean that r has positive inner product with this negative norm vector. We will prove

Theorem 10.1. Suppose that f(τ) = ∑_{n} c(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 ρ ∈ L such that

Φ(v) = e^{−2πi(ρ,v)} ∏_{r>0}(1 − e^{−2πi(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 Φ lie on rational quadratic divisors, and the multiplicity of the zero of Φ at the rational quadratic divisor of the primitive positive norm vector r ∈ M (see section 5) is

∑_{n>0} c(−n^{2}(r, r)/2).

In particular if this number is always nonnegative then Φ is holomorphic.

Proof. We write L = K ⊕ II_{1,1} where K is the lattice E_{8}^{s/8}. We let ϕ(τ, z) be the nearly holomorphic Jacobi form f(τ)θ_{K}(z, τ). We define a vector system on E_{8}^{s/8} to be the multiset of vectors v ∈ 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) ∈ K ⊕ Z with multiplicities c((v, n)) = c(−(v, v)/2). By theorem 6.5 the function ψ(z, τ) associated to V satisfies the following functional equations

\begin{aligned}ψ(z, τ + 1) & = ψ(z, τ) \\ ψ(z/τ, −1/τ) & = τ^{k}e^{2πim(z,z)/2τ} ψ(z, τ) \\ ψ(z + μ, τ) & = ψ(z, τ) \\ ψ(z + λτ, τ) & = q^{−m(λ,λ)/2}ζ^{−mλ}ψ(z, τ) \end{aligned}

for any λ, μ ∈ K. (This follows because the integer k = s/2 is divisible by 4 and d is divisible by 24, and ρ ∈ K by the remark after lemma 6.4.) In particular ψ is a Jacobi form of weight k and index m for J(Z)^{+}, and therefore

p^{m}ψ(z, τ)

transforms like an automorphic form of weight k under all elements of J(Z)^{+}. On the other hand ϕ|V_{ℓ} is a Jacobi form of weight 0 and index ℓ, so that

exp(∑_{ℓ>0} p^{ℓ}(ϕ|V_{ℓ})(z, τ))

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

Φ(z, τ, σ) = p^{m}ψ(z, τ) exp(∑_{ℓ>0} p^{ℓ}(ϕ|V_{ℓ})(z, τ))

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 Φ 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 Φ 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 Φ. We first check that it is invariant under the element g_{1} taking (z, α, −δ, γ, β) to (z, −δ, α, γ, β). Under this transformation the factor e^{−2πi(ρ,v)} of ψ is multiplied by

e^{2πi(ρ−g_{1}(ρ),v)},

and the factor ∏(1 − e^{−2πi(r,v)})^{c(−(r,r)/2)} of ψ is multiplied by a factor of ∏_{r>0,g_{1}(r)<0} −e^{2πic(−(r,r)/2)(r,v)}. Hence to prove (anti)invariance under g_{1} we have to show that

ρ − g_{1}(ρ) − ∑_{r>0,g_{1}(r)<0} c(−(r, r)/2)r = 0.

Before proceeding further with the proof of invariance of Φ under g_{1} we need to calculate the Weyl vector ρ and check some of its properties, which we do in 10.2 to 10.7.

Lemma 10.2. If f(τ), g(τ) 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(τ)g′(τ) vanishes.

Proof. The SL_{2}(Z) invariant differential form f(τ)g′(τ)dτ has only one pole on the compactification of the upper half plane modulo SL_{2}(Z) (which is at the cusp i∞) and therefore its residue there must vanish. But its residue is just the constant term of f(τ)g′(τ). This proves lemma 10.2.

Lemma 10.3. Suppose that θ(τ) is a nearly holomorphic modular form of weight s/2 and f(τ) is a nearly holomorphic modular form of weight −s/2 (both of level 1). Then the constant term of the q expansion of

sθ(τ)f(τ)E_{2}(τ)/24 − θ′(τ)f(τ)

is zero.

Proof. This follows by applying lemma 10.2 to the modular functions f(τ)η(τ)^{s} and θ(τ)η(τ)^{−s}, since η′(τ)/η(τ) = E_{2}(τ)/24. (Alternatively we can observe that θ′(τ) − sθ(τ)E_{2}(τ)/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.

Theorem 10.4. The Weyl vector ρ is equal to

( ∑_{(r,v)>0} c(−r^{2}/2)r/2, m, d/24)

where m is the constant coefficient of θ_{K}(τ)f(τ)E_{2}(τ)/24, and d is the constant coefficient of θ_{K}(τ)f(τ).

Proof. The Weyl vector is (ρ_{K}, m, d/24) where ρ_{K}, m, and d are as in section 6. The formulas for ρ_{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 θ′_{K}(τ)f(τ)/s, so we have to show that the constant term of

θ′_{K}(τ)f(τ)/s − θ_{K}(τ)f(τ)E_{2}(τ)/24

vanishes. But this follows from lemma 10.3. This proves theorem 10.4.

Now we check that the Weyl vector ρ lies in L (and not just L ⊗ Q).

