Skip to content

Advertisement

  • Research
  • Open Access

Weakly holomorphic modular forms on \(\Gamma _{0}(4)\) and Borcherds products on unitary group \(\mathrm{U}(2,1)\)

Research in Number Theory20184:2

https://doi.org/10.1007/s40993-018-0096-z

Received: 13 January 2017

Accepted: 16 November 2017

Published: 31 January 2018

Abstract

In this note, we construct canonical bases for the spaces of weakly holomorphic modular forms with poles supported at the cusp \(\infty \) for \(\Gamma _{0}(4)\) of integral weight k for \(k\le -1\), and we make use of the basis elements for the case \(k=-1\) to construct explicit Borcherds products on unitary group \(\mathrm{U}(2,1)\).

Keywords

  • Borcherds product
  • Unitary modular form
  • Heegner divisor
  • Unitary modular variety

Mathematics Subject Classification

  • 11F27
  • 11F41
  • 11F55
  • 11G18
  • 14G35

1 Introduction

In 1998, Borcherds developed a new method to produce meromorphic modular forms on an orthogonal Shimura variety from weakly holomorphic classical modular forms via regularized theta liftings. These meromorphic modular forms have two distinct properties. The first one is the so-called Borcherds product expansion at a cusp of the Shimura variety—his original motivation to prove the Moonshine conjecture. The second is that the divisor of these modular forms are known to be a linear combination of special divisors dictated by the principal part of the input weakly holomorphic forms. The second feature has been extended to produce so-called automorphic green functions for special divisors using harmonic Maass forms via regularized theta lifting by Bruinier [1] and Bruinier-Funke [2], which turned out to be very useful to generalization of the well-known Gross–Zagier formula [3] and the beautiful Gross–Zagier factorization formula of singular moduli [4] to Shimura varieties of orthogonal type (n, 2) and unitary type (n, 1) (see for example [511]). On the other hand, the Borcherds product expansion and in particular its integral structure is essential to prove modularity of some generating functions of arithmetic divisors on these Shimura varieties [12, 13]. Borcherds products are also closely related to Mock theta functions (see for example [14] and references there).

We should mention that the analogue of the Borcherds product to unitary Shimura varieties of type (n, 1) has been worked out by Hofmann [15] (see also [12]). The Borcherds product expansion in the unitary case is a little more complicated as it is a Fourier–Jacobi expansion rather than Fourier expansion; the coefficients are theta functions rather than numbers. The purpose of this note is to give some explicit examples of these Borcherds product expansion in concrete term. For this reason, we focus on the Picard modular surface \(X_{\Gamma _{L}}=\Gamma _{L}\backslash \mathcal {H}\) associated to the Hermitian lattice \(L={\mathbb {Z}}[i] \oplus {\mathbb {Z}}[i] \oplus \frac{1}{2} {\mathbb {Z}}[i]\) with Hermitian form
$$\begin{aligned} \langle x, y \rangle = x_1 {\bar{y}}_3 + x_3 {\bar{y}}_1 + x_2 {\bar{y}}_2. \end{aligned}$$
Here
$$\begin{aligned} {\mathcal {H}}=\{ (\tau , \sigma ) \in {\mathbb {H}} \times {\mathbb {C}}| \, 4 \text{ Im }(\tau ) > |\sigma |^2\}, \end{aligned}$$
and \(\Gamma _{L}\) is a subgroup of \(\mathrm{U}(L)\) defined by (3.4). Our inputs are weakly holomorphic modular forms for \(\Gamma _0(4)\) of weight \(-1\), character \(\chi _{-4}:=\left( \frac{-4}{}\right) \) which have poles only at the cusp \(\infty \), which we denote by \(M_{-k}^{!, \infty }(\Gamma _0(4), \chi _{-4}^k)\) with \(k=1\). Our first result (Theorem 2.1) is to give a canonical basis \(F_{k,m}\) (\(m \ge 1\)) for the infinitely dimensional vector space for every \(k \ge 1\). The even k case was given by Haddock and Jenkins in [16] in a slightly different fashion. Similar method can be applied to yield a canonical basis for the space of weakly holomorphic forms of \(\Gamma _0(4)\) with weight \(-k\), character \(\chi _{-4}^k\), and having poles only at the cusp 0 (resp. \(\frac{1}{2}\)).

Next, we use a standard induction procedure to produce vector-valued weakly holomorphic modular forms for \({\text {SL}}_2({\mathbb {Z}})\) using our lattice L which will be used to construct Picard modular forms on \(\mathrm{U}(2,1)\) (described above). Although the resulting vector-valued modular forms for \({\text {SL}}_2({\mathbb {Z}})\) from the three different scalar valued spaces \(M_{-k}^{!, P}(\Gamma _0(4), \chi _{-4}^k)\), \(P=\infty , 0, \frac{1}{2}\) are linearly independent, they don’t generate the whole space. This concludes Part I of our note, which should be of independent interest.

In Part II, we focus on the unitary group \(\mathrm{U}(2, 1)\) associated to the above Hermitian form and give explicit Borcherds product expansion of the Picard modular forms constructed from \(F_{m}=F_{1, m}\). The delicate part is to choose a proper Weyl chamber, which is a dimensional 3 real manifold and described it explicitly and carefully. Our main formula is Theorem 3.5. We remark that the same method also applies to high dimensional unitary Shimura varieties of unitary type (n, 1) using forms in \(M_{1-n}^{!, P}(\Gamma _0(4), \chi _{-4}^k)\) where P is a cusp for \(\Gamma _0(4)\). We restrict to \(\mathrm{U}(2,1)\) for being as explicit as possible.

2 Part I: vector-valued modular forms

In this part, we derive a canonical basis for the space \(M_{-k}^{!,\infty }(\Gamma _{0}(4),\chi _{-4}^{k})\) for any integer \(k\ge 0\), and investigate the properties of the vector-valued modular forms arising from \(M_{-k}^{!,\infty }(\Gamma _{0}(4),\chi _{-4}^{k})\). For completeness, we will also give canonical bases for \(M_{-k}^{!,0}(\Gamma _{0}(4),\chi _{-4}^{k})\) and \(M_{-k}^{!,\frac{1}{2}}(\Gamma _{0}(4),\chi _{-4}^{k})\).

2.1 A canonical basis for \(M_{-k}^{!,\infty }(\Gamma _{0}(4),\chi _{-4}^{k})\)

Let \(\chi _{-4}(\cdot ):=\left( \frac{-4}{\cdot }\right) \) be the Kronecker symbol modulo 4. Recall that \(X_0(4)\) has 3 cusps, represented by \(\infty \), 0, and \(\frac{1}{2}\). For each cusp P, let \(M_{-k}^{!,P}(\Gamma _{0}(4),\chi _{-4}^{k})\) denote the space of weakly holomorphic modular forms, which are holomorphic everywhere except at the cusp P, of weight \(-k\) on \(\Gamma _0(4)\) with character \(\chi _{-4}^{k}\). We will focus mainly on the cusp \(\infty \) and will remark on other cusps (very similar) in the end. We will also denote \(M_{-k}^!(\Gamma _0(4), \chi _{-4}^k)\) for the space of weakly holomorphic modular forms for \(\Gamma _0(4)\) of weight \(-k\) and character \(\chi _{-4}^k\).

Let \(\tau \) be a complex number with positive imaginary part, and set \(q=e(\tau )=e^{2\pi i\tau }\), and \(q_{r}=e^{2\pi i\tau /r}\). The Dedekind eta function is defined by
$$\begin{aligned} \eta (\tau )=q^{1/24}\prod _{n=1}^{\infty }(1-q^{n}). \end{aligned}$$
Throughout this paper, we write \(\eta _{m}\) for \(\eta (m\tau )\). The well known Jacobi theta functions are defined by
$$\begin{aligned} \vartheta _{00}(\tau )=\sum _{n=-\infty }^{\infty }q^{n^{2}},\quad \vartheta _{01}(\tau )=\sum _{n=-\infty }^{\infty }(-q)^{n^{2}},\quad \vartheta _{10}(\tau )=\sum _{n=-\infty }^{\infty }q^{\left( n+\frac{1}{2}\right) ^{2}}. \end{aligned}$$
Now we define three functions as follows.
$$\begin{aligned} \theta _{1}=\theta _{1}(\tau )&:=\frac{1}{16}\vartheta ^{4}_{10}(\tau )=\frac{\eta _{4}^{8}}{\eta _{2}^{4}}=q+O(q^{2}), \end{aligned}$$
(2.1)
$$\begin{aligned} \theta _{2}=\theta _{2}(\tau )&:=\vartheta ^{2}_{00}(\tau )=\frac{\eta _{2}^{10}}{\eta _{1}^{4}\eta _{4}^{4}}=1+O(q), \end{aligned}$$
(2.2)
$$\begin{aligned} \varphi _{\infty }=\varphi _{\infty }(\tau )&:=\left( \frac{\eta _{1}}{\eta _{4}}\right) ^{8}=q^{-1}+O(1). \end{aligned}$$
(2.3)
Here are some basic facts [16] about the functions \(\theta _{1}\), \(\theta _{2}\) and \(\varphi _{\infty }\).
  1. (1)

    \(\theta _{1}(\tau )\) is a holomorphic modular form of weight 2 on \(\Gamma _{0}(4)\) with trivial character, has a simple zero at the cusp \(\infty \), and vanishes nowhere else.

     
  2. (2)

    \(\theta _{2}(\tau )\) is a holomorphic modular form of weight 1 on \(\Gamma _{0}(4)\) with character \(\chi _{-4}\), has a zero of order \(\frac{1}{2}\) at the irregular cusp \(\frac{1}{2}\), and vanishes nowhere else.

     
  3. (3)

    \(\varphi _{\infty }(\tau )\) is a modular form of weight 0 on \(\Gamma _{0}(4)\) with trivial character, has exactly one simple pole at the cusp \(\infty \) and a simple zero at the cusp 0.

     
The following is a variant of [16] where the case even k has been treated by Haddock and Jenkins. We should mention that similar results for the space of weakly holomorphic modular forms for \(\mathrm{SL}_{2}({\mathbb {Z}})\) were first obtained in [17] by Duke and Jenkins.

Theorem 2.1

  1. (1)
    For \(k \ge 1 \) odd, there is a (canonical) basis \(F_{k, m}\) (\(m \ge 1\)) of \(M^{!,\infty }_{-k}(\Gamma _{0}(4),\chi _{-4})\) whose Fourier expansion has the following form:
    $$\begin{aligned} F_{k, m} = q^{-\frac{k+1}{2} -m+1 } + \sum _{n \ge -\frac{k-1}{2}} c(n) q^n. \end{aligned}$$
     
  2. (2)
    For \(k >1\) even, there is a (canonical) basis \(F_{k, m}\) (\(m \ge 1\)) of \(M^{!,\infty }_{-k}(\Gamma _{0}(4))\) whose Fourier expansion has the following form:
    $$\begin{aligned} F_{k,m} =q^{-\frac{k}{2}-m+1}+\sum _{n\ge -\frac{k}{2}+1}c(n)q^{n}, \end{aligned}$$
     

Proof of Theorem 2.1

The proof is similar to those given in [17] and [16], and we include it for completeness. We prove (1) first. Notice that \(X_0(4)\) has no elliptic points [18, Section 3.9]. For \(F \in M_{-k}^{!, \infty }(\Gamma _0(4), \chi _{-4})\), the valence formula for \(\Gamma _{0}(4)\) asserts that
$$\begin{aligned} \sum _{z\in \Gamma _{0}(4)\backslash \mathbb {H}}\mathrm{ord}_{z}(F)+\mathrm{ord}_{\infty }(F)+\mathrm{ord}_{0}(F)+\mathrm{ord}_{1/2}(F)=-\frac{k}{2}. \end{aligned}$$
This implies \(\mathrm{ord}_{1/2} F \ge \frac{1}{2}\) (1 / 2 is the unique irregular cusp), \(\mathrm{ord}_{\infty }(F)\le -\frac{k+1}{2}\). This implies the uniqueness of the basis \(\{F_{k, m}\}\) if it exists. We prove the existence by inductively constructing a sequence of monic polynomials \(P_{k, m}(x)\) of degree m (\(m \ge 0\)) such that \(F_{k, m+1}=\theta _2\theta _1^{-\frac{k+1}{2}} P_{k, m} (\varphi _\infty )\) give the basis we seek, i.e., with the following property
$$\begin{aligned} F_{k, m+1}=\theta _2\theta _1^{-\frac{k+1}{2}} P_{k, m} (\varphi _\infty ) = q^{-\frac{k+1}{2} -m } + \sum _{n \ge -\frac{k-1}{2}} c(n) q^n. \end{aligned}$$
(2.4)
  1. (1)
    Notice that \(\theta _2\theta _1^{-\frac{k+1}{2}} \in M_{-k}^{!, \infty }(\Gamma _{0}(4),\chi _{-4})\) with
    $$\begin{aligned} \theta _2\theta _1^{-\frac{k+1}{2}} = q^{-\frac{k+1}{2}} + \sum _{n \ge -\frac{k-1}{2}} c(n) q^n. \end{aligned}$$
    So we can and will first define \(P_{k, 0}=1\).
     
  2. (2)
    For \(m \ge 1\), assume that \(P_{k, m-1}(x) \in {\mathbb {C}}[x] \) is constructed with degree \(m-1\), leading coefficient 1, and the property
    $$\begin{aligned} F_{k, m}=\theta _2\theta _1^{-\frac{k+1}{2}} P_{k, m-1} (\varphi _\infty ) = q^{-\frac{k+1}{2} -m +1} + \sum _{n \ge -\frac{k-1}{2}} c(n) q^n. \end{aligned}$$
    Then it is easy to see
    $$\begin{aligned} F_{k, m} \varphi _\infty =q^{-\frac{k+1}{2}-m} + \sum _{n > -\frac{k+1}{2} -m} d(n) q^n. \end{aligned}$$
    Let
    $$\begin{aligned} P_{k, m} =x P_{k, m-1} - \sum _{n=-\frac{k+1}{2} -m+1}^{-\frac{k+1}{2}} d(n) P_{k, -n}, \end{aligned}$$
    and
    $$\begin{aligned} F_{k, m+1} =\theta _2\theta _1^{-\frac{k+1}{2}} P_{k, m} (\varphi _\infty ). \end{aligned}$$
    Then \(F_{k, m+1}\) satisfies (2.4). By induction, we prove the existence of the basis \(\{F_{k, m}\}\), and (1).
     
