On the non-existence of singular Borcherds products

In this paper we prove some nice properties of holomorphic automorphic products of singular weight on orthogonal groups $\mathop{\hbox{}\mathrm{O}}\nolimits(l,2)$, and resolve an open problem posed by Borcherds in 1995.

Let $M$ be an even integral lattice of signature $(l,2)$ with $l\geq 3$. Let $(-,-)$ denote the bilinear form on $M$ and let $M^{\prime}$ be the dual lattice of $M$. Either of the two connected components of the space

defines the attached Hermitian symmetric domain $\mathcal{D}(M)$; we fix a component once and for all. We denote by $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(M)$ the subgroup of the orthogonal group of $M$ that preserves $\mathcal{D}(M)$. Let $k$ be an integer, $\Gamma$ be a finite-index subgroup of $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(M)$ and $\chi$ be a character of $\Gamma$. A modular form of weight $k$ and character $\chi$ for $\Gamma$ is a holomorphic function $F$ on the affine cone

over $\mathcal{D}(M)$ which satisfies

The symmetric domain $\mathcal{D}(M)$ can be realized as a tube domain around any cusp. The action of $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(M)$ on the tube domain induces an automorphy factor, with respect to which one can also define modular forms of half-integral weight. If $F$ is nonzero, then either $k=0$ (in which case $F$ is constant), or $k\geq l/2-1$. The smallest possible weight $l/2-1$ of a non-constant modular form is called the singular weight. On any tube domain realization, modular forms can be expanded into Fourier series, and singular-weight modular forms are characterized by the fact that their Fourier series is supported only on norm-zero vectors.

In 1995 and 1998 Borcherds [2, 3] established a remarkable lift to construct orthogonal modular forms with nice properties. The Borcherds lift is a multiplicative map and its input $f$ is a weakly holomorphic modular form of weight $1-l/2$ for the Weil representation of $\mathop{\mathrm{SL}}\nolimits_{2}(\mathbb{Z})$ attached to $M^{\prime}/M$. The image $\mathbf{B}(f)$ is a meromorphic modular form for the discriminant kernel

whose divisor is a linear combination of rational quadratic divisors

The function $\mathbf{B}(f)$ has an infinite product expansion around any cusp, so we call it a Borcherds product or an automorphic product.

Holomorphic Borcherds products of singular weight are very exceptional. They are usually the denominators of generalized Kac–Moody algebras that have a natural construction as the BRST cohomology related to a superconformal field theory (e.g. [4, 5, 19]). For example, the fake monster Lie algebra, which was constructed by Borcherds [4] in 1990, is the BRST cohomology related to the Leech lattice vertex operator algebra. This infinite dimensional Lie algebra describes the physical states of a bosonic string moving on a 26-dimensional torus, and its denominator is given by the singular-weight Borcherds product $\Phi_{12}$ on $\mathop{\mathrm{II}}\nolimits_{26,2}$, the even unimodular lattice of signature $(26,2)$. Motivated by this connection, in 1995 Borcherds [2, Open problem 3 in Section 17] posed the following problems:

1. (1)

   Are there a finite or infinite number of singular-weight modular forms which can be written as automorphic products?

2. (2)

   Are there holomorphic automorphic products of singular weight on $\mathop{\hbox{}\mathrm{O}}\nolimits(l,2)$ when $l>26$?

These problems have remained open, although there are some partial results in the literature. Dittmann, Hagemeier and Schwagenscheidt [11] and Opitz and Schwagenscheidt [18] classified holomorphic Borcherds products of singular weight on simple lattices (i.e. lattices on which there is no obstruction to constructing Borcherds products). Note that there are only finitely many simple lattices and a full classification was given in [9]. Scheithauer [21] classified singular Borcherds products on unimodular lattices, and derived a conditional bound on the signature of lattices of prime level with singular Borcherds products.

This paper gives a complete solution to problem (2) and a weak solution to problem (1). To state the main results more conveniently, we introduce the following definition.

Bruinier’s converse theorem [7, 8] shows that every special modular form for the discriminant kernel of $M$ is a Borcherds product on $M$ if $M$ splits as $U\oplus U(m)\oplus L$, where $U$ is the unique even unimodular lattice of signature $(1,1)$. It is not known whether this holds for arbitrary $M$ (of signature $(l,2)$ with $l\geq 3$).

In this paper, we first describe zeros of special modular forms of singular weight. The proof follows from studying the Laplace operator on the tube domain around any cusp.

Following Borcherds [3] and Gritsenko–Nikulin [13], we define reflective modular forms.

As a corollary of Theorem 1.2, we prove the following result, which is a conjecture formulated by the first named author in 2019 (see [26, Conjecture 9.8]).

Theorem 1.4 reduces the difficult problem of classifying singular automorphic products to the easier problem of classifying reflective modular forms. A full classification of reflective modular forms is not yet available, but many partial results have been obtained over the past two decades [13, 12, 1, 20, 21, 15, 16, 10, 22, 14, 26, 23, 24, 25]. In particular, the first named author [22, 26, 25] classified lattices of large rank which has a reflective modular form. This yields a complete solution to Borcherds’ problem (2).

Special modular forms of singular weight on $\mathop{\hbox{}\mathrm{O}}\nolimits(l,2)$ do exist for $l=3$, $4$, $6$, $8$, $10$, $12$, $14$, $18$. They do not appear to exist on $\mathop{\hbox{}\mathrm{O}}\nolimits(l,2)$ for any other $l\leq 20$.