Lemma 10.5. For any nonzero integer n the constant term of Δ(τ)^{n} is divisible by 24.

Proof. Δ′(τ)/Δ(τ) = 1 − 24 ∑_{m>0} σ_{1}(m)q^{m} is congruent to 1 mod 24, so
(Δ(τ)^{n})′ = nΔ(τ)^{n−1}Δ′(τ) ≡ nΔ(τ)^{n} mod 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 ≠ 0.

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 Δ^{m}E_{4}^{n} with m < 0, and lemma 10.6 then follows from lemma 10.5 and the fact that E_{4} ≡ 1 mod 24.

Corollary 10.7. The Weyl vector ρ_{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 ρ_{K} = ∑_{(r,v)>0} c(−r^{2}/2)r/2 lies in K. We know that ρ_{K} ∈ K by the remark after lemma 6.4. The constant term d of f(τ)θ_{K}(τ) is divisible by 24 by lemma 10.6 and the fact that θ_{K}(τ) = E_{4}(τ)^{s/8} ≡ 1 mod 24, so d/24 is integral. Also, E_{2} ≡ 1 mod 24, so m ≡ d/24 mod 1 is an integer. This proves corollary 10.7.

We now continue with the proof that Φ is invariant under g_{1}, which was interrupted by the calculation of ρ. We know that ρ = (ρ_{K}, m, d/24) by theorem 10.4, so that g_{1}(ρ) = (ρ_{K}, d/24, m). The vector (κ, a, b) is positive if a > 0, or a = 0, b > 0, or a = b = 0, κ > 0, so that (κ, a, b) > 0 and g_{1}(κ, a, b) = (κ, b, a) < 0 if and only if a > 0, b < 0. Hence

∑_{r>0,g_{1}(r)<0} c(−(r, r)/2)r = ∑_{κ} ∑_{a>0} ∑_{b<0} c(ab − (κ, κ)/2)(κ, a, b)

= ∑_{κ} ∑_{n>0} ∑_{a|n} c(−n − (κ, κ)/2)(0, a, −a)

= ∑_{κ} ∑_{n>0} c(−n − (κ, κ)/2)(0, σ_{1}(n), −σ_{1}(n))

= (0, x, −x)

where x is the constant term of θ_{K}(τ)f(τ) ∑_{n>0} σ_{1}(n)q^{n}. By theorem 10.4 x = m − d/24 (as E_{2}(τ) = 1 − 24 ∑_{n>0} σ_{1}(n)q^{n}). By comparing this with the expression for ρ we see that we have proved that

ρ − g_{1}(ρ) − ∑_{r>0,g_{1}(r)<0} c(−(r, r)/2)r = 0.

This completes the proof that Φ is (anti)invariant under g_{1}.

Next we see that it transforms like an automorphic form under the element g_{2} taking (z, α, −δ, γ, β) to (z, α, −δ, β, γ), because this is conjugate to g_{1} under an element of the Jacobi group. This element g_{2} acts as v → 2v/(v, v) on L⊗C, and in particular commutes with of O_{L}(Z)^{+}. If g_{3} is any element of O_{L}(Z)^{+} then Φ(g_{3}(v)) = e^{2πi(λ,v)}Φ(v) for some λ depending on g_{3} because of the expression for Φ as an infinite product. We now see that

((v, v)/2)^{k}e^{2πi(λ,v)}Φ(v)
=((g_{3}(v), g_{3}(v))/2)^{k}Φ(g_{3}(v))
=Φ(g_{2}(g_{3}(v)))
=Φ(g_{3}(g_{2}(v)))
=e^{2πi(λ,g_{2}(v))}Φ(g_{2}(v))
=e^{2πi(λ,2v/(v,v))}((v, v)/2)^{k}Φ(v)

so that

e^{2πi(λ,v)} = e^{2πi(λ,2v/(v,v))}

for all v, which implies that λ = 0 and hence that Φ is invariant or antiinvariant under O_{L}(Z)^{+}.

We have shown that Φ transforms correctly under both F (Z)^{+} and J(Z)^{+}, so by theorem 8.1 Φ transforms like an automorphic form under the whole of O_{M}(Z)^{+}.

Next we have to find the singularities and zeros of Φ, which we know lie on rational quadratic divisors by theorem 5.1. If v ∈ 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 Φ 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 Φ along the divisor of r is just

∑_{n>0} c(−(nr, nr)/2),

coming from the factors

∏_{n>0}(1 − e^{−2πi(nr,v)})^{c(−(nr,nr)/2)}

in the infinite product for Φ. This completes the proof of theorem 10.1.

Example 1. If we take L to be II_{1,1} and f to be j(τ)−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/Δ(τ). Then we find that Φ 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 Φ. By the Koecher boundedness principle the quotient is an invariant automorphic form. Hence multiplication by Φ 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 Φ 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 Φ 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

Φ(v) = e^{−2πi(ρ,v)} ∏_{r>0}(1 − e^{−2πi(r,v)})^{p_{24}(1−r^{2}/2)} = ∑_{w∈W} ∑_{n>0} det(w)τ(n)e^{−2πin(w(ρ),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.