The proof of (2) is similar and is left to the reader. In this case, the basis \(\{ F_{k,m+1}\}\), \(m \ge 0\), has the form
$$\begin{aligned} F_{k, m+1} = \theta _{1}^{-\frac{k}{2}}Q_{k,m}(\varphi _{\infty })=q^{-\frac{k}{2}-m}+\sum _{n=-\frac{k}{2}+1}^{\infty }c(n)q^{n} \end{aligned}$$
(2.5)
for a unique monic polynomial \(Q_{k, m}\) of degree m. \(\square \)

The following corollary follows directly from the proof of Theorem 2.1(1).

Corollary 2.2

Every weakly holomorphic modular form \(f(\tau )\in M_{-k}^{!,\infty }(\Gamma _{0}(4),\chi _{-4}^{k})\) with k odd, vanishes at the cusp 1 / 2.

2.2 Vector-valued modular form arising from \(M_{-k}^{!,\infty }(\Gamma _{0}(4),\chi _{-4}^{k})\)

Let L be an even lattice over \(\mathbb {Z}\) with symmetric non-degenerate bilinear form \((\cdot ,\cdot )\) and associated quadratic form \(Q(x)=\frac{1}{2}(x, x)\). Let \(L'\) be the dual lattice of L. Assume that L has rank \(2m+2\) and signature (2m, 2). Then the Weil representation of the metaplectic group \({\text {Mp}}_{2}(\mathbb {Z})\) on the group algebra \(\mathbb {C}[L'/L]\) factors through \({\text {SL}}_{2}(\mathbb {Z})\). Thus we have a unitary representation \(\rho _{L}\) of \({\text {SL}}_{2}(\mathbb {Z})\) on \(\mathbb {C}[L'/L]\), defined by
$$\begin{aligned} \rho _{L}(T)\phi _{\mu }&=e(-Q(\mu ))\phi _{\mu }, \end{aligned}$$
(2.6)
$$\begin{aligned} \rho _{L}(S)\phi _{\mu }&=\frac{\sqrt{i}^{2m-2}}{\sqrt{|L'/L|}}\sum _{\beta \in L'/L}e((\mu ,\beta ))\phi _{\beta } \end{aligned}$$
(2.7)
where \(T=\begin{pmatrix}1&{}\quad 1\\ 0&{}\quad 1\end{pmatrix}\), \(S=\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}\), \(\phi _{\mu }\) for \(\mu \in L'/L\) are the standard basis elements of \(\mathbb {C}[L'/L]\) and \(e(z)=e^{2\pi iz}\). We remark that the Weil representation \(\rho _L\) depends only on the finite quadratic module \((L'/L, Q)\) (called the discriminant group of L), where \(Q(x+L) =Q(x) \pmod 1 \in {\mathbb {Q}}/{\mathbb {Z}}\).
Let k be an integer and \(\vec {F}\) be a \(\mathbb {C}[L'/L]\) valued function on \(\mathbb {H}\) and let \(\rho =\rho _{L}\) be a representation of \({\text {SL}}_{2}(\mathbb {Z})\) on \(\mathbb {C}[L'/L]\). For \(\gamma \in {\text {SL}}_{2}(\mathbb {Z})\) we define the slash operator by
$$\begin{aligned} \left( \left. \vec {F}\right| _{k,\rho }\gamma \right) (\tau )=(c\tau +d)^{-k}\rho (\gamma )^{-1}\vec {F}(\gamma \tau ), \end{aligned}$$
where \(\gamma =\begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix}\) acts on \(\mathbb {H}\) via \(\gamma \tau =\frac{a\tau +b}{c\tau +d}\).

Definition 2.3

Let k be an integer. A function \(\vec {F}:\mathbb {H}\rightarrow \mathbb {C}[L'/L]\) is called a weakly holomorphic vector-valued modular form of weight k with respect to \(\rho =\rho _{L}\) if it satisfies
  1. (1)

    \(\left. \vec {F}\right| _{k,\rho }\gamma =F\) for all \(\gamma \in {\text {SL}}_{2}(\mathbb {Z})\),

     
  2. (2)

    \(\vec {F}\) is holomorphic on \(\mathbb {H}\),

     
  3. (3)

    \(\vec {F}\) is meromorphic at the cusp \(\infty \).

     
The space of such forms is denoted by \(M^{!}_{k,\rho }\).
The invariance of T-action implies that \(\vec {F}\in M^{!}_{k,\rho }\) has a Fourier expansion of the form
$$\begin{aligned} \vec {F}=\sum _{\mu \in L'/L}\sum _{\begin{array}{c} n\in \mathbb {Q}\\ n\gg -\infty \end{array}}c(n,\phi _\mu )q^{n}\phi _{\mu }. \end{aligned}$$
Note that \(c(n,\phi _\mu )=0\) unless \(n\equiv -Q(\mu )\pmod {1}\).

From now on, we focus on the special case with the discriminant group \(L'/L\cong \mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z}\) with quadratic form \(Q(x,y)=\frac{1}{4}(x^{2}+y^{2})\pmod {1}\). For our purpose (in Sect. 3), it is convenient to identify \(\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z} \cong \mathbb {Z}[i]/2\mathbb {Z}[i]\), where \(Q(z) =\frac{1}{4} z {{\bar{z}}} \in {\mathbb {Q}}/{\mathbb {Z}}\). We write \(\phi _{0}\), \(\phi _{1}\), \(\phi _{i}\) and \(\phi _{1+i}\) for the basis elements of \(\mathbb {C}[L'/L]\) corresponding to (0, 0), (1, 0), (0, 1) and (1, 1) respectively.

Let \(F=F(\tau )\in M_{-k}^{!,\infty }(\Gamma _0(4),\chi _{-4})\) with k odd and positive. Then using \(\Gamma _{0}(4)\)-lifting, we can construct a vector-valued modular form \(\vec {F}=\vec {F}(\tau )\) arising from \(F(\tau )\) as follows:
$$\begin{aligned} \vec {F}(\tau )=\sum _{\gamma \in \Gamma _0(4)\backslash \mathrm{SL}_{2}(\mathbb {Z})}(\left. F\right| _{-k}\gamma )\rho _{L}(\gamma )^{-1}\phi _{0} =\frac{1}{2}\sum _{\gamma \in \Gamma _1(4)\backslash \mathrm{SL}_{2}(\mathbb {Z})}(\left. F\right| _{-k}\gamma )\rho _{L}(\gamma )^{-1}\phi _{0}. \end{aligned}$$
(2.8)
Define modular forms \(F_{0}\), \(F_{2}\) and \(F_{3}\) as follows. Recall that \(q_{r}=e^{2\pi i\tau /r}\). Let
$$\begin{aligned} \left. F\right| _{-k}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}=\sum _{n=0}^{\infty }a(n)q_{4}^{n}. \end{aligned}$$
Then for \(j\in \{0, 2,3\}\), we write
$$\begin{aligned} F_{j}=\sum _{n=0}^{\infty }a(4n+j)q_{4}^{4n+j}. \end{aligned}$$
(2.9)
We also define \(F_{1/2}\) to be
$$\begin{aligned} F_{1/2}=\left. F\right| _{-k}\begin{pmatrix}1&{}\quad 0\\ 2&{}\quad 1\end{pmatrix}=\sum _{n=0}^{\infty }b(n)q_{2}^{n}. \end{aligned}$$
(2.10)
In addition, taking the coset representatives \(\{I,S,ST^{-1},ST,ST^{2},ST^{2}S^{-1}\}\) for \(\Gamma _{0}(4)\backslash \mathrm{SL}_{2}({\mathbb {Z}})\), it is easy to check by (2.6)–(2.7) that
$$\begin{aligned} \rho _{L}(S)^{-1}\phi _{0}&=-\frac{i}{2}\left( \phi _{0}+\phi _{1}+\phi _{i}+\phi _{1+i}\right) ,\\ \rho _{L}(ST^{-1})^{-1}\phi _{0}&=-\frac{i}{2}\left( \phi _{0}-i\phi _{1}-i\phi _{i}-\phi _{1+i}\right) ,\\ \rho _{L}(ST)^{-1}\phi _{0}&=-\frac{i}{2}\left( \phi _{0}+i\phi _{1}+i\phi _{i}-\phi _{1+i}\right) ,\\ \rho _{L}(ST^{2})^{-1}\phi _{0}&=-\frac{i}{2}\left( \phi _{0}-\phi _{1}-\phi _{i}+\phi _{1+i}\right) ,\\ \rho _{L}(ST^{2}S^{-1})^{-1}\phi _{0}&=\phi _{1+i}. \end{aligned}$$
Finally, direct calculations yield
$$\begin{aligned} \vec {F}(\tau )&=\left( -2iF_{0}+F\right) \phi _{0}-2iF_{3}\phi _{1}-2iF_{3}\phi _{i}+\left( -2iF_{2}-F_{1/2}\right) \phi _{1+i}. \end{aligned}$$
(2.11)
The following theorem gives some basic facts about \(F_{0}\), \(F_{2}\), \(F_{3}\) and \(F_{1/2}\).

Theorem 2.4

With the above definitions, we have
$$\begin{aligned} F_{0}&\in M_{-k}^{!}(\Gamma _{0}(4),\chi _{-4}), \end{aligned}$$
(2.12)
$$\begin{aligned} F_{3}&\in M_{-k}^{!}(\Gamma _{0}(4),\chi _{1}) \end{aligned}$$
(2.13)
where \(\chi _{1}(\gamma )=\chi _{-4}(d)e(-ab/4)\) for \(\gamma =\begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix}\in \Gamma _{0}(4)\),
$$\begin{aligned} (2iF_{2}+F_{1/2})&\in M^{!}_{-k}(\Gamma _{0}(4),\chi _{2}) \end{aligned}$$
(2.14)
where \(\chi _{2}(\gamma )=\chi _{-4}(d)e(-ab/2)\) for \(\gamma =\begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix}\in \Gamma _{0}(4)\),
and
$$\begin{aligned} F_{1/2}&\in M_{-k}^{!}(\delta ^{-1}\Gamma _{0}(4)\delta ,\chi _{-4}) \end{aligned}$$
(2.15)
where \(\delta =\begin{pmatrix}1&{}\quad 0\\ 2&{}\quad 1\end{pmatrix}\).

Proof

By (2.11), and [19, Section 3, p. 6] or [20, Proposition 4.5], we can show that for \(\gamma =\begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix}\in \Gamma _{0}(4)\),
$$\begin{aligned} \left. (-2iF_{0}+F)\right| _{-k}\gamma&=\chi _{-4}(d)(-2iF_{0}+F), \end{aligned}$$
(2.16)
$$\begin{aligned} \left. F_{3}\right| _{-k}\gamma&=\chi _{-4}(d)e(-ab/4)F_{3},\end{aligned}$$
(2.17)
$$\begin{aligned} \left. (-2iF_{2}-F_{1/2})\right| _{-k}\gamma&=\chi _{-4}(d)e(-ab/2)(-2i F_{2}-F_{1/2}). \end{aligned}$$
(2.18)
Since \(F\in M_{-k}^{!}(\Gamma _{0}(4),\chi _{-4})\), then (2.16) implies (2.12). Relations (2.13) and (2.14) follow directly from (2.17) and (2.18), respectively. The last relation (2.15) follows from the definition of \(F_{1/2}\),
$$\begin{aligned} F_{1/2}=\left. F\right| _{-k}\begin{pmatrix}1&{}\quad 0\\ 2&{}\quad 1\end{pmatrix}. \end{aligned}$$
\(\square \)

Theorem 2.5

Let k be odd. Let \(F=F(\tau )\in M^{!,\infty }_{-k}(\Gamma _{0}(4),\chi _{-4})\) with
$$\begin{aligned} F(\tau )=\sum _{n=-m}^{\infty }c(n)q^{n}. \end{aligned}$$
Write
$$\begin{aligned} \left. F\right| _{-k}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}=\sum _{n=0}^{\infty }a(n)q_{4}^{n}\quad \text{ and }\quad \left. F\right| _{-k}\begin{pmatrix}1&{}\quad 0\\ 2&{}\quad 1\end{pmatrix}=\sum _{n=0}^{\infty }b(n)q_{2}^{n}. \end{aligned}$$
And let the \(\Gamma _{0}(4)\)-lifting of F be
$$\begin{aligned} \vec {F}(\tau )=\sum _{\mu \in L'/L}\sum _{\begin{array}{c} n\in \mathbb {Q}\\ n\gg -\infty \end{array}}c(n,\phi _{\mu })q^{n}\phi _{\mu }. \end{aligned}$$
Then we have
  1. (i)
    $$\begin{aligned} c(n,\phi _{0})&=-2ia(4n)+c(n),\\ c(n,\phi _{1})=c(n,\phi _{i})&=-2ia(4n),\\ c(n,\phi _{1+i})&=-2ia(4n)-b(2n), \end{aligned}$$
     
  2. (ii)
    the principal part of the vector-valued modular form \(\vec {F}(\tau )\) is
    $$\begin{aligned} \left( c(-m)q^{-m}+\cdots +c(-1)q^{-1}\right) \phi _{0}, \end{aligned}$$
     
  3. (iii)
    the constant term of the \(\phi _{0}\)-component of \(\vec {F}(\tau )\) is
    $$\begin{aligned} c(0,\phi _{0})=-(8i)^{k+1}\sum _{n=\frac{k+1}{2}}^{m}c(-n)P_{k,n-\frac{k+1}{2}}(0)+c(0), \end{aligned}$$
    where \(P_{k,n}(x)\) are the polynomials defined as in the proof of Theorem 2.1.
     