The finiteness of lattices with reflective modular forms is known in certain specific cases [15, 16, 25]. However, the finiteness of reflective modular forms is unknown in general. Therefore, we cannot give a full solution to Borcherds’ problem (1). In 2018 Ma [16, Corollary 1.10] proved that up to scaling there are only finitely many lattices of signature $(l,2)$ with $l\geq 4$ which has a reflective modular form with simple zeros. Theorem 1.2, Theorem 1.4 and Ma’s result together yield the following weak solution to Borcherds’ problem (1).

This paper is organized as follows. In Section 2 we introduce the Laplace operator and prove Theorem 1.2. In Section 3 we prove Theorem 1.4 and give a corollary. Section 4 is devoted to the proof of Theorem 1.5.

In this section we use the Laplace operator on a tube domain to prove Theorem 1.2.

Let $M$ be an even lattice of signature $(l,2)$ with $l\geq 3$. Let $c\in M$ be a primitive norm-zero vector and $c^{\prime}\in M^{\prime}$ with $(c,c^{\prime})=1$. Let $e_{1},...,e_{l}$ be any $\mathbb{R}$-basis of the signature $(l-1,1)$ lattice

and define the Gram matrix $S=(s_{ij})_{i,j=1}^{l}$ with inverse $S^{-1}=(s^{ij})_{i,j}$, where $s_{ij}:=(e_{i},e_{j})$. The holomorphic Laplace operator is defined as

This operator acts on functions defined on the tube domain $\mathbb{H}_{c,c^{\prime}}$ which is a connected component of the space

It is clear that $\mathbf{\Delta}_{c,c^{\prime}}$ is independent of the basis $e_{i}$.

For any $\lambda\in M_{c,c^{\prime}}\otimes\mathbb{Q}$, we have

Therefore, a function $F$ that is represented near $c$ by a Fourier series

satisfies $\mathbf{\Delta}F=0$ if and only if it is singular at $c$, that is, $c(\lambda)=0$ for every $\lambda$ of nonzero norm.

For any meromorphic function $F:\mathbb{H}_{c,c^{\prime}}\rightarrow\mathbb{C}$, matrix elements $g\in\mathrm{O}^{+}(M\otimes\mathbb{R})$ and positive integer $m$, we have (see e.g. [27, Lemma 2.4])

In particular, if $F$ transforms like a modular form of weight $l/2-m$ then $\mathbf{\Delta}^{m}F$ transforms like a modular form of weight $l/2+m$. If $F$ is in fact a non-constant (holomorphic) modular form, then $m$ must be 1, i.e. $F$ is of singular weight, and thus $\mathbf{\Delta}F=0$.

Theorem 1.2 is a particular case of the following result.

We divide the proof of Theorem 1.4 into two lemmas. The notation is the same as Theorem 1.4.

For lattices that split a unimodular plane, Theorem 1.4 sharply restricts the principal parts of vector-valued modular forms that lift to holomorphic Borcherds products of singular weight:

In this section we prove Theorem 1.5 as a corollary of Theorem 1.4 using the following results:

Let $M$ be an even lattice of signature $(l,2)$ with $l\geq 21$ and suppose $F$ is a special modular form of singular weight for some finite-index subgroup $\Gamma$ of $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(M)$. By Theorem 1.4, Lemma 3.1 and Lemma 3.2, $F$ is a reflective modular form for $\Gamma_{F}<\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(\mathbb{M})$. Theorem 4.1 implies that $l=22$ or $26$. By [16, Corollory 3.2], there exists a lattice $\mathbb{M}_{1}$ in $\mathbb{M}\otimes\mathbb{Q}$ such that $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(\mathbb{M})\subset\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(\mathbb{M}_{1})$ and such that $\mathbb{M}_{1}$ is a scaling of an even lattice of type $\mathbb{M}_{2}=2U\oplus L$. The subgroup $\Gamma_{F}$ has finite index in $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(\mathbb{M}_{2})$, and we denote the index by $a$. Then the function

defines a reflective modular form of weight $a(l/2-1)$ for $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(\mathbb{M}_{2})$.

When $l=26$, we conclude from Theorem 4.1 that $\mathbb{M}_{2}\cong\mathop{\mathrm{II}}\nolimits_{26,2}$ and $\hat{F}=\Phi_{12}^{a}$ up to a constant factor. By Theorem 1.2, $F$ has only simple zeros. Therefore, $\Phi_{12}/F$ is a (holomorphic) modular form of weight $0$ for $\Gamma_{F}$ and thus constant.

When $l=22$, Theorem 4.1 implies that $\mathbb{M}_{2}\cong 2U\oplus D_{20}$. Borcherds [6] constructed a reflective modular form $\Psi_{24}$ of weight $24$ for $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(2U\oplus D_{20})$ which vanishes on $v^{\perp}$ with multiplicity $1$ and vanishes on $u^{\perp}$ with multiplicity $8$, where $v\in 2U\oplus D_{20}$ with $v^{2}=2$ and $u\in 2U\oplus D_{20}^{\prime}$ with $u^{2}=1$. We know from [25, Lemma 3.2] that $\Psi_{24}$ is the unique reflective modular form for $\mathop{\hbox{}\mathrm{O}}\nolimits^{+}(2U\oplus D_{20})$ up to a power. It follows that $\hat{F}=\Psi_{24}^{d}$ for some positive integer $d$. Recall that $F$ has only simple zeros. By comparing the weights and zero orders of $F$ and $\Psi_{24}$, we find

which leads to $10a/3\leq a$, a contradiction. This finishes the proof of Theorem 1.5.