11. The negative weight case.

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 = ∑ 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 Φ 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/dτ)^{1−k}f(τ).

We would expect the Fourier coefficients A(v) (v ∈ L) of Φ to look something like

A(v) = ∑_{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

∑_{d|v} d^{k−1}(−(v, v)/2)^{1−k}c(−(v, v)/2d^{2}).

On the other hand, if we apply the operator \frac{d}{dτ} ^{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 ≤ 0 is a meromorphic modular form of weight 2 − k. The automorphic form associated to \frac{d}{dτ} ^{1−k}f in theorem 9.3 has Fourier coefficients

∑_{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 Φ we do seem to get a meromorphic automorphic form (assuming we can define Φ), but the bad news is that this is not a new automorphic form.

We can of course still apply powers of the Laplacian to Φ if f has weight ≥ −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 Φ 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.

12. Invariant modular products

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

Φ(y) = e^{−2πi(ρ,y)} ∏_{x>0}(1 − e^{−2πi(x,y)})^{c(x)}

is a modular product for some Lorentzian lattice L which defines a holomorphic automorphic form. In particular Φ 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) ≠ 0 divide up the cone C into closed chambers that we will call the Weyl chambers of Φ. If W is a Weyl chamber of Φ, we define the Weyl vector ρ_{W} of W by

Φ(y) = ±e^{−2πi(ρ_{W} ,y)} ∏_{(x,−W)>0}(1 − e^{−2πi(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 Φ.

1 If g ∈ G then ρ_{g(W)} = g(ρ_{W}).

2 If W_{1} and W_{2} are Weyl chambers then

ρ_{W_{1}} = ρ_{W_{2}} + ∑_{(x,−W_{1})<0,(x,−W_{2})>0} c(x)x.

3 Any Weyl vector has coefficient ±1 in the Fourier expansion of Φ. 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.

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 ρ_{W} lies in the interior of W if ρ_{W} has negative norm, and on the boundary if it has zero norm. In either case, W is the only Weyl chamber containing ρ_{W}. (If ρ_{W} does not have maximal norm I do not know whether or not it necessarily lies in W.)

5 If W_{1} and W_{2} are 2 adjacent chambers separated only by the hyperplane x^{⊥} with (x, W_{1}) > 0, and x is a root of G, then (ρ_{W_{1}}, x) = ∑_{r∈Q,r>0} c(rx)(rx, rx)/2. This follows by applying property 1 with g equal to reflection in x^{⊥}. In particular if c(rx) is 1 when r = 1 and 0 otherwise then (ρ_{W} , x) = −(x, x)/2, so that ρ_{W} does indeed behave like a Weyl vector with respect to the simple root x.

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/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 Φ. 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 Φ(y) be the automorphic form

Φ(y) = e^{−2πi(ρ_{W} ,y)} ∏_{x>0}(1 − e^{−2πi(x,y)})^{c(x)}

where c(x) is the coefficient of q^{−(x,x)/2} in some nearly holomorphic modular form f(τ) = ∑_{n} c(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 ⊕ II_{1,1}, so that vectors of II_{s+1,1} can be written in the form (v, m, n) with v ∈ K, m, n ∈ Z, and (v, m, n)^{2} = v^{2} − 2mn. We choose the vector u to be (0, 0, 1). We choose a vector v ∈ 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).

Theorem 12.1. If 24|s and s > 0 then the constant term of θ_{K}(τ)/Δ(τ)^{s/24} is divisible by 24.

Proof. We take f to be Δ^{−s/24}. By theorem 10.4 the constant term of θ_{K}(τ)/Δ(τ)^{s/24} is equal to d = 24(ρ, u_{1}) where ρ is the Weyl vector of the Weyl chamber containing u and u_{1} = (0, −1, 0). By corollary 10.7, ρ ∈ L so the inner product (ρ, u) is an integer, and this proves theorem 12.1.

For example, when s = 24 and f(τ) = Δ(τ)^{−1} = q^{−1} + 24 + …, 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 Φ(y) is a cusp form if and only if there are no Weyl vectors of zero norm, which is true unless f(τ) 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:

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 ⊗ 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 Φ for the group O_{L}(Z)^{+} in theorem 10.1 for L = II_{25,1} and f(τ) = 1/Δ(τ) = q^{−1} + 24 + 324q + . . .. 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 ρ 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}.

13. Heights of vectors.

Suppose that we have fixed an automorphic form Φ 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 height of a vector λ in the positive cone of L = II_{s+1,1} to be the inner product −(ρ, λ) where ρ is the Weyl vector of any Weyl chamber containing λ. 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 λ has norm −2 or −2p for p a prime. We do this by looking at the restriction of f to multiples τλ of λ. This is a modular form in τ for the group Γ_{0}(p) = \{\binom{ab}{cd} ∈ SL_{2}(Z)|c ≡ 0 mod 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 λ.