In particular, when \(k=1\), the constant term of the \(\phi _{0}\)-component of \(\vec {F}(\tau )\) is
$$\begin{aligned} c(0,\phi _{0})=\sum _{n=1}^{m}c(-n)\left( \sum _{d|n}\left( 64\chi _{-4}(n/d)+4\chi _{-4}(d)\right) d^{2}\right) . \end{aligned}$$
(2.19)

Proof

Assertion (i) follows directly from (2.11). For the assertion (ii), since F is holomorphic at 0 and \(\frac{1}{2}\), then \(F_{j}\) for \(j\in \{0,2,3\}\) and \(F_{1/2}\) will not contribute anything to the principal part of \(\vec {F}\). So the principal part of \(\vec {F}\) is given by
$$\begin{aligned} \left( c(-m)q^{-m}+\cdots +c(-1)q^{-1}\right) \phi _{0}. \end{aligned}$$
For the assertion (iii), we first note by (i) that
$$\begin{aligned} c(0,\phi _0)=-2ia(0)+c(0). \end{aligned}$$
By Theorem 2.1(1), we have
$$\begin{aligned} F=c(-m)\theta _{2}\theta _{1}^{-\frac{k+1}{2}}P_{k,m-\frac{k+1}{2}}(\varphi _{\infty })+\cdots +c\left( -\frac{k+1}{2}\right) \theta _{2}\theta _{1}^{-\frac{k+1}{2}}P_{k,0}(\varphi _{\infty }) \end{aligned}$$
(2.20)
Since \(\theta _{1}\) and \(\theta _{2}\) do not vanish at the cusp 0, and \(\varphi _{\infty }\) has a simple zero at 0 of width 4, then we have
$$\begin{aligned} \left. \theta _{2}\theta _{1}^{-\frac{k+1}{2}}\varphi _{\infty }^{l}\right| _{-k}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}=O(q^{\frac{l}{4}}), \end{aligned}$$
and thus \(\left. \theta _{2}\theta _{1}^{-\frac{k+1}{2}}\varphi _{\infty }^{l}\right| _{-1}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}\) will not contribute anything to the constant term of \(F_{0}\) when \(l\ge 1\). Moreover, simple calculation using the transformation formula for the Dedekind eta function shows that the constant term of the Fourier expansion at the cusp 0 of \(\theta _{2}\theta _{1}^{-\frac{k+1}{2}}\) is \(-(8i)^{k+1}\). Therefore,
$$\begin{aligned} a(0)&=\left( \left. \sum _{n=\frac{k+1}{2}}^{m}c(-n)P_{k,n-\frac{k+1}{2}}(0)\theta _{2}\theta _{1}^{-\frac{k+1}{2}}\right| _{-k}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}\right) _{0}\\&=-(8i)^{k+1}\sum _{n=\frac{k+1}{2}}^{m}c(-n)P_{k,n-\frac{k+1}{2}}(0) \end{aligned}$$
where \((f)_{0}\) denotes the constant term of the q-expansion of f. Hence, we have
$$\begin{aligned} c(0,\phi _{0})=-(8i)^{k+1}\sum _{n=\frac{k+1}{2}}^{m}c(-n)P_{k,n-\frac{k+1}{2}}(0)+c(0). \end{aligned}$$
For (2.19), according to (iii), we need to show that
$$\begin{aligned} P_{1,m}(0)=\sum _{d|(m+1)}\chi _{-4}((m+1)/d)d^{2}\quad \text{ and }\quad c(0)=\sum _{n=1}^{m}c(-n)\left( 4\sum _{d|n}\chi _{-4}(d)d^{2}\right) . \end{aligned}$$
For the first formula, we first observe that
$$\begin{aligned} \theta _{2}\theta _{1}^{-1}\varphi _{\infty }^{\ell }=q^{-\ell -1}+\sum _{j=1}^{\ell }c_{\ell }(-j)q^{-j}+O(1) \end{aligned}$$
for \(0\le \ell \le m\). Thus there are \(b_{1},\ldots ,b_{m-1}\) such that
$$\begin{aligned} h(\tau )&:=\theta _{2}\theta _{1}^{-1}\varphi _{\infty }^{m}+b_{m-1}\theta _{2}\theta _{1}^{-1} \varphi _{\infty }^{m-1}+\cdots +b_{1}\theta _{2}\theta _{1}^{-1}\varphi _{\infty }\\&=q^{-m-1}+a(-1)q^{-1}+O(1) \end{aligned}$$
for some constant \(a(-1)\). Let \(g(\tau )\) be defined by
$$\begin{aligned} g(\tau )=\sum _{n=1}^{\infty }\left( \sum _{d|n}\chi _{-4}(n/d)d^{2}\right) q^{n}=\sum _{n=1}^{\infty }d_{n}q^{n}. \end{aligned}$$
It is known [21] that \(g(\tau )\) is a weight 3 modular form on \(\Gamma _{0}(4)\) with character \(\chi _{-4}\). We note by the basic facts about \(\theta _{1}\), \(\theta _{2}\) and \(\varphi _{\infty }\) that \(h(\tau )\) vanishes at the cusps 1 / 2 and 0. Then by [22, Theorem 3.1], we have
$$\begin{aligned} d_{m+1}+a(-1)=0,\,\, i.e., \,\, d_{m+1}=-a(-1). \end{aligned}$$
Therefore
$$\begin{aligned} P_{1,m}(0)=d_{m+1}=\sum _{d|(m+1)}\chi _{-4}((m+1)/d)d^{2}. \end{aligned}$$
This proves the first formula. For the second one, the proof is similar by noting that
$$\begin{aligned} h_1(\tau ):=\theta _{2}\theta _{1}^{-1}P_{1,m}(\varphi _{\infty })=q^{-m-1}+C+O(q) \end{aligned}$$
and
$$\begin{aligned} g_1(\tau )=1+4\sum _{n=1}^{\infty }\left( \sum _{d|n}\chi _{-4}(d)d^{2}\right) q^{n} \end{aligned}$$
is [21] a weight 3 modular form on \(\Gamma _{0}(4)\) with character \(\chi _{-4}\). Then again [22, Theorem 3.1] shows that
$$\begin{aligned} C=4\sum _{d|(m+1)}\chi _{-4}(d)d^{2}. \end{aligned}$$
This together with (2.20) proves the second formula. \(\square \)

Example 2.6

Let \(k=1\) and \(F(\tau )=\theta _{2}\theta _{1}^{-1}=\frac{\eta _{2}^{14}}{\eta _{1}^{4}\eta _{4}^{12}}\in M_{-1}^{!,\infty }(\Gamma _{0}(4),\chi _{-4})\). Then we have
$$\begin{aligned} \vec {F}(\tau )=\left( -2iF_{0}+F\right) \phi _{0}-2iF_{3}\phi _{1}-2iF_{3}\phi _{i}+\left( -2iF_{2}-F_{1/2}\right) \phi _{1+i} \end{aligned}$$
(2.21)
where \(F_{0}\), \(F_{2}\), \(F_{3}\) and \(F_{1/2}\) are defined as in (2.9) and (2.10). We have
$$\begin{aligned} \left. F\right| _{-1}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}&=32i\frac{\eta (\tau /2)^{14}}{\eta (\tau /4)^{4}\eta (\tau )^{12}}\\&=32i\big (1+12q^{1/4}+76q^{2/4}+352q^{3/4}+1356q+4600q^{5/4}\\&\quad +14176q^{6/4}+40512q^{7/4}+\cdots \big )\\&=32i\left( 1+1356q+O(q^{2})\right) \\&\quad +32i\left( 12q^{1/4}+4600q^{5/4}+O(q^{9/4})\right) \\&\quad +32i\left( 76q^{2/4}+14176q^{6/4}+O(q^{10/4})\right) \\&\quad +32i\left( 352q^{3/4}+40512q^{7/4}+O(q^{11/4})\right) , \end{aligned}$$
then
$$\begin{aligned} F_{0}&=32i\left( 1+1356q+O(q^{2})\right) ,\\ F_{2}&=32i\left( 76q^{2/4}+14176q^{6/4}+O(q^{10/4})\right) \\ F_{3}&=32i\left( 352q^{3/4}+40512q^{7/4}+O(q^{11/4})\right) . \end{aligned}$$
And
$$\begin{aligned} F_{1/2}=\left. F\right| _{-1}\begin{pmatrix}1&{}\quad 0\\ 2&{}\quad 1\end{pmatrix}=64\left( q^{1/2}-8q^{3/2}+42q^{5/2}+O(q^{7/2})\right) . \end{aligned}$$
From (2.21), we note that the principal part of F is \(e(-\tau )\phi _{0}\) and the constant term of the \(\phi _{0}\)-component is \(c(0,\phi _0)=68\).

2.3 Canonical bases for \(M_{-k}^{!,0}(\Gamma _{0}(4),\chi _{-4}^{k})\) and \(M_{-k}^{!,\frac{1}{2}}(\Gamma _{0}(4),\chi _{-4}^{k})\)

We complete this section by giving canonical bases for the other two companions of \(M^{!,\infty }_{-k}(\Gamma _{0}(4),\chi _{-4}^{k})\).

Let \(\theta _{3}(\tau )\), \(\varphi _{0}(\tau )\) and \(\varphi _{1/2}(\tau )\) be defined by
$$\begin{aligned} \theta _{3}=\theta _{3}(\tau )&:=\vartheta _{01}^{4}(\tau )=\frac{\eta _{1}^{8}}{\eta _{2}^{4}}=1+O(q), \end{aligned}$$
(2.22)
$$\begin{aligned} \varphi _{0}=\varphi _{0}(\tau )&:=\left( \frac{\eta _{4}}{\eta _{1}}\right) ^{8}=q+O(q^{2}),\end{aligned}$$
(2.23)
$$\begin{aligned} \varphi _{1/2}=\varphi _{1/2}(\tau )&:=\frac{\eta _{1}^{8}\eta _{4}^{16}}{\eta _{2}^{24}}=q+O(q^{2}). \end{aligned}$$
(2.24)
Here are some basic facts about \(\theta _{3}\), \(\varphi _{0}\) and \(\varphi _{1/2}\):
  1. (1)

    \(\theta _{3}(\tau )\) is a weight 2 modular form on \(\Gamma _{0}(4)\) with trivial character, has a simple zero at the cusp 0, and vanishes nowhere else;

     
  2. (2)

    \(\varphi _{0}(\tau )\) is a weight 0 modular form on \(\Gamma _{0}(4)\) with trivial character, has a simple pole at the cusp 0 and a simple zero at the cusp \(\infty \), and vanishes nowhere else;

     
  3. (3)

    \(\varphi _{1/2}(\tau )\) is a weight 0 modular form on \(\Gamma _{0}(4)\) with trivial character, has a simple pole at the cusp \(\frac{1}{2}\) and a simple zero at the cusp \(\infty \), and vanishes nowhere else.

     

Theorem 2.7

Let \(\theta _{2}\), \(\theta _{3}\) and \(\varphi _{0}\) be as defined in (2.2), (2.22) and (2.23), respectively.
  1. (1)
    For k odd, the set \(\{\theta _{2}\theta _{3}^{-\frac{k+1}{2}}P_{k,m}(\varphi _{0})\}_{m=0}^{\infty }\) where \(P_{k,m}\) is a monic polynomial of degree m such that
    $$\begin{aligned} \left. \theta _{2}\theta _{3}^{-\frac{k+1}{2}}P_{k,m}(\varphi _{0})\right| _{-k}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}=q_{4}^{-\frac{k+1}{2}-m}+\sum _{n=-\frac{k-1}{2}}^{\infty }c(n)q_{4}^{n}, \end{aligned}$$
    is a canonical basis for \(M^{!,0}_{-k}(\Gamma _{0}(4),\chi _{-4})\).
     
  2. (2)
    For k even, the set \(\{\theta _{3}^{-\frac{k}{2}}P_{k,m}(\varphi _{0})\}_{m=0}^{\infty }\) where \(P_{k,m}\) is a monic polynomial of degree m such that
    $$\begin{aligned} \left. \theta _{3}^{-\frac{k}{2}}P_{k,m}(\varphi _{0})\right| _{-k}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}=q_{4}^{-\frac{k}{2}-m}+\sum _{n=-\frac{k}{2}+1}^{\infty }c(n)q_{4}^{n}, \end{aligned}$$
    is a canonical basis for \(M^{!,0}_{-k}(\Gamma _{0}(4))\).
     

Theorem 2.8

Let \(\theta _{2}\) and \(\varphi _{1/2}\) be as defined in (2.2) and (2.24), respectively. Then the set \(\{\theta _{2}^{-k}P_{k,m}(\varphi _{1/2})\}_{m=0}^{\infty }\) where \(P_{k,m}\) is a monic polynomial of degree m such that
$$\begin{aligned} \left. \theta _{2}^{-k}P_{k,m}(\varphi _{1/2})\right| _{-k}\begin{pmatrix}1&{}\quad 0\\ 2&{}\quad 1\end{pmatrix}=q^{-\frac{k}{2}-m}+\sum _{n=-\frac{k}{2}+1}^{\infty }c(n)q^{n}, \end{aligned}$$
is a canonical basis for \(M^{!,\frac{1}{2}}_{-k}(\Gamma _{0}(4),\chi _{-4}^{k})\).

Proofs of Theorems 2.7 and 2.8 are similar to that of Theorem 2.1, so we omit the details.