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(τ) = ∑_{n} c(n)q^{n}, where f is the nearly holomorphic modular form of weight −s/2 from which Φ is constructed as in section 10.

We define an isomorphism from (L ⊕ II_{1,1}) ⊗ R to v^{⊥} ⊗ R ⊕ sl_{2}(R) which takes (λ, m, n) to

λ̄ ⊕ ( \begin{matrix}−(λ, v) & 2n \\ 2mN & (λ, v) \end{matrix} )

where λ̄ is the projection of λ into v^{⊥} and sl_{2}(R) is the set of real 2 × 2 matrices of trace 0. We let the group SL_{2}(R) act on sl_{2}(R) by conjugation, and on v^{⊥} by the trivial action, which induces an action of SL_{2}(R) on (L ⊕ II_{1,1}) ⊗ R. This action is given by

\begin{matrix}\binom{ab}{cd}(λ, m, n) = \\ (λ + (bdm − b\frac{c}{N} (λ, v) − a\frac{c}{N} n)v, d^{2}m − \frac{c}{N} d(λ, v) − \frac{c^{2}}{N} n, ab(λ, v) + a^{2}n − b^{2}mN) . \end{matrix}

In particular, if \binom{ab}{cd} ∈ Γ_{0}(N) (so that N|c) then this maps L ⊕ II_{1,1} into L ⊕ II_{1,1}. This defines an action of Γ_{0}(N) on L ⊕ II_{1,1}, and hence an action on the Hermitian symmetric space H.

We embed the upper half plane into H by mapping τ to τv, which is represented by the point (τv, 1, −τ^{2}N) ∈ M ⊗ C. The action of Γ_{0}(N) on H restricts to the usual action \binom{ab}{cd}(τ) = (aτ + b)/(cτ + 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 Γ_{0}(N).

The restriction of Φ to multiples of v will often be identically 0. We define Φ_{0} to be Φ divided by all the factors in the product defining Φ which are identically zero on multiples of v, so that

Φ_{0}(y) = e^{−2πi(ρ,y)} ∏_{x∈L,x>0,(x,v)≠0}(1 − e^{−2πi(x,y)})^{c(x)}.

We define H_{N,j}(−D) to be the number of complex numbers τ in a fundamental domain of Γ_{0}(N) satisfying a nonzero quadratic equation of the form aτ^{2} + bτ + c = 0 with a¯,c ∈ Z, N|a, b ≡ j mod 2N, and b^{2} − 4ac = D < 0 if D < 0, and define H_{N,j}(0) to be −|SL_{2}(Z)/Γ_{0}(N)|/12 if j ≡ 0 mod 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}(τ) by

H_{N,j}(τ) = ∑_{n} H_{N,j}(n)q^{n}.

We put H_{N}(n) = ∑_{j mod 2N} H_{N,j}(n) and H_{N}(n) = ∑_{j mod 2N} H_{N,j}(n). For example, the first few functions are

\begin{aligned}H_{1}(τ) & = −1/12 + (1/3)q^{3} + (1/2)q^{4} + q^{7} + q^{8} + q^{11} + (4/3)q^{12} + ⋯ \\ H_{2}(τ) & = −1/4 + (1/2)q^{4} + 2q^{7} + q^{8} + 2q^{12} + ⋯ \\ H_{3}(τ) & = −1/3 + (1/3)q^{3} + 2q^{8} + 2q^{11} + (4/3)q^{12} + ⋯ \end{aligned}

Theorem 13.1. Suppose v is a vector of norm −2N < 0 and let Φ_{0}(vτ) the function Φ_{0} defined above, restricted to the multiples vτ of v for τ ∈ C, ℑ(τ) > 0. Then Φ_{0}(vτ) is a modular form for Γ_{0}(N) of weight k_{0} equal to the constant term of f(τ)θ_{v^{⊥}}(τ). If τ is a root of an equation aτ^{2} + bτ + c = 0 with a/N, b, c ∈ Z, (a/N, b, c) = 1, b^{2} − 4ac = D < 0, then the order of the zero of Φ_{0}(τv) is

∑_{d>0} ∑_{(λ,v)=db} c(d^{2}D/4N − λ̄^{2}/2).

The sum of the orders of the zeros of Φ_{0}(τv) at the cusps of a fundamental domain of Γ_{0}(N) is the constant term of

− ∑_{b mod 2N} f(τ)θ_{v^{⊥},b}H_{N,b}(τ/4N).

Under the Fricke involution τ → −1/Nτ Φ_{0}(τv) transforms as

Φ_{0}(−v/Nτ) = ±(\sqrt{N}τ)^{k_{0}}Φ_{0}(τv).

Proof. Under the group Γ_{0}(N), direct calculation shows that the function Φ transforms as

Φ (\frac{−2Nȳ + (ay + bv, cy + dv)v}{(cy + dv, cy + dv)} ) = ± (\frac{(cy + dv, cy + dv)}{−2N})^{k} Φ(y).

From this we find that the function Φ_{0}(τv) transforms as

Φ_{0}(v(aτ + b)/(cτ + d)) = ±(cτ + d)^{2k+2 ∑_{x>0,(x,v)=0} c(−(x,x)/2)}Φ_{0}(vτ)

for \binom{ab}{cd} ∈ Γ_{0}(N), so that Φ_{0}(τv) is a nonzero modular form for Γ_{0}(N) of weight equal to

2k + 2 ∑_{x>0,(x,v)=0} c(−(x, x)/2) = ∑_{(x,v)=0} c(−(x, x)/2)

which is the constant term of f(τ)θ_{v^{⊥}}(τ). This follows because Φ_{0}(τv) is essentially the first nonvanishing coefficient in the Taylor series expansion of Φ orthogonal to Cv and is obtained by differentiating Φ c(−(x, x)/2) times for each positive norm vector x orthogonal to v, and each differentiation contributes 2 to the weight of Φ_{0}(τv).

We have to calculate the zeros of Φ_{0}(τv). The zeros of Φ_{0}(y) are the divisors of positive norm vectors (λ, m, n) of M, each with multiplicity c(−λ^{2}/2 + mn). We have to remember only to count the zeros from one of the vectors (λ, m, n) and −(λ, m, n), and also remember to count the zeros from positive multiples of (λ, m, n). This gives a contribution of c(−λ^{2}/2 + 2mn) to the zero of Φ_{0}(τv) at τ whenever ((vτ, 1, v^{2}τ^{2}/2), (λ, m, n)) = 0, or in other words when mNτ^{2} + (v, λ)τ − n = 0. In particular τ must be an imaginary quadratic irrational. Suppose that aτ^{2} + bτ + c = 0 with a/N, b, c ∈ Z, (a/N, b, c) = 1, b^{2} − 4ac = D. Then we must have m = da, (v, λ) = db, −n = dc for some integer d. But then d^{2}D/4N = (v, λ)^{2}/4N + mn = λ̄^{2}/2 + (mn − λ^{2}/2) where λ̄ = λ − v(λ, v)/(v, v) is the projection of λ into v^{⊥}. Hence the multiplicity of the zero of Φ_{0}(τv) at τ is

∑_{d>0} ∑_{λ,(λ,v)=bd} c(d^{2}D/4N − λ̄^{2}/2).

As Φ_{0}(τv) is a modular form for Γ_{0}(N), the total number of zeros in a fundamental domain is equal to |SL_{2}(Z)/Γ_{0}(N)|/12 times its weight, which is the constant term of −H_{N,b} times the constant term of f(τ)θ_{v^{⊥}}(τ). Hence the number of zeros at all the cusps is this number minus the number of complex zeros of Φ_{0}(τv) in a fundamental domain. We have just worked out the multiplicity of a zero at any complex number τ, 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 Φ_{0}(τv) under the Fricke involution follows from the formula Φ(2y/(y, y)) = ±((y, y)/2)^{k}Φ(y). This proves theorem 13.1.

Corollary 13.2. Suppose that v is a vector of norm −2N ≤ 0.

If N = 0 and v is primitive then the height of v is the constant term of

θ_{v^{⊥}/v}(τ)f(τ)E_{2}(τ)/24.

If N = 1 then the height of v is the constant term of

−θ_{v^{⊥}′}(τ)f(τ)H(τ).

If N is prime then the height of v is the constant term of

−\frac{1}{2} ∑_{b mod 2N} θ_{v^{⊥},b}(τ)f(τ)H_{N,b}(τ).

Proof. The case N = 0 follows from theorem 10.4. If N = 1 then the height is just the order of the zero of Φ_{0} at the cusp i∞ and as SL_{2}(Z) has only one cusp the corollary then follows from theorem 13.1. If N is prime then Γ_{0}(N) has 2 cusps represented by 0 and i∞. These two cusps are exchanged by the Fricke involution, so the Φ_{0}(τv) has zeros of the same order at both cusps. Therefore the order of the zero at i∞ 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^{⊥} is isomorphic to the lattice of vectors of even norm in the corresponding 25-dimensional unimodular lattice. We put N = 2 and f(τ) = 1/Δ(τ) and find that the constant term of

−\frac{1}{2} ∑_{b mod 4} θ_{v^{⊥},b}H_{2,b}(τ)f(τ)

is the height of v and therefore is an integer. The first few coefficients of H_{2,b} are given by H_{2,0}(τ/8) = −1/4 + q + ⋯, H_{2,1}(τ/8) = H_{2,3}(τ/8) = q^{7/8} + ⋯, and H_{2,2}(τ/8) = (1/2)q^{1/2} + · · ·. The coefficients of the theta functions are given by θ_{v^{⊥},0}(τ) + θ_{v^{⊥},2}(τ) = θ_{K}(τ), and θ_{v^{⊥},1}(τ) = θ_{v^{⊥},3}(τ) = aq^{1/8} + ⋯, 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

8height(v) = 20 + r_{2} − 2r_{1} − 8a

where r_{n} is the number of vectors of norm n in K. In particular

r_{2} ≡ 2r_{1} + 4 mod 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 ≡ 1 mod 24 with s > 0 then the constant term of