Remark 2.9

For a cusp P, denote by \(M^{!,P}_{-k,\rho _{L}}\) the space of vector-valued modular forms induced from \(M^{!,P}_{-k}(\Gamma _{0}(4),\chi _{-4}^k)\) via \(\Gamma _{0}(4)\)-lifting. We have, by (2.11),
$$\begin{aligned} M^{!,\infty }_{-k,\rho _{L}}+M^{!,0}_{-k,\rho _{L}}+M^{!,\frac{1}{2}}_{-k,\rho _{L}}=M^{!,\infty }_{-k,\rho _{L}}\oplus M^{!,0}_{-k,\rho _{L}}\oplus M^{!,\frac{1}{2}}_{-k,\rho _{L}}. \end{aligned}$$
Clearly, \(M^{!,\infty }_{-k,\rho _{L}}+M^{!,0}_{-k,\rho _{L}}+M^{!,\frac{1}{2}}_{-k,\rho _{L}}\) is a subspace of \(M^{!}_{-k,\rho _{L}}\). In general, the former space may not be equal to the latter one. We first note that every vector-valued modular form in \(M^{!,\infty }_{-k,\rho _{L}}+M^{!,0}_{-k,\rho _{L}}+M^{!,\frac{1}{2}}_{-k,\rho _{L}}\) must have the same component functions at \(\phi _{1}\) and \(\phi _{i}\). We now give an example of functions in \(M^{!}_{-1,\rho _{L}}\) that does not have this property. Let \(F(\tau )=\theta _{2}\theta _{1}^{-1}\in M_{-1}^{!,\infty }(\Gamma _{0}(4),\chi _{-4})\). Then as above we write the \(\Gamma _{0}(4)\)-lifting of \(F(\tau )\) as
$$\begin{aligned} \vec {F}(\tau )&=\left( -2iF_{0}+F\right) \phi _{0}-2iF_{3}\phi _{1}-2iF_{3}\phi _{i}+\left( -2iF_{2}-F_{1/2}\right) \phi _{1+i} \end{aligned}$$
where
$$\begin{aligned} F_{j}=\sum _{n=0}^{\infty }a(4n+j)q_{4}^{4n+j},\\ \left. F\right| _{-k}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}=\sum _{n=0}^{\infty }a(n)q_{4}^{n} \end{aligned}$$
and
$$\begin{aligned} F_{1/2}=\left. F\right| _{-1}\begin{pmatrix}1&{}\quad 0\\ 2&{}\quad 1\end{pmatrix}. \end{aligned}$$
By (2.13), we know that \(F_{3}(\tau )\in M^{!}_{-1}(\Gamma _{1}(4),\chi )\) where \(\chi \left( \begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix}\right) =e(-b/4)\). Now we do \(\Gamma _{1}(4)\)-lifting on \(F_{3}(\tau )\) against \(\phi _{1}\), namely,
$$\begin{aligned} \vec {F}_{3}(\tau )=\sum _{\gamma \in \Gamma _{1}(4)\backslash \mathrm{SL}_{2}({\mathbb {Z}})}\left( \left. F_{3}\right| _{-1}\gamma \right) \rho _{L}(\gamma )^{-1}\phi _{1}, \end{aligned}$$
and get
$$\begin{aligned} \vec {F}_{3}(\tau )=-4i{f}_{0}\phi _{0}+(2F_{3}+4if_{3})\phi _{1}+(-4if_{3}-2f_{1/2})\phi _{i}+4if_{2}\phi _{1+i} \end{aligned}$$
where
$$\begin{aligned} f_{j}=\sum _{\begin{array}{c} n\in \mathbb {Z}\\ n\gg -\infty \end{array}}{\tilde{a}}(4n+j)q_{4}^{4n+j},\\ \left. F_{3}\right| _{-1}\begin{pmatrix}0&{}\quad -1\\ 1&{}\quad 0\end{pmatrix}=\sum _{\begin{array}{c} n\in \mathbb {Z}\\ n\gg -\infty \end{array}}{\tilde{a}}(n)q_{4}^{n} \end{aligned}$$
and
$$\begin{aligned} f_{1/2}=\left. F_{3}\right| _{-1}\begin{pmatrix}1&{}\quad 0\\ 2&{}\quad 1\end{pmatrix}. \end{aligned}$$
Now the component functions at \(\phi _{1}\) and \(\phi _{i}\) are \(2F_{3}+4if_{3}\) and \(-4if_{3}-2f_{1/2}\), respectively. We can compute and verify that they are not the same. Therefore, \(\vec {F}_{3}(\tau )\) is not in the space \(M^{!,\infty }_{-k,\rho _{L}}+M^{!,0}_{-k,\rho _{L}}+M^{!,\frac{1}{2}}_{-k,\rho _{L}}\).

3 Part II: Borcherds products on \(\mathrm{U}(2,1)\)

It is well-known that the vector-valued weakly modular forms construction in Part I can be used to construct memomophic modular forms on Shimura varieties of orthogonal type (n, 2) and unitary type (n, 1) with Borcherds product formulas and known divisors. In this part, we focus on one special case to make it very explicitly—the Picard modular surfaces over . In particular, we describe a Weyl chamber explicitly and write down the Borcherds product expression concretely.

This part is devoted to deriving Borcherds products lifted from a vector-valued modular form arising from \(M^{!,\infty }_{-1}(\Gamma _{0}(4),\chi _{-4})\).

3.1 Picard modular surfaces over

Let \((V, \langle \, , \rangle )\) be a Hermitian vector space over of signature (2, 1) and let \(H=\mathrm{U}(V)\), where \(\mathrm{U}(V)\) denotes the unitary group associated to V. Let , and
$$\begin{aligned} {\mathcal {L}}=\{ w \in V_{\mathbb {C}}|\, \langle w, w \rangle < 0\}. \end{aligned}$$
Then \({\mathcal {K}} ={\mathcal {L}}/{\mathbb {C}}^\times \) is the Hermitian domain for \(H({\mathbb {R}})\), and \({\mathcal {L}}\) is the tautological line bundle over \({\mathcal {K}}\). For a congruence subgroup \(\Gamma \) of \(H({\mathbb {Q}})\), the associated Picard modular surface \(X_\Gamma =\Gamma \backslash {\mathcal {K}}\) is defined over some number field.
Given an isotropic line (i.e., a cusp), choose another isotropic element \(\ell '\) with \(\langle \ell , \ell '\rangle \ne 0\). Let , and let
$$\begin{aligned} {\mathcal {H}}={\mathcal {H}}_{\ell , \ell '}=\left\{ (\tau , \sigma )\in {\mathbb {H}}\times V_{0, {\mathbb {C}}}\left| \, \mathop {\hbox {Im}}\nolimits {\tau }>\frac{\langle \sigma ,\sigma \rangle }{4|\langle \ell ',\ell \rangle |^{2}}\right. \right\} . \end{aligned}$$
Then the map
$$\begin{aligned} {\mathcal {H}} \rightarrow {\mathcal {L}}, \,\, (\tau , \sigma ) \mapsto z(\tau , \sigma ) = 2i\langle \ell ',\ell \rangle \tau \ell +\sigma + \ell ' \end{aligned}$$
(3.1)
gives rise to an isomorphism \({\mathcal {H}} \cong {\mathcal {K}}\). It is also a nowhere vanishing section of the line bundle \({\mathcal {L}}\). Using this map, we can define action of \(H({\mathbb {R}})\) on \({\mathcal {H}}\) and automorphy factor \(j(\gamma , \tau , \sigma )\) via the equation
$$\begin{aligned} \gamma z(\tau , \sigma ) =j(\gamma , \tau , \sigma ) z(\gamma (\tau , \sigma )). \end{aligned}$$
(3.2)
Indeed, both \(\gamma z(\tau ,\sigma )\) and \(z(\gamma (\tau ,\sigma ))\) are in \(\mathcal {L}\) and they become the same in \(\mathcal {K}\), so they are different by a multiplication constant, namely, the automorphy factor \(j(\tau ,\sigma )\).

Definition 3.1

Let \(\Gamma \) be a unitary modular group. A holomorphic automorphic form of weight k and with character \(\chi \) for \(\Gamma \) is a function \(g:\mathcal {H}\rightarrow \mathbb {C}\), with the following properties:
  1. (1)

    g is holomorphic on \(\mathcal {H}\),

     
  2. (2)

    \(g(\gamma (\tau ,\sigma ))=j(\gamma ;\tau ,\sigma )^{k}\chi (\gamma )g(\tau ,\sigma )\) for all \(\gamma \in \Gamma \).

     

We remark that a holomorphic modular form g for \(\Gamma \) is automatically holomorphic at the cusps.

Now we make everything concrete and explicit. First choose a basis \(\{\mathbf{e _1}, \mathbf{e _2}, \mathbf{e _3}\}\) of V with Gram matrix
$$\begin{aligned} J=\begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 \end{pmatrix} \end{aligned}$$
so with Hermitian form
$$\begin{aligned} \langle {x}, {y}\rangle =x_{1}{\bar{y}}_{3}+x_{2}{\bar{y}}_{2}+x_{3}{\bar{y}}_{1}={}^{t}xJ{\bar{y}}, \end{aligned}$$
(3.3)
and
We take the lattice
$$\begin{aligned} L= \mathbb {Z}[i]\oplus \mathbb {Z}[i]\oplus \frac{1}{2}\mathbb {Z}[i] \end{aligned}$$
(instead of the typical \({\mathbb {Z}}[i]^3\)). Its \({\mathbb {Z}}\)-dual lattice is
So \(L'/L \cong \frac{1}{2}{\mathbb {Z}}[i]/{\mathbb {Z}}[i]\) with quadratic form \(Q(x) =x {{\bar{x}}} \in \frac{1}{4}{\mathbb {Z}}/{\mathbb {Z}}\), which is the same finite quadratic module considered in Part I. Let
$$\begin{aligned} \mathrm{U}(L)&=\{g\in H|\,gL=L\}\\&=H\cap \left\{ \begin{pmatrix}\mathbb {Z}[i]&{}\quad \mathbb {Z}[i]&{}\quad 2\mathbb {Z}[i]\\ \mathbb {Z}[i]&{}\quad \mathbb {Z}[i]&{}\quad 2\mathbb {Z}[i]\\ \frac{1}{2}\mathbb {Z}[i]&{}\quad \frac{1}{2}\mathbb {Z}[i]&{}\quad \mathbb {Z}[i]\end{pmatrix}\right\} . \end{aligned}$$
be the stabilizer of L in H, and \(\Gamma _L\) be the subgroup of \(\mathrm{U}(L)\) which acts on the discriminant group \(L'/L\) trivially:
$$\begin{aligned} \Gamma _{L}=\mathrm{U}(L)\cap \left\{ \begin{pmatrix}\mathbb {Z}[i]&{}\quad 2\mathbb {Z}[i]&{}\quad 2\mathbb {Z}[i]\\ \mathbb {Z}[i]&{}\quad 1+2\mathbb {Z}[i]&{} \quad 2\mathbb {Z}[i]\\ \mathbb {Z}[i]&{}\quad 2\mathbb {Z}[i]&{}\quad \mathbb {Z}[i]\end{pmatrix}\right\} . \end{aligned}$$
(3.4)
Take the cusp \(\ell =\mathbf e _1\) and \(\ell '=\mathbf e _3\). Then with Hermitian form \(\langle x, y\rangle =x {{\bar{y}}}\), and
$$\begin{aligned} {\mathcal {H}}=\{ (\tau , \sigma ) \in {\mathbb {H}} \times {\mathbb {C}}| \, 4 \text{ Im }(\tau ) > |\sigma |^2\}. \end{aligned}$$
Moreover, one has for \(\gamma =(a_{ij}) \in H\)
$$\begin{aligned} \gamma (\tau ,\sigma )=\bigg (\frac{a_{11}\tau +(2i)^{-1}a_{12}\sigma +(2i)^{-1}a_{13}}{2i a_{31}\tau +a_{32}\sigma +a_{33}},\frac{2i a_{21}\tau +a_{22}\sigma +a_{23}}{2i a_{31}\tau +a_{32}\sigma +a_{33}}\bigg ). \end{aligned}$$
and
$$\begin{aligned} j(\gamma , \tau ,\sigma )=\frac{\langle \gamma z,\ell \rangle }{\langle \ell ',\ell \rangle }=2i\tau a_{31}+a_{32}\sigma +a_{33}. \end{aligned}$$
Our Picard modular surface is the quotient space \(X_{\Gamma _{L}}=\Gamma _{L}\backslash \mathcal {H}\) of \(\mathcal {H}\) modulo the action of \(\Gamma _{L}\).
Let \(P_\ell \) be the stabilizer of the cusp in H. Then \(P_\ell =N_\ell M_\ell \) with
where is the norm one group. Notice that \(N_\ell \) is a Heisenberg group action on \({\mathcal {H}}_{\ell , \ell '}\) via
$$\begin{aligned} n(b, c) (\tau , \sigma ) = ( \tau + c+ i{{\bar{b}}} (\sigma +b), \sigma +b). \end{aligned}$$
In particular
$$\begin{aligned} n(0, c)(\tau , \sigma )= (\tau +c, \sigma ). \end{aligned}$$
Let
$$\begin{aligned} \Gamma _{L, \ell } =\Gamma _L \cap N_\ell = \{ n(b, c)|\, b \in {\mathbb {Z}}[i], c \in {\mathbb {Z}}\}. \end{aligned}$$
Then for a holomorphic modular form \(f(\tau , \sigma )\) for \(\Gamma _L\), we have a Fourier–Jacobi expansion at the cusp :
$$\begin{aligned} f(\tau , \sigma )=\sum _{n \ge 0} f_n(\sigma ) q^n. \end{aligned}$$
(3.5)