(H_{2,0}(τ/8) + H_{2,2}(τ/8))θ_{K}(τ)/Δ(τ)^{s/24}

is divisible by 2. As the Fricke involution acts on a fundamental domain of Γ_{0}(2) fixed point freely except at the images of the points τ = i and τ = \sqrt{2}i the coefficients of the series H_{2,0} and H_{2,2} are usually even; more precisely H_{2,0}(τ) ≡ −1/4 + ∑_{n>0} q^{8n^{2}} mod 2 and H_{2,2}(τ) ≡ (1/2) ∑_{n>0} q^{4n^{2}} mod 2. This implies that the constant term of

(2θ(τ) + θ(τ/2))θ_{K}(τ)/Δ(τ)^{s/24}

is divisible by 8. We can get similar congruences for unimodular lattices whose dimension is not 1 mod 24 by adding on copies of the 1-dimensional unimodular lattice until the dimension is 1 mod 24. These congruences can probably also be deduced by constructing automorphic forms for the groups O_{K⊕I_{2,2}}(Z)^{+}.

14. Product formulas for modular forms.

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 ∏_{[σ]}(j(τ) − j(σ)) obviously satisfies the conditions in theorem 14.1, and has zeros corresponding to a function f_{0}(τ) 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 ∑ H(n)q^{n} = −1/12 + q^{3}/3 + q^{4}/2 + q^{7} + q^{8} + q^{11} + (4/3)q^{12} + · · ·. )

Theorem 14.1. Suppose that f_{0}(τ) = ∑ c_{0}(n)q^{n} is a nearly holomorphic modular form of weight 1/2 for Γ_{0}(4) with integer coefficients whose coefficients c_{0}(n) vanish unless n is 0 or 1 mod 4. We put

Ψ(τ) = q^{−h} ∏_{n>0}(1 − q^{n})^{c_{0}(n^{2})}

where h is the constant term of f_{0}(τ) ∑ H(n)q^{n}. Then Ψ(τ) 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 Ψ. Under this isomorphism, the weight of the modular form Ψ is c_{0}(0), and the multiplicity of the zero of Ψ at a quadratic irrational τ of discriminant D < 0 is ∑_{d>0} c_{0}(Dd^{2}). (The discriminant of τ is D = b^{2} − 4ac, where a, b, and c are integers with no common factor such that aτ^{2} + bτ + c = 0.)

Example 1. Under this isomorphism f_{0}(τ) = 12θ(τ) = 12 + 24q + 24q^{4} + 24q^{9} + ⋯ corresponds to Ψ(τ) = Δ(τ) = q ∏_{n>0}(1 − q^{n})^{24} of weight c_{0}(0) = 12; this is the usual product formula for the Δ function.

Example 2. Put

F (τ) = ∑_{n>0,n odd} σ_{1}(n)q^{n} = q + 4q^{3} + 6q^{5} ⋯
θ(τ) = ∑_{n∈Z} q^{n^{2}} = 1 + 2q + 2q^{4} + ⋯
f_{0}(τ) = F (τ)θ(τ)(θ(τ)^{4} − 2F (τ))(θ(τ)^{4} − 16F (τ))E_{6}(4τ)/Δ(4τ) + 56θ(τ)

= q^{−3} − 248q + 26752q^{4} − ⋯

= ∑_{n} c_{0}(n)q^{n}.

Then the corresponding infinite product Ψ(τ) 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(τ)^{1/3}. The Fourier series of 3f_{0}(τ) is

3f_{0}(τ) = 3q^{−3} − 744q + 80256q^{4} − 257985q^{5} + 5121792q^{8} − 12288744q^{9} + ⋯

so

j(τ) = q^{−1} + 744 + 196884q + 21493760q^{2} + ⋯

= q^{−1} ∏_{n>0}(1 − q^{n})^{3c_{0}(n^{2})}

= q^{−1}(1 − q)^{−744}(1 − q^{2})^{80256}(1 − q^{3})^{−12288744} ⋯

Example 3. The Eisenstein series E_{4}, E_{6}, E_{8}, E_{10}, and E_{14} all satisfy the condition on Ψ and so can be written as infinite products corresponding to some functions f_{0}(τ). An explicit formula for E_{4} follows easily from the infinite product expansions of j and Δ because j = E_{4}^{3}/Δ, 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_{4}E_{6}, and E_{14} = E_{4}^{2}E_{6}. The remaining Eisenstein series cannot be written as modular products.

Example 4. There are exactly 13 integers n for which j(τ) − n satisfies the conditions on Ψ(τ) and hence can be written as a modular product; these are the values of j(τ) at values of imaginary quadratic τ for which j(τ) 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^{3}5^{3}, j(i\sqrt{2}) = 2^{6}5^{3}, j((1+i\sqrt{11})/2) = −2^{15}, j(i\sqrt{3}) = 2^{4}3^{3}5^{3}, j(2i) = 2^{3}3^{3}11^{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^{3}5^{3}, j((1+i\sqrt{67})/2) = −2^{15}3^{3}5^{3}11^{3}, j((1 + i\sqrt{163})/2) = −2^{18}3^{3}5^{3}23^{3}29^{3}. There are therefore exactly 14 modular forms of weight 12 with integer coefficients for SL_{2}(Z) which are modular products: the forms Δ(τ)(j(τ) − n) and the form Δ(τ).

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 Ψ satisfying the conclusion of theorem 14.1 can be written as a product of the modular products constructed from the forms f_{0}.

Lemma 14.2. Every sequence of integers c_{0}(n) for n ≤ 0, n ≡ 0, 1 mod 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}(τ) = ∑_{n} c_{0}(n)q^{n} is a modular form satisfying the conditions of theorem 14.1 with c_{0}(n) = 0 for n ≤ 0 then we define h_{0}(τ) = ∑_{n} c_{0}(4n)q^{n} and h_{1}(τ) = ∑_{n} c_{0}(4n+1)q^{n+1/4}. If we use the fact that f_{0}(τ) = h_{0}(4τ) + h_{1}(4τ) satisfies the relation f_{0}(σ/(4σ + 1)) = \sqrt{(4σ + 1)}f_{0}(4σ) and put τ = 4σ + 1 we find that h_{0} and h_{1} satisfy the relation

h_{0}(−1/τ) − ih_{1}(−1/τ) = \sqrt{τ}(h_{0}(τ) − ih_{1}(τ)).

If we let τ be imaginary and take real and imaginary parts of this we find that h_{0} and h_{1} satisfy the relations

\begin{aligned}h_{0}(τ + 1) & = h_{0}(τ) \\ h_{1}(τ + 1) & = ih_{1}(τ) \\ h_{0}(−1/τ) & = (1/2 − i/2)\sqrt{τ}(h_{0}(τ) + h_{1}(τ)) \\ h_{1}(−1/τ) & = (1/2 − i/2)\sqrt{τ}(−h_{0}(τ) + h_{1}(τ)). \end{aligned}

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}/η and h_{1}/η 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} = ⋯ for every n ≤ 0 with n ≡ 0, 1 mod 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(4τ). For n = 0 we can use the function ∑_{n∈Z} q^{n^{2}}, and for n = −3 we can use the function F (τ)θ(τ)(θ(τ)^{4} − 2F (τ))(θ(τ)^{4} − 16F (τ))E_{6}(4τ)/Δ(4τ) + 56θ(τ) where F (τ) is defined in example 2 above. This proves lemma 14.2.

Lemma 14.3. There is a norm −2 vector v ∈ II_{25,1} such that θ_{v^{⊥}}′(τ) = 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} ⊕ (Leech lattice).

Lemma 14.4. There is a norm −2 vector v ∈ II_{25,1} such that θ_{v^{⊥}}′(τ) = 1+6q+O(q^{5/4}) = 1 + O(q).

Proof. If we take ρ to be a primitive norm 0 vector of II_{25,1} corresponding to the Leech lattice, then ρ 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 = ρ + 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^{⊥} are just r_{1} and r_{2} so that v^{⊥} 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^{⊥}′. This proves lemma 14.4.

Lemma 14.5. The Z-module of functions f_{0} satisfying the conditions of theorem 14.1 is spanned by functions of the form θ_{v^{⊥}}′(4τ)f(4τ), where v is a norm -2 vector of a lattice II_{s+1,1} and f = ∑_{n} c(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} + … or 2q^{−4n−3} + … 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

Φ_{0}(y) = e^{−2πi(ρ,y)} ∏_{x∈L,x>0,(x,v)≠0}(1 − e^{−2πi(x,y)})^{c(x)}

is the function defined in section 13. By theorem 13.1 Ψ(τ) = Φ_{0}(τv) is a modular form for SL_{2}(Z) of weight equal to 2k + 2 ∑_{x>0,(x,v)=0} c(−(x, x)/2) = ∑_{(x,v)=0} c(−(x, x)/2) which is the constant term of f(τ)θ_{v′}(τ). By theorem 13.1 again the zeros of Ψ are as stated in theorem 14.1.

We can find the infinite product decomposition of Φ_{0}(τv) from that of Φ_{0}(y) by restriction, and we see that

Φ_{0}(τv) = q^{h} ∏_{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(4τ)θ_{v^{⊥}}′(4τ) = f_{0}(τ). 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 Ψ in theorem 14.1 is an isomorphism. If the image of f_{0} is 1, then Ψ has no zeros and weight 0, so the coefficients c_{0}(n) are 0 if n ≤ 0, so f_{0} is 0 by lemma 14.2, which proves that the map is injective. If Ψ 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 Ψ. By taking the quotient we can assume that Ψ has no zeros or poles except at cusps. But this implies that Ψ must be a positive or negative power of η(τ)^{2} (as Ψ has integral weight), and we obtain these functions Ψ by taking f_{0} to be an integral multiple of θ(τ) = 1 + 2q + 2q^{4} + · · ·. 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

∑_{t∈Z} H(4m − t^{2}) = ∑_{d|m} max(d, m/d)

for positive integers m.