3.2 The hermitian space V as a quadratic space

As mentioned in the previous subsection, the hermitian space V can be viewed as a quadratic space \(V_{\mathbb {Q}}\) of signature (4,2) associated with bilinear form induced from the hermitian form:
Then the lattice L can be considered as a quadratic \(\mathbb {Z}\)-lattice in \(V_{\mathbb {Q}}\). Denote by
$$\begin{aligned} {\text {SO}}(V_{\mathbb {Q}})=\{g\in {\text {SL}}(V_{\mathbb {Q}})|\,(gx,gy)=(x,y)\hbox { for all}\ x,y\in V_{\mathbb {Q}}\} \end{aligned}$$
the special orthogonal group of \(V_{\mathbb {Q}}\) and its set of real points as \({\text {SO}}(V_{\mathbb {Q}})(\mathbb {R})\cong {\text {SO}}(4,2)\). A model for the symmetric domain of \({\text {SO}}(V_{\mathbb {Q}})(\mathbb {R})\) is the Grassmannian of two-dimensional negative definite subspaces of \(V_{\mathbb {Q}}\), denoted by \(\mathrm{Gr}_{O}\). It can be realized as a tube domain \(\mathcal {H}_{O}\) as follows. Denote by \(V_{\mathbb {Q}}(\mathbb {C})\) the complex quadratic space \(V_{\mathbb {Q}}\otimes _{\mathbb {Q}}\mathbb {C}\) with \((\cdot ,\cdot )\) extended to a \(\mathbb {C}\)-valued bilinear form.
Now we view L as a \(\mathbb {Z}\)-lattice. Let \(e_{1}\in L\) be a primitive isotropic lattice vector and choose an isotropic dual vector \(e_{2}\in L'\) with \((e_{1},e_{2})=1\). Denote by K the Lorentzian \(\mathbb {Z}\)-sublattice \(K=L\cap e_{1}^{\perp }\cap e_{2}^{\perp }\) with respect to \((\cdot ,\cdot )\). The tube domain model \(\mathcal {H}_{O}\) is one of the the two connected components of the following subset of \(K\otimes _{\mathbb {Z}}\mathbb {C}\)
$$\begin{aligned} \{Z=X+iY|\,X,Y\in K\otimes _{\mathbb {Z}}\mathbb {R},\,Q(Y)<0\}. \end{aligned}$$
Recall that \(\ell =\mathbf e _{1}\) and \(\ell '=\mathbf e _{3}\). We define
$$\begin{aligned} e_{1}=\ell ,\,e_{2}=\hat{\frac{1}{2}}\ell ',\,e_{3}=-{\hat{i}}\ell ,\,e_{4}={-\frac{{{\hat{i}}}}{2}}\ell ' \end{aligned}$$
where we denote by \({\hat{\mu }}\) the endomorphism of \(V_{\mathbb {Q}}(\mathbb {R})\) induced from the scalar multiplication with \(\mu \). Then we can check that \(\{e_{1},e_{2},e_{3},e_{4}\}\) is a basis for \((\mathbb {Z}[i]\ell +\mathbb {Z}[i]\ell ')\otimes _{\mathbb {Z}}\mathbb {Q}\) and we can see that \(K\otimes _{\mathbb {Z}}\mathbb {R}=((\mathbb {Q}e_{3}+\mathbb {Q}e_{4})\otimes _{\mathbb {Z}}\mathbb {R})\oplus (V_{0}\otimes _{\mathbb {Z}}\mathbb {R})\). Thus we can identify Y with \(y_{1}e_{3}+y_{2}e_{4}+\sigma \in K\otimes _{\mathbb {Z}}\mathbb {R}\). Now denote by \(\mathcal {C}\) the set of \(Y=y_{1}e_{3}+y_{2}e_{4}+\sigma \) with \(y_{1}y_{2}+Q(\sigma )<0\), \(y_{1}<0\) and \(y_{2}>0\). We can fix \(\mathcal {H}_{O}\) as the component for which \(Y\in \mathcal {C}\). Therefore, \(\mathcal {H}_{O}=K\otimes _{\mathbb {Z}}\mathbb {R}+i\mathcal {C}\).
In addition, the tube domain \(\mathcal {H}_{O}\) can be mapped biholomorphically to any one of the two connected components of a negative cone of \(\mathbb {P}^{1}(V_{\mathbb {Q}})(\mathbb {C})\) given by
$$\begin{aligned} \{[Z_{L}]\in \mathbb {P}^{1}(V_{\mathbb {Q}})(\mathbb {C})|\,(Z_{L},Z_{L})=0,\, (Z_{L},{\bar{Z}}_{L})<0\}. \end{aligned}$$
We fix this component and denote it by \(\mathcal {K}_{O}\). For each \([Z_{L}]\), we can uniquely represent it as
$$\begin{aligned} Z_{L}=e_{2}-q(Z)e_{1}+Z \end{aligned}$$
with \(Z\in \mathcal {H}_{O}\).

3.3 Embedding of \(\mathcal {H}\) into \(\mathcal {H}_{O}\)

As in [15, Section 4], we can embed \(\mathcal {H}\) into \(\mathcal {H}_{O}\) via
$$\begin{aligned} (\tau ,\sigma )\rightarrow \iota (\tau , \sigma )= -\tau e_{3}+ie_{4}+\mathfrak {z}(\sigma ) \end{aligned}$$
(3.6)
where
$$\begin{aligned} \mathfrak {z}(\sigma )=\frac{{\hat{1}}}{2}\sigma +i\left( -\frac{{\hat{i}}}{2}\right) \sigma . \end{aligned}$$
(3.7)
Similarly, \(\mathcal {K}_{U}\) can be embedded into \(\mathcal {K}_{O}\) through the identifications between \(\mathcal {K}_{U}\) and \(\mathcal {H}\), and between \(\mathcal {K}_{O}\) and \(\mathcal {H}_{O}\). Namely,
$$\begin{aligned} z=\ell '+2i\tau \ell +\sigma \rightarrow Z_{L}=-i\tau e_{1}+e_{2}-\tau e_{3}+ie_{4}+\mathfrak {z}(\sigma ). \end{aligned}$$
(3.8)

3.4 Weyl chambers of \(K\otimes _{\mathbb {Z}}\mathbb {R}\)

In Theorem 2.1 (1), we have shown that \(F_{1, m} =q^{-m}+O(1)\) for \(m\ge 1\), form a canonical basis for \(M^{!,\infty }_{-1}(\Gamma _{0}(4),\chi _{-4})\). Therefore, to study the Borcherds product lifted from \(M^{!,\infty }_{-1,\rho _{L}}\), it suffices to start with \(F_{1, m}\). Since we only deal with weight \(-1\) in the rest of this paper, we will simply write \(F_m=F_{1, m}\), and \(\vec {F}_m=\vec {F}_{1, m}\).

For general definitions of the following, we refer the reader to [1, Chapter 3.1]. For \(\kappa \in K\) with \(q(\kappa ) >0\), denote by \(\kappa ^{\perp }\) the orthogonal complement of \(\kappa \) in \(K\otimes _{\mathbb {Z}}\mathbb {R}\). Denote by \(\mathcal {D}_{K}\) the Grassmannian of negative 1-lines of \(K\otimes _{\mathbb {Z}}\mathbb {R}\), which can be realized as
$$\begin{aligned} \mathcal {D}_{K}&=\{ {\mathbb {R}} w \subset K_{{\mathbb {R}}}|\, q(w)<0\}\\&\cong \{w=y_{1}e_{3}+e_{4}+(y_{3}+iy_{4})|\, y_i \in \mathbb {R}, q(w)<0\}. \end{aligned}$$
Then by considering the Grassmannian of negative 1-lines of \(\kappa ^{\perp }\), it corresponds to a codimension 1 sub-manifold of the Grassmannian \(\mathcal {D}_{K}\) of \(K\otimes _{\mathbb {Z}}\mathbb {R}\).
In our case, a Heegner divisor of index (m, 0), \(H_{K}(m,0)\), is a locally finite union of codimension 1 sub-manifolds of \(\mathcal {D}_{K}\), namely,
$$\begin{aligned} H_{K}(m,0)=\{ z \in {\mathcal {D}}_K|\, \exists \kappa \in K \hbox { with } q(\kappa )=m \hbox { and } (z, \kappa ) =0 \} \end{aligned}$$
Let \(\vec {F}_{m}(\tau )\) be the vector-valued modular form arising from \(F_m\). It is known by Theorem 2.5 that the principal part of \(\vec {F}_{m}(\tau )\) is \(q^{-m}\phi _{0}\). The Weyl chambers attached to \(\vec {F}_{m}(\tau )\) are the connected components \(W_{m}\) of
$$\begin{aligned} \mathcal {D}_{K}-H_{K}(m,0). \end{aligned}$$
Fix a Weyl chamber \(W_{m}\) of \(\mathcal {D}_{K}\), we can also define the corresponding Weyl chambers of \(K\otimes _{\mathbb {Z}}\mathbb {R}\) and \(\mathcal {H}\) by
$$\begin{aligned} W_{m,K}&=\{w\in K\otimes _{\mathbb {Z}}\mathbb {R}|\,\,\mathbb {R}w\in W_{m}\},\\ W_{m,U}&=\left\{ (\tau ,\sigma )\in \mathcal {H}\left| \,\mathop {\hbox {Im}}\nolimits (\iota (\tau , \sigma )) =-\mathop {\hbox {Im}}\nolimits \tau e_{3}+e_{4}-\frac{{\hat{i}}}{2}\sigma \in W_{m,K}\right. \right\} , \end{aligned}$$
respectively. In the following lemma, we give an explicit description of the Weyl chamber that we use to construct Borcherds product in Theorem 3.5.

Lemma 3.2

  1. (1)
    Let
    $$\begin{aligned} W_{m}&=\left\{ y_{1}e_{3}+e_{4}+(y_{3}+iy_{4})\in \mathcal {D}_{K}\left| \begin{array}{c} y_{1}<r^{2}+s^{2}-m+2ry_{3}+2sy_{4}\,\,\forall \,r,\,s\in \mathbb {Z},\\ \\ 1+2ty_{3}+2hy_{4}>0\,\,\forall \,t,\,h\in \mathbb {Z},\,t^{2}+h^{2}=m,\\ \\ ty_{3}+hy_{4}>0,\,\,\forall \,t,\,h\in \mathbb {Z},\,t^{2}+h^{2}=m,\,t>0,\\ \\ y_{4}>0\,\,\text {if }m\text { is a square.} \end{array}\right. \right\} \nonumber \\&\subset \left\{ y_{1}e_{3}+e_{4}+(y_{3}+iy_{4})\in \mathcal {D}_{K}\left| \begin{array}{c} k_{2}y_{1}<-k_{1}+2k_{3}y_{3}+2k_{4}y_{4}\,\,\forall \,k_{i}\in \mathbb {Z},\,k_{2}>0,\,k_{1}k_{2}+k_{3}^{2}+k_{4}^{2}=m,\\ \\ k+2ty_{3}+2hy_{4}>0\,\,\forall \,k,\,t,\,h\in \mathbb {Z},\,k>0,\,t^{2}+h^{2}=m,\\ \\ ty_{3}+hy_{4}>0,\,\,\forall \,t,\,h\in \mathbb {Z},\,t^{2}+h^{2}=m,\,t>0,\\ \\ y_{4}>0\,\,\text {if } m\text { is a square} \end{array}\right. \right\} . \end{aligned}$$
    (3.9)
    Then \(W_{m}\) is a Weyl chamber containing \(e_{3}\).
     
  2. (2)
    Let
    $$\begin{aligned} K_{m}=\left\{ \lambda =\lambda _{1}e_{3}-\lambda _{2}e_{4}+\frac{1}{2}(\lambda _{3}+i\lambda _{4})\in K'\left| \begin{array}{c} (\lambda ,W_{m})>0,\\ \\ \,\,\left( Q(\lambda )=m\,\,with\,\,\lambda _{3},\lambda _{4}\in 2\mathbb {Z}\right) \\ \\ \,or\,\,\left( Q(\lambda )\le 0\right) \end{array}\right. \right\} \end{aligned}$$
    where \((\lambda ,W_{m})>0\) means that \((\lambda ,w)>0\) for all \(w\in W_{m}\). Then
    $$\begin{aligned} K_{m}=\left\{ \lambda =\lambda _{1}e_{3}-\lambda _{2}e_{4}+\frac{1}{2}(\lambda _{3}+i\lambda _{4})\left| \begin{array}{c} \lambda _{1},\,\lambda _{2},\,\lambda _{3},\,\lambda _{4}\in \mathbb {Z},\\ \lambda _{2}>0,\\ {or\,\, (\lambda _{2}=0\,\, and\,\, \lambda _{1}>0),}\\ {or \,\,(\lambda _{2}=\lambda _{1}=0 \,\,and\,\, \lambda _{3}>0),}\\ {or\,\,(\lambda _{2}=\lambda _{1}=\lambda _{3}=0\,\, and \,\, \lambda _{4}>0)} \end{array}\right. \right\} . \end{aligned}$$
     

Proof

For Assertion (1), it is clear that \(W_{m}\) contains \(e_{3}\) since the set of \((y_{3},y_{4},y_{1})\) determined by the inequalities in \(W_{m}\) contains \(y_{1}=-\infty \). We only need to show \(W_{m}\) is actually a Weyl chamber.