15. Generalized Kac-Moody algebras.

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(τ) is the Hauptmodul of an element of the monster simple group then f(σ)−f(τ) 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(Aσ) − f(Bτ) 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

E_{6}(τ) = 1 − 504 ∑_{n>0} σ_{5}(n)q^{n}

= 1 − 504q − 16632q^{2} − 122976q^{3} − ⋯

= ∏_{n>0}(1 − q^{n})^{a(n^{2})}

= (1 − q)^{504}(1 − q^{2})^{143388}(1 − q^{3})^{51180024} ⋯

where

∑_{n} a(n)q^{n}

=q^{−4} + 6 + 504q + 143388q^{4} + 565760q^{5} + 18473000q^{8} + 51180024q^{9} + O(q^{12})
=(j(4τ) − 852)θ(τ) − 2F (τ)θ(τ)(θ(τ)^{4} − 2F (τ))(θ(τ)^{4} − 16F (τ))E_{6}(4τ)/Δ(4τ)

(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 α of norm −2, the simple roots are nα (α > 0) with multiplicity 504σ_{3}(α), and the multiplicity of the roots nα 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 504σ_{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_{Λ}(Z) (where Λ 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

∑_{m,n∈Z}(−1)^{m+n}p^{m^{2}}q^{n^{2}}r^{mn} = ∏_{a+b+c>0} (\frac{1 − p^{a}q^{c}r^{b}}{1 + p^{a}q^{c}r^{b}} )^{f(ac−b^{2})}

where f(n) is defined by ∑ f(n)q^{n} = 1/(∑_{n}(−1)^{n}q^{n^{2}}) = 1 + 2q + 4q^{2} + 8q^{3} + 14q^{4} + . . .. 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.

16. Hyperbolic reflection groups

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}(τ)^{2}/Δ(τ) = q^{−1} + 504 + · · ·. 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}(τ)/Δ(τ) = q^{−1} + 256 + · · ·. 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 Φ for O_{II_{26,2}}(Z)^{+} and divided it by the factors vanishing on L, we get a function Φ_{0} which restricts to an automorphic form for O_{L⊕II_{1,1}}(Z)^{+}. If we take Φ to be the automorphic form associated to 1/Δ(τ) 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 Λ of II_{25,1} whose components are all of the form d_{4}, d_{n} for n ≥ 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}.

17. Open problems.

1. Can the methods for constructing automorphic forms as infinite products be used for semisimple groups other than O_{s+2,2}(R)?

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?

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.

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.

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?

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 Δ(τ) 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 (Δ) is an eigenform.

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.

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.

9. Is there any connection between the 3 product formulas for j(τ) − j(σ)? (The three formulas are the Gross-Zagier formula when both of σ, τ 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.)

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 Φ_{0}, and the coefficients of q^{n^{2}} appear in the infinite product expansion of Φ_{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 ≡ 0, (−1)^{(k−1)/2} mod 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.)

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.

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)?

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.

References.

[B] R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent.

Math. 109, 405-444 (1992). [B-M] A. Borel, G. D. Mostow, “Algebraic groups and discontinuous subgroups”, Proc.

Symp. Pure Math. vol. IX, American Mathematical Society 1966. [C-S] J. H. Conway, N. J. A. Sloane. “Sphere packings, lattices and groups”. Grundlehren

der mathematischen Wissenschaften 290, Springer-Verlag New York 1988.

[E] A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, “Higher transcendental functions”, vols. 1-3. McGraw-Hill Book Company Inc, New York, Toronto, London 1953.

[E-Z] M. Eichler, D. Zagier, “The theory of Jacobi forms”, Progress in mathematics vol. 55,

Birkhäuser, Boston, Basel, Stuttgart 1985.

[F] E. Freitag, “Siegelsche Modulfunktionen”, Die Grundlehren der mathematischen Wissenschaften 254, Springer–Verlag, Berlin Heidelberg New York 1983.

[G] V. G. Gritsenko, Jacobi functions of n variables. J. Soviet Math. 53 (1991), 243-252.

(The original Russian is in Zap. Nauch. Seminars of LOMI v. 168 (1988), p. 32-45.) [G-Z] B. H. Gross, D. B. Zagier, On singular moduli, Journal für die reine und angewandte

Mathematik, Band 355, p. 191-220, 1985.

[K] V. G. Kac, “Infinite dimensional Lie algebras”, third edition, Cambridge University Press, 1990.

[Ko] W. Kohnen, Modular forms of half integral weight on Γ_{0}(4), Math. Annalen 248 (1980) 249-266.

[M] H. Maass, Über eine Spezialschar von Modulformen zweiten Grades I, II, III, Inv. Math. 52 (1979) 95-104, Inv. Math 53 (1979) 249-253, and Inv. Math. 53 (1979) 255-265.

[R] H. Rademacher, “Topics in analytic number theory”, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Band 169, Springer Verlag Berlin, Heidelberg, New York 1973.