Write \(\kappa =k_1 e_3 + k_2 e_4 + k_3 + i k_4 \in K\) with \(k_i \in {\mathbb {Z}}\). Since \((-\kappa )^{\perp }=\kappa ^\perp \), we can assume \(k_{2}\ge 0\). By the definition of Weyl chamber \(W_{m}\), we can see that a Weyl chamber \(W_{m}\) can be viewed as a connected component of \(\mathbb {R}^{3}\) cut out by the planes
$$\begin{aligned} k_{2}y_{1}+k_{1}+2k_{3}y_{3}+2k_{4}y_{4}=0 \end{aligned}$$
for all \(k_{1},\ldots ,k_{4}\in \mathbb {Z}\) with \(k_{2}\ge 0\) and \(k_{1}k_{2}+k_{3}^{2}+k_{4}^{2}=m\).
When \(k_{2}=0\) and m is representable as a sum of two squares, then we have planes
$$\begin{aligned} k_{1}+2k_{3}y_{3}+2k_{4}y_{4}=0 \end{aligned}$$
perpendicularly passing through the \((y_{3},y_{4})\)-plane. In this case, the connected components are determined by the connected components of the \(y_{3}-y_{4}\) plane cut out by the lines
$$\begin{aligned} k_{1}+2k_{3}y_{3}+2k_{4}y_{4}=0, \end{aligned}$$
and it is easy to find that one of the connected components \(\mathcal {C}_{1}\) can be identified as
$$\begin{aligned} \left\{ (y_{3},y_{4})\in \mathbb {R}^{2}\left| \begin{array}{c} 1+2ty_{3}+2hy_{4}>0\,\,\forall \,t,\,h\in \mathbb {Z},\,t^{2}+h^{2}=m,\\ \\ ty_{3}+hy_{4}>0,\,\,\forall \,t,\,h\in \mathbb {Z},\,t^{2}+h^{2}=m,\,t>0,\\ \\ y_{4}>0\,\,\text {if }m\text { is a square} \end{array}\right. \right\} \end{aligned}$$
which is a subset of
$$\begin{aligned} \left\{ (y_{3},y_{4})\in \mathbb {R}^{2}\left| \begin{array}{c} k+2ty_{3}+2hy_{4}>0\,\,\forall \,k,\,t,\,h\in \mathbb {Z},\,k>0,\,t^{2}+h^{2}=m,\\ \\ ty_{3}+hy_{4}>0,\,\,\forall \,t,\,h\in \mathbb {Z},\,t^{2}+h^{2}=m,\,t>0,\\ \\ y_{4}>0\,\,\text {if }m\text { is a square} \end{array}\right. \right\} . \end{aligned}$$
When \(k_{2}>0\), with the aid of MAPLE, we can check that there is a connected component \(\mathcal {C}_{2}\) of \(\mathbb {R}^{3}\) covered by
$$\begin{aligned} y_{1}=r^{2}+s^{2}-m+2ry_{3}+2sy_{4} \end{aligned}$$
for \(r,\,s\in \mathbb {Z}\). Such a connected component contains \(y_{1}<-m\), and all the other planes
$$\begin{aligned} k_{2}y_{1}=-k_{1}+2k_{3}y_{3}+2k_{4}y_{4} \end{aligned}$$
for \(k_{1},\ldots ,k_{4}\in \mathbb {Z}\) with \(k_{2}>0\) and \(k_{1}k_{2}+k_{3}^{2}+k_{4}^{2}=m\). In conclusion, \(W_{m}=\mathcal {C}_{1}\cap \mathcal {C}_{2}\) is a connected component of \(\mathbb {R}^{3}\) cut out by the planes
$$\begin{aligned} k_{2}y_{1}+k_{1}+2k_{3}y_{3}+2k_{4}y_{4}=0 \end{aligned}$$
for all \(k_{1},\ldots ,k_{4}\in \mathbb {Z}\) with \(k_{2}\ge 0\) and \(k_{1}k_{2}+k_{3}^{2}+k_{4}^{2}=m\), and thus \(W_{m}\) is a Weyl chamber. For the case \(m=1\), we can visualize it by a 3D-plot. See Fig. 1.
Figure 1
Fig. 1

A 3D-plot for the Weyl chamber \(W_{1}\)

Now let us prove Assertion (2).
  1. (i)
    Suppose that \(Q(\lambda )=m\) and \(\lambda _{3},\,\lambda _{4}\in 2\mathbb {Z}\) which imply that \(\lambda \in K\). By (3.9), we note that \(y_{1}e_{3}+e_{4}+(y_{3}+iy_{4})\in W_{m}\) implies that
    $$\begin{aligned} k_{2}y_{1}<-k_{1}+2k_{3}y_{3}+2k_{4}y_{4} \end{aligned}$$
    for all \(k_{i}\in \mathbb {Z}\) with \(k_{2}>0\) and \(k_{1}k_{2}+k_{3}^{2}+k_{4}^{2}=m\), which is equivalent to
    $$\begin{aligned} k_{2}y_{1}+k_{1}+2k_{3}y_{3}+2k_{4}y_{4}>0 \end{aligned}$$
    for all \(k_{i}\in \mathbb {Z}\) with \(k_{2}<0\) and \(k_{1}k_{2}+k_{3}^{2}+k_{4}^{2}=m\). Therefore, when \(\lambda _{2}\ne 0\) and \(Q(\lambda )=m\), that is, \(\lambda _{1}(-\lambda _{2})+\frac{1}{4}(\lambda _{3}^{2}+\lambda _{4}^{2})=m\), \((\lambda ,W_{m})>0\) if and only if \(-\lambda _{2}<0\), that is, \(\lambda _{2}>0\). Similarly, by the other conditions given in (3.9), we can conclude that when \(Q(\lambda )=m\), \((\lambda ,W_{m})>0\) if and only if \(\lambda _{2}<0\), or (\(\lambda _{2}=0\) and \(\lambda _{1}>0\)), or (\(\lambda _{2}=\lambda _{1}=0\) and \(\lambda _{3}>0\)), or (\(\lambda _{2}=\lambda _{1}=\lambda _{3}=0\) and \(\lambda _{4}>0\)).
     
  2. (ii)
    Now suppose that \(Q(\lambda )\le 0\), that is, \(\lambda _{1}\lambda _{2}+\frac{1}{4}(\lambda _{3}^{2}+\lambda _{4}^{2})\le 0\). By (3.9), we know that
    $$\begin{aligned} y_{1}<r^{2}+s^{2}-m+2ry_{3}+2sy_{4} \end{aligned}$$
    for all \(r,\,s\in \mathbb {Z}\). By [1, Lemma 3.2], it is known that if \((\lambda ,w_{0})>0\) for a \(w_{0}\in W_{m}\), then \((\lambda ,W_{m})>0\). Thus \((\lambda ,W_{m})>0\) if and only if \(\lambda _{2}>0\). When \(\lambda _{2}=0\), since \(Q(\lambda )\le 0\), then \(\lambda _{3}=\lambda _{4}=0\), and thus \((\lambda ,w)=\lambda _{1}\) for \(w\in W_{m}\). This implies that \((\lambda ,W_{m})>0\) if and only if \(\lambda _{1}>0\) when \(\lambda _{2}=0\).\(\square \)
     

3.5 The Weyl vector for \(\vec {F}_{m}\)

In this subsection, we aim to compute the Weyl vector \(\rho (W_{m},\vec {F}_{m})\). We first recall a nice summary of the explicit computations of Weyl vector given in [11, Subsection 2.1] (also see [23, Thm. 10.4] for original definitions).

Let L be a \(\mathbb {Z}\)-lattice with quadratic form \(Q(\cdot )\) of a quadratic space V of type (n, 2) and \(L'\) be its dual lattice. Take \(\ell _{L}\in L\) and \(\ell _{L}'\in L'\) to be such that \(Q(\ell _{L})=Q(\ell _{L}')=0\) and \((\ell _{L},\ell _{L}')=1\). Assume that \((\ell _{L},L)=N_{L}\mathbb {Z}\) for some unique positive integer and choose \(\xi \in L\) with \((\ell _{L},\xi )=N_{L}\). Let \(K=L\cap (\mathbb {Q}\ell _{L}+{\mathbb {Q}}\ell _{L}')^{\perp }\) and let
$$\begin{aligned} L_{0}'=\{x\in L'|\,(\ell _{L},x)\equiv 0\pmod {N_{L}}\}\subset L'. \end{aligned}$$
Then there is a projection
$$\begin{aligned} p:L_{0}'\rightarrow K',\quad p(x)=x_{K}+\frac{(x,\ell _{L})}{N_{L}}\xi _{K}, \end{aligned}$$
where \(x_{K}\) and \(\xi _{K}\) are the orthogonal projections of \(x,\xi \in V\) to \(K_{\mathbb {Q}}=K\otimes _{{\mathbb {Z}}}{\mathbb {Q}}\). So it induces a projection from \(L'_{0}/L\) to \(K'/K\). Next, for
$$\begin{aligned} \vec {f}=\sum f_{\mu }\phi _{\mu }=\sum c(m,\phi _{\mu })q^{m}\phi _{\mu }\in M^{!}_{1-\frac{n}{2},\rho _{L}}, \end{aligned}$$
define
$$\begin{aligned} \vec {f}_{K}=\sum _{\lambda \in K'/K}f_{\lambda }\phi _{\lambda ,K}=\sum c_{K}(m,\lambda )q^{m}\phi _{\lambda ,K}, \end{aligned}$$
where \(\phi _{\lambda ,K}\) is the basis element associated to \(\lambda \) of \(\mathbb {C}[K'/K]\), and
$$\begin{aligned} f_{\lambda }=\sum _{\begin{array}{c} \mu \in L_{0}'/L\\ p(\mu )=\lambda \end{array}}f_{\mu }. \end{aligned}$$
For a Weyl chamber W, take \(\ell _{K}\in K\cap {\overline{W}}\), where \({\overline{W}}\) denotes the closure of the Weyl chamber W, and \(\ell _{K}'\in K'\) with \(Q(\ell _{K})=Q(\ell _{K}')=0\) and \((\ell _{K},\ell _{K}')=1\), and let \(P=K\cap ({\mathbb {Q}}\ell _{K}+{\mathbb {Q}}\ell _{K}')^{\perp }\), which is positive definite of rank \(n-2\). Similar to the projection p from \(L_{0}'/L\) to \(K'/K\), one also has a projection, also denoted by p, from \(K_{0}'/K\) to \(P'/P\) defined in the same way. Similarly, we have a weakly holomorphic modular form \(\vec {f}_{P}\) induced by \(\vec {f}_{K}\). Then we can compute and express the Weyl vector \(\rho (W,\vec {f})\) associated to W and \(\vec {f}\) as
$$\begin{aligned} \rho (W,\vec {f})=\rho _{\ell _{K}}\ell _{K}+\rho _{\ell _{K}'}\ell _{K}'+\rho _{P}, \end{aligned}$$
where
$$\begin{aligned} \rho _{\ell _{K}}&=-\frac{1}{4}\sum _{\begin{array}{c} \lambda \in K_{0}'/K\\ p(\lambda )=0+P \end{array}}c_{K}(0,\lambda )B_{2}((\lambda ,\ell _{K}'))\nonumber \\&\quad -\frac{1}{2}\sum _{\begin{array}{c} \gamma \in P'\\ (\gamma ,W)>0 \end{array}}\sum _{\begin{array}{c} \lambda \in K_{0}'/K\\ p(\lambda )=\gamma +P \end{array}}c_{K}(-Q(\gamma ),\lambda )B_{2}((\lambda ,\ell _{K}')), \end{aligned}$$
(3.10)
\(B_{2}(x):=\{x\}^{2}-\{x\}+\frac{1}{6}\) is the second Bernoulli polynomial, \(\{x\}\) is the fractional part of x,
$$\begin{aligned} \rho _{\ell _{K}'}&=\hbox {constant term of } \langle \vec {\theta }_{P},\vec {f}_{P}\rangle E_{2}/24,\nonumber \\ \vec {\theta }_{P}&:=\sum _{\gamma \in P'/P}\sum _{\lambda \in \gamma +P}e(Q(\lambda )\tau )\phi _{\gamma }, \end{aligned}$$
(3.11)
\(E_{2}:=1-24\sum _{n=1}^{\infty }\sigma _{1}(n)q^{n}\) is the holomorphic Eisenstein series of weight 2,
$$\begin{aligned} \rho _{P}&=-\frac{1}{2}\sum _{\begin{array}{c} \gamma \in P'\cap K'\\ (\gamma ,W)>0 \end{array}}c_{K}(-Q(\gamma ),\gamma )\gamma . \end{aligned}$$
(3.12)
Now for our case, we set \(L={\mathbb {Z}}[i]\oplus {\mathbb {Z}}[i]\oplus \frac{1}{2}{\mathbb {Z}}[i]\), \(\ell _{L}=e_{1}\), \(\ell _{L}'=e_{2}\), \(\vec {f}=\vec {F}_{m}\), \(W=W_{m}\), \(\ell _{K}=e_{3}\) and \(\ell '_{K}=e_{4}\), where \(e_{1}\), \(e_{2}\), \(e_{3}\) and \(e_{4}\) are defined as in Subsection 3.2. It is easy to check that \(K={\mathbb {Z}}e_{3}\oplus {\mathbb {Z}}e_{4}\oplus P\) and \(P={\mathbb {Z}}[i]\mathbf e _{2}\) where \(\mathbf e _{2}\) is defined as in Subsection 3.1. Direct calculations show that \(L_{0}'=L'\), \(K_{0}'=K'\) and \(L'/L\cong K'/K\cong P'/P\). Write
$$\begin{aligned} \vec {F}_{m}=\sum _{\mu \in L'/L}F_{m,\mu }\phi _{\mu }=\sum _{\mu \in L'/L}\sum _{n\in \mathbb {Q}}c(n,\mu )q^{n}\phi _{\mu }. \end{aligned}$$
Then under \(L'/L\cong K'/K\cong P'/P\), direct calculations show that
$$\begin{aligned} \vec {F}_{m,K}=\vec {F}_{m,P}=\vec {F}_{m}. \end{aligned}$$
Note that by Theorem 2.5, for \(n>0\), \(c_{K}(-n,\lambda )\ne 0\) if and only if \(c_{K}(-n,\lambda )=c_{K}(-m,0)\), which equals 1. Also, we can compute and express \(\vec {\theta }_{P}\) as
$$\begin{aligned} \vec {\theta }_{P}&=\left( \sum _{r,s\in {\mathbb {Z}}}e\left( \left( r^{2}+s^{2}\right) \tau \right) \right) \phi _{0}+ \left( \sum _{r,s\in {\mathbb {Z}}}e\left( \left( \frac{1}{4}+r+r^{2}+s^{2}\right) \tau \right) \right) \phi _{1}\nonumber \\&\quad +\left( \sum _{r,s\in {\mathbb {Z}}}e\left( \left( r^{2}+s^{2}+s+\frac{1}{4}\right) \tau \right) \right) \phi _{i} +\left( \sum _{r,s\in {\mathbb {Z}}}e\left( \left( \frac{1}{2}+r+s+r^{2}+s^{2}\right) \tau \right) \right) \phi _{1+i}. \end{aligned}$$
(3.13)
Now let us first compute \(\rho _{e_{3}}\). Since \(K_{0}'/K=K'/K\cong P'/P\), then \(\lambda \in K_{0}'/K\) such that \(p(\lambda )=0+P\) if and only if \(\lambda =0+K\). In addition, as we point out above that for \(n>0\), \(c_{K}(-n,\lambda )\ne 0\) if and only if \(c_{K}(-n,\lambda )=c_{K}(-m,0)=1\), then we can see that \(c_{K}(-Q(\gamma ),\lambda )=1\) if and only if \(\gamma \in P\) with \(Q(\gamma )=m\), i.e., \(\gamma =(r+si)\mathbf e _{2}\) with \(r,s\in {\mathbb {Z}}\) and \(r^{2}+s^{2}=m\). In addition, by Lemma 3.9, to have \((\gamma ,W_{m})>0\), we must have \((r>0)\) or (\(r=0\) and \(s>0\)). Now by the definition of \(\rho _{e_{3}}\) and the above analysis, we have
$$\begin{aligned} \rho _{e_{3}}&=-\frac{1}{24}c_{K}(0,0)-\frac{1}{2}\sum _{\begin{array}{c} r^{2}+s^{2}=m\\ r>0\\ \text {or } (r=0\text { and } s>0) \end{array}}\frac{1}{6}\\&=-\frac{1}{6}\sum _{d|m}\left( 16\chi _{-4}(m/d)+\chi _{-4}(d)\right) d^{2}-\frac{1}{6}\sigma _{\chi _{-4}}(m), \end{aligned}$$
where \(\sigma _{\chi _{-4}}(m)=\sum _{d|m}\chi _{-4}(d)\) follows from the well known fact (see, e.g., [24, Thm. 3.2.1]) that the number of integral solutions of \(r^{2}+s^{2}=m\) is given by \(4\sigma _{\chi _{-4}}(m)\).
For the \(e_{4}\)-component \(\rho _{e_{4}}\), we first note that the non-\(\phi _{0}\)-component functions of \(\theta _{P}\) have no constant terms, and the non-\(\phi _{0}\)-component functions of \(\vec {F}_{P}\) have no negative power terms. In addition, the \(\phi _{0}\)-component function of \(\theta _{P}\) is
$$\begin{aligned} \sum _{r,s\in {\mathbb {Z}}}e\left( (r^{2}+s^{2})\tau \right) =1+4\sum _{n=1}^{\infty }\sigma _{\chi _{-4}}(n)q^{n}, \end{aligned}$$
and the \(\phi _{0}\)-component function of \(\vec {F}_{P}\) is \(q^{-m}+c(0,0)+O(q)\). Therefore, the constant term of \(\langle \vec {\theta }_{P},\vec {f}_{P}\rangle E_{2}\) is the constant term of
$$\begin{aligned} \left( 1+4\sum _{n=1}^{\infty }\sigma _{\chi _{-4}}(n)q^{n}\right) \left( q^{-m}+c(0,0)+O(q)\right) \left( 1-24\sum _{n=1}^{\infty }\sigma _{1}(n)q^{n}\right) , \end{aligned}$$
which is
$$\begin{aligned} 4\sigma _{\chi _{-4}}(m)-24\sigma _{1}(m)-96\left( \sum _{\begin{array}{c} k+l=m\\ k,l\ge 1 \end{array}}\sigma _{\chi _{-4}}(k)\sigma _{1}(l)\right) +c(0,0). \end{aligned}$$
Thus by Theorem 2.5, we have
$$\begin{aligned} \rho _{e_{4}}&=\frac{1}{6}\Bigg [\sigma _{\chi _{-4}}(m)-6\sigma _{1}(m)-24\left( \sum _{\begin{array}{c} k+l=m\\ k,l\ge 1 \end{array}}\sigma _{\chi _{-4}}(k)\sigma _{1}(l)\right) \\&\qquad +\sum _{d|m}\left( 16\chi _{-4}(m/d)+\chi _{-4}(d)\right) d^{2}\Bigg ]. \end{aligned}$$
For \(\rho \), notice that for \(n>0\), \(c_{K}(-n,\lambda )\ne 0\) if and only if \(\gamma \in P\) with \(Q(\gamma )=m\), i.e., \(\gamma =(r+si)\mathbf e _{2}\) with \(r,s\in {\mathbb {Z}}\) and \(r^{2}+s^{2}=m\). Notice also that \((\gamma ,W_{m})>0\) implies \((r>0)\) or (\(r=0\) and \(s>0\)). So by similar calculations, we have
$$\begin{aligned} \rho _{P}&=-\frac{1}{2}\sum _{\begin{array}{c} r^{2}+s^{2}=m\\ r>0\\ \text {or } (r=0\text { and }s>0) \end{array}}(r+si)\mathbf e _{2}. \end{aligned}$$
Summing up, we conclude with the following proposition.

Proposition 3.3

Let \(\vec {F}_{m}\) be defined as in Subsection 3.4, and let \(W_{m}\) be the Weyl chamber given by (3.9). Then the Weyl vector associated to the Weyl chamber \(W_{m}\) and the weakly holomorphic modular form \(\vec {F}_{m}\) is
$$\begin{aligned} \rho (W_{m},\vec {F}_{m})=\rho _{e_{3}}e_{3}+\rho _{e_{4}}e_{4}+\rho _{P}, \end{aligned}$$
where
$$\begin{aligned} \rho _{e_{3}}&=-\frac{1}{6}\sum _{d|m}\left( 16\chi _{-4}(m/d)+\chi _{-4}(d)\right) d^{2}-\frac{1}{24}\sigma _{\chi _{-4}}(m),\\ \rho _{e_{4}}&=\frac{1}{6}\Bigg [\sigma _{\chi _{-4}}(m)-6\sigma _{1}(m)-24\left( \sum _{\begin{array}{c} k+l=m\\ k,l\ge 1 \end{array}}\sigma _{\chi _{-4}}(k)\sigma _{1}(l)\right) \\&\qquad +\sum _{d|m}\left( 16\chi _{-4}(m/d)+\chi _{-4}(d)\right) d^{2}\Bigg ],\\ \rho _{P}&=-\frac{1}{2}\sum _{\begin{array}{c} r^{2}+s^{2}=m\\ r>0\\ \text {or } (r=0\text { and }s>0) \end{array}}(r+si)\mathbf e _{2}, \end{aligned}$$
and \(\sigma _{\chi _{-4}}(m)=\sum _{d|m}\chi _{-4}(d)\).

3.6 Heegner divisors for \(\Gamma _{L}\)

Let \(\lambda \in L'\) be a lattice vector with positive norm, i.e., \(\langle \lambda ,\lambda \rangle >0\). The orthogonal complement of \(\lambda \) in \(\mathcal {K}_{U}\) is a closed analytic subset of comdimension 1, which we denote as follows
$$\begin{aligned} \mathbf H (\lambda )=\{[z]\in \mathcal {K}_{U}|\,\langle z,\lambda \rangle =0\}. \end{aligned}$$
By identification between \(\mathcal {K}_{U}\) and \(\mathcal {H}\), \(\mathbf H (\lambda )\) can also be considered as a closed analytic subset of \(\mathcal {H}\), and we call such set a prime Heegner divisor on \(\mathcal {H}\). Given \(\beta \in L'/L\) and \(m\in \mathbb {Z}_{>0}\), a Heegner divisor of index \((m,\beta )\) in \({\mathcal {H}}\) is defined as the locally finite sum
$$\begin{aligned} \mathbf H (m,\beta )=\sum _{\begin{array}{c} \lambda \in \beta +L\\ Q(\lambda )=m \end{array}}{} \mathbf H (\lambda ). \end{aligned}$$
The associated Heegner divisor in \(X_{\Gamma _L} =\Gamma _L \backslash {\mathcal {H}}\) is \(\mathbf Z (m, \beta ) = \Gamma _L \backslash \mathbf H (m,\beta )\).

3.7 Borcherds products

In this section, we give a family of new Borcherds products explicitly by using the results of Hofmann [15, Thms . 4, 5 and Cor. 1]. We first summarize Hofmann’s results as follows.

Theorem 3.4

(Hofmann) Let \(\mathbb {F}\) be an imaginary quadratic field. Let L be an even hermitian lattice of signature (m, 1) with \(m\ge 1\), and \(\ell \in L\) a primitive isotropic vector. Let \(\ell '\in L'\) an isotropic vector with \(\langle \ell ,\ell '\rangle \ne 0\). Further assume that L is the direct sum of a hyperbolic plane \(H\cong \mathcal {O}_{\mathbb {F}}\oplus \partial ^{-1}_{\mathbb {F}}\) and a definite part D with \(\langle D,H\rangle =0\).

Given a weakly holomorphic modular form \(f\in M_{1-m,\rho _{L}}^{!}\) with Fourier coefficients \(c(n,\beta )\) satisfying \(c(n,\beta )\in \mathbb {Z}\) for \(n<0\), there is a meromorphic function \(\Psi (\tau ,\sigma ;f)\) on \(\mathcal {H}\) with the following properties:
  1. (1)

    \(\Psi (\tau ,\sigma ;f)\) is an automorphic form of weight \(c(0,\phi _{0})/2\) for \(\Gamma _{L}\) with some multiplier system \(\chi \) of finite order.

     
  2. (2)
    The zeros and poles of \(\Psi (\tau ,\sigma ;f)\) lie on Heegner divisors. The divisor of \(\Psi (\tau ,\sigma ;f)\) on \(X_{\Gamma _{L}}=\Gamma _{L}\backslash \mathcal {H}\) is given by
    $$\begin{aligned} div\left( \Psi (\tau ,\sigma ;f)\right) =\frac{1}{2}\sum _{\beta \in L'/L}\sum _{\begin{array}{c} n\in \mathbb {Z}-Q(\beta )\\ n>0 \end{array}}c(-n,\phi _{\beta })\mathbf H (n,\beta ). \end{aligned}$$
    The multiplicities of \(\mathbf H (n,\beta )\) are 2 if \(2\beta =0\) in \(L'/L\), and 1 otherwise.
     
  3. (3)
    For a Weyl chamber W whose closure contains the cusp \(\mathbb {Q}e_{3}\), \(\Psi (\tau ,\sigma ;f)\) has an infinite product expansion of the form
    $$\begin{aligned} \Psi (\tau ,\sigma ;f)=Ce\left( \frac{\langle z,\rho (W,f)\rangle }{\langle \ell ,\ell '\rangle }\right) \prod _{\begin{array}{c} \lambda \in K'\\ (\lambda ,W)>0 \end{array}}\left[ 1-e\left( \frac{\langle z,\lambda \rangle }{\langle \ell ,\ell '\rangle }\right) \right] ^{c(-Q(\lambda ),\lambda )}, \end{aligned}$$
    where \(z=z(\tau ,\sigma )=\ell '+\delta \langle \ell ,\ell '\rangle \tau \ell +\sigma \), \(\delta \) is the square root of the discriminant of \(\mathbb {F}\), the constant C has absolute value 1 and \(\rho (W,f)\) is the Weyl vector attached to W and f.
     
  4. (4)

    The lifting is multiplicative: \(\Psi (\tau ,\sigma ;f+g)=\Psi (\tau ,\sigma ;f)\Psi (\tau ,\sigma ;g)\).

     
  5. (5)
    Let W be a Weyl chamber such that the cusp corresponding to \(\ell \) is contained in the closure of W. If this cusp is neither a pole nor a zero of \(\Psi (\tau ,\sigma ;f)\), then we have
    $$\begin{aligned} \lim _{\tau \rightarrow \infty }\Psi (\tau ,\sigma ;f)=Ce\left( \overline{\rho (W,f)_{\ell }}\right) \prod _{\begin{array}{c} \lambda \in K'\\ \lambda =\frac{1}{2}\kappa \delta \ell \\ \kappa \in \mathbb {Q}_{>0} \end{array}}\left( 1-e\left( -\frac{1}{2}\kappa {\bar{\delta }}\right) \right) ^{c(0,\lambda )} \end{aligned}$$
    where \(\overline{\rho (W,f)_{\ell }}\) denotes the complex conjugate of the \(\ell \)-component of the Weyl vector \(\rho (W,f)\).
     

By specializing Theorem 3.4 in our case, we obtain the main result of this note.

Theorem 3.5

Let \(L=\mathbb {Z}[i]\oplus \mathbb {Z}[i]\oplus \frac{1}{2}\mathbb {Z}[i]\) with respect to the standard basis over \(\mathbb {Z}[i]\) with hermitian form defined in (3.3). We set \(\ell =(1,0,0)\) and \(\ell '=(0,0,1)\). Let \(\vec {F}_{m}\) be the vector-valued modular form arising from \(F_m=\theta _{2}\theta _{1}^{-1}P_{1,m-1}(\varphi _{\infty })\) and denote by \(c(n,\phi _{\mu })\) the Fourier coefficient of index \((n,\phi _{\mu })\) of \(\vec {F}_{m}\). Then there is a meromorphic function \(\Psi (\tau ,\sigma ; F_{m})= \Psi (\tau ,\sigma ; \vec {F}_{m})\) on \(\mathcal {H}\) with the following properties:
  1. (1)
    \(\Psi (\tau ,\sigma ;\vec {F}_{m})\) is an automorphic form of weight
    $$\begin{aligned} 32\sum _{d|m}\chi _{-4}(n/d)d^{2}+2\sum _{d|m}\chi _{-4}(d)d^{2} \end{aligned}$$
    for \(\Gamma _{L}\), with some multiplier system \(\chi \) of finite order.
     
  2. (2)
    The zeros and poles of \(\Psi (\tau ,\sigma ;\vec {F}_{m})\) lie on Heegner divisors. The divisor of \(\Psi (\tau ,\sigma ;\vec {F}_{m})\) on \(X_{\Gamma _L}=\Gamma _{L}\backslash \mathcal {H}\) is given by
    $$\begin{aligned} div(\Psi (\tau ,\sigma ;\vec {F}_{m}))=\mathbf Z (m,0) =\Gamma _{L}\backslash \mathbf H (m,0), \end{aligned}$$
    where
    $$\begin{aligned} \mathbf H (m,0)=\sum _{\begin{array}{c} (r_1,s_1,r_2,s_2,r_3,s_3)\in \mathbb {Z}^{6}\\ r_1r_3+s_1s_3+r_2^2 +s_2^2 =m \end{array}}\left\{ (\tau ,\sigma )\in \mathcal {H}\left| \begin{array}{c} r_1+2r_2\mathop {\hbox {Re}}\nolimits {\sigma }+2s_2\mathop {\hbox {Im}}\nolimits {\sigma }+s_3\mathop {\hbox {Re}}\nolimits {\tau }-r_3\mathop {\hbox {Im}}\nolimits {\tau }=0,\\ \\ s_1+2r_2\mathop {\hbox {Im}}\nolimits {\sigma }-2s_2\mathop {\hbox {Re}}\nolimits {\sigma }+s_3\mathop {\hbox {Im}}\nolimits {\tau }+r_3\mathop {\hbox {Re}}\nolimits {\tau }=0 \end{array}\right. \right\} . \end{aligned}$$
     
  3. (3)
    For the Weyl chamber \(W_{m}\) described in (3.9), \(\Psi (\tau ,\sigma ;\vec {F}_{m})\) has an infinite product expansion near the cusp \(\mathbb {Q}e_{3}\) (precisely, when \((\tau ,\sigma )\in W_{m,U}\) with \(\mathop {\hbox {Im}}\nolimits {\tau }\) sufficiently large):
    $$\begin{aligned} \Psi (\tau ,\sigma ;F_{m})=A_1(\tau , \sigma ) A_2(\sigma ) A_3(\sigma ) A_4(\sigma ) A_5(\tau , \sigma ), \end{aligned}$$
    (3.14)
    where
    1. (i)
      $$\begin{aligned} A_1(\tau , \sigma )=e(i\rho _{e_{3}}-\rho _{e_{4}}\tau +{\bar{\rho }}\sigma ) \end{aligned}$$
      where \(\rho _{e_{3}}\), \(\rho _{e_{4}}\) and \(\rho \) are defined as in Proposition 3.3,
       
    2. (ii)
      $$\begin{aligned} A_2(\sigma )&={\left\{ \begin{array}{ll}\displaystyle { \left[ 1-e\left( -i\sigma \sqrt{m}\right) \right] }&{}\hbox {if } m \hbox { is a square,}\\ \qquad \qquad \qquad 1&{}\text {otherwise,} \end{array}\right. } \end{aligned}$$
       
    3. (iii)
      $$\begin{aligned} A_3(\sigma ) = \prod _{\begin{array}{c} (k_{3},k_{4})\in \mathbb {Z}_{>0}^{2}\\ k_{3}^{2}+k_{4}^{2}=m \end{array}} \big [1-e\left( \sigma (k_{3}+ik_{4})\right) \big ]\big [1-e\left( \sigma \left( k_{3}-ik_{4}\right) \right) \big ], \end{aligned}$$
       
    4. (iv)
      $$\begin{aligned} A_4(\sigma )&=\prod _{\begin{array}{c} n_3, n_4 \in {\mathbb {Z}}\\ n_3^2+n_4^2= m \end{array}} \prod _{ n_{2}\in {\mathbb {Z}}_{>0}} \big [1-e(i n_{2})e\left( \sigma \left( {n_{3}}-i{n_{4}}\right) \right) \big ]\\&\quad \times \prod _{ n_{2}\in {\mathbb {Z}}_{>0}}\left( 1-e(i n_2)\right) ^{c(0,0)} \end{aligned}$$
      with
      $$\begin{aligned} c(0, 0)=c(0, \phi _0)=\sum _{d|m}\left( 64\chi _{-4}(m/d)+4\chi _{-4}(d)\right) d^{2}, \end{aligned}$$
       
    5. (v)
      $$\begin{aligned} A_5(\tau , \sigma )&=\prod _{\begin{array}{c} (n_{1},n_{2},n_{3},n_{4})\in \mathbb {Z}^{4}\\ n_{1}>0 \end{array}}\\&\quad \left[ 1-e\left( n_{1}\tau +\sigma \left( \frac{n_{3}}{2}-i\frac{n_{4}}{2}\right) +in_{2}\right) \right] ^{c(n_1n_2- \frac{1}{4} (n_3^2+n_4^2),\phi _{\vec {n}})} \end{aligned}$$
      with \(\vec {n}=n_{2}e_{3}-n_1 e_4+\frac{1}{2}(n_{3}+in_{4})\).
       
     
  4. (4)
    If the cusp corresponding to \(\ell \) is neither a pole nor a zero of \(\Psi (\tau ,\sigma ;\vec {F}_{m})\), then we have
    $$\begin{aligned} \lim _{\tau \rightarrow i\infty }\Psi (\tau ,\sigma ;\vec {F}_{m})&=e(i\rho _{e_{3}})\prod _{k=1}^{\infty }\left( 1-e(ki)\right) ^{c(0,\phi _0)} \end{aligned}$$
    where
    $$\begin{aligned} \rho _{e_{3}}=-\frac{1}{6}\sum _{d|m}\left( 16\chi _{-4}(m/d)+\chi _{-4}(d)\right) d^{2}-\frac{1}{24}\sigma _{\chi _{4}}(m) \end{aligned}$$
    is defined as in Proposition 3.3, and
    $$\begin{aligned} c(0,\phi _0)=\sum _{d|m}\left( 64\chi _{-4}(m/d)+4\chi _{-4}(d)\right) d^{2} \end{aligned}$$
    is defined as in Theorem 2.5.
     

Proof

Assertion (1) follows from Theorem 2.5 and Theorem 3.4 (1). Assertion (2) follows directly from Theorem 3.4 (2).

Then by Theorem 3.4 (3) together with Lemma 3.2 and Proposition 3.3, we have that \(\Psi (\tau ,\sigma ;\vec {F}_{m})\) has the following infinite product expansion near the cusp \(\mathbb {Q}e_{3}\)
$$\begin{aligned}&\psi (\tau ,\sigma ;\vec {F}_{m})\\&=e(i\rho _{e_{3}}-\rho _{e_{4}}\tau +{\bar{\rho }}\sigma )\\&\times \prod _{\begin{array}{c} (\lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4})\in \mathbb {Z}^{4}\\ \lambda _{2}>0,\\ {or\,\, \lambda _{2}=0\,\, and\,\, \lambda _{1}>0,}\\ {or \,\,\lambda _{2}=\lambda _{1}=0 \,\,and\,\, \lambda _{3}>0,}\\ {or\,\,\lambda _{2}=\lambda _{1}=\lambda _{3}=0\,\, and \,\, \lambda _{4}>0.} \end{array}}\left[ 1-e\left( \lambda _{2}\tau +\sigma \left( \frac{\lambda _{3}}{2}-i\frac{\lambda _{4}}{2}\right) +i\lambda _{1}\right) \right] ^{c\left( \lambda _{1}\lambda _{2}-\frac{1}{4}(\lambda _{3}^{2}+\lambda _{4}^{2}),\,\phi _{\lambda }\right) } \end{aligned}$$
where \(\lambda =\lambda _{1}e_{3}-\lambda _{2}e_{4}+\frac{1}{2}(\lambda _{3}+i\lambda _{4})\), and \(\rho _{e_{3}}\), \(\rho _{e_{4}}\) and \(\rho \) are given as in Proposition 3.3. We first set \(A_1(\tau , \sigma )=e(i\rho _{e_{3}}-\rho _{e_{4}}\tau +{\bar{\rho }}\sigma )\). Then by decomposing the infinite product according to the four cases in its product index set, we can easily rewrite it as (3.14).

Finally, for Assertion (4), we first note that in our case, \(K'=\mathbb {Z}i\oplus \mathbb {Z}[i]\oplus \frac{1}{2}\mathbb {Z}i\) and \(\delta =2i\), then \(\lambda \in K'\) and \(\lambda =\frac{1}{2}\kappa \delta \ell =\kappa i\ell \) with \(\kappa \in \mathbb {Q}_{>0}\) imply that \(\kappa \in \mathbb {Z}_{>0}\) and \(c(0,\lambda )=c(0,\phi _{0})\). Together with the Weyl vector attached to \(W_{m}\) and \(\vec {F}_{m}\) shown in Subsection 3.5, Theorem 3.4 (5) proves Assertion (4). \(\square \)

Declarations

Author's contributions

Acknowlegements

The authors thank the anonymous referees for their helpful comments. The research of the first author is supported by a NSF Grant DMS-1500743.

Authors’ Affiliations

(1)
Department of Mathematics, University of Wisconsin, Madison, USA

References

  1. Bruinier, J.H.: Borcherds products on \(O(2, l)\) and Chern classes of Heegner divisors, volume 1780 of Lecture Notes in Mathematics. Springer, Berlin (2002)MATHGoogle Scholar
  2. Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125, 45–90 (2004)MathSciNetView ArticleMATHGoogle Scholar
  3. Gross, B., Zagier, D.: Heegner points and derivatives of L-series. Invent. Math. 84, 225–320 (1986)MathSciNetView ArticleMATHGoogle Scholar
  4. Gross, B., Zagier, D.: On singular moduli. J. Reine Angew. Math. 355, 191–220 (1985)MathSciNetMATHGoogle Scholar
  5. Andreatta, F., Goren, E., Howard, B., Madapusi Pera, K.: Height pairings on orthogonal Shimura varieties. Compos. Math. 153, 474–534 (2017)MathSciNetView ArticleMATHGoogle Scholar
  6. Andreatta, F., Goren, E., Howard, B., Madapusi Pera, K.: Faltings heights of abelian varieties with complex multiplication, preprint (2016)Google Scholar
  7. Bruinier, J.H., Howard, B., Yang, T.H.: Heights of Kudla–Rapoport divisors and derivatives of \(L\)-functions. Invent. Math. 201, 1–95 (2015)MathSciNetView ArticleMATHGoogle Scholar
  8. Bruinier, J.H., Kudla, S.S., Yang, T.H.: Special values of Green functions at big CM points. Int. Math. Res. Notes. 9, 1917–1967 (2012)MathSciNetMATHGoogle Scholar
  9. Bruinier, J.H., Yang, T.H.: Faltings heights of CM cycles and derivatives of \(L\)-functions. Invent. Math. 177, 631–681 (2009)MathSciNetView ArticleMATHGoogle Scholar
  10. Schofer, J.: Borcherds forms and generalizations of singular modul. J. Reine Angew. Math. 629, 1–36 (2009)MathSciNetView ArticleMATHGoogle Scholar
  11. Yang, T.H., Yin, H.B.: Difference of modular functions and their CM value factorization, to appear in Trans. Am. Math. Soc [arXiv:1711.02983]
  12. Bruinier, J.H., Howard, B., Kuda, S.S., Rapoport, M., Yang, T.H.: Modularity of generating series of divisors on unitary Shimura varieties, preprint (2016)Google Scholar
  13. Howard, B., Matapusi-Peri, K.: Arithmetic of Borcherds products, in progressGoogle Scholar
  14. Ono, K., Unearthing the visions of a master: harmonic Maass forms and number theory. In: Current developments in mathematics, vol. 2008, pp. 347–454. International Press, Somerville (2009)Google Scholar
  15. Hofmann, E.: Borcherds products on unitary groups. Math. Ann. 358, 799–832 (2014)MathSciNetView ArticleMATHGoogle Scholar
  16. Haddock, A., Jenkins, P.: Zeros of weakly holomorphic modular forms of level 4. Int. J. Number Theory 10, 455–470 (2014)MathSciNetView ArticleMATHGoogle Scholar
  17. Duke, W., Jenkins, P.: On the zeros and coefficients of certain weakly holomorphic modular forms. Pure Appl. Math. Q. 4, 1327–1340 (2008)MathSciNetView ArticleMATHGoogle Scholar
  18. Diamond, F., Shurman, J.: A first course in modular forms, Graduate Texts in Mathematics 228. Springer, Berlin (2005)MATHGoogle Scholar
  19. Scheithauer, N.R.: Some constructions of modular forms for the Weil representation of \(\text{ SL }_{2}(\mathbb{Z})\). Nagoya Math. J. 220, 1–43 (2015)MathSciNetView ArticleGoogle Scholar
  20. Scheithauer, N.R.: The Weil representation of \(\text{ SL }_{2}(\mathbb{Z})\) and some applications. Int. Math. Res. Not. 8, 1488–1545 (2009)View ArticleMATHGoogle Scholar
  21. Kolberg, O.: Note on the Eisenstein series of \(\Gamma _{0}(p)\) Arbok for Universitet i Bergen Mat. Nat. Ser. 1968(1968), 20 (1969)MathSciNetGoogle Scholar
  22. Borcherds, R.E.: The Gross–Kohnen–Zagier theorem in higher dimensions. Duke Math. J. 97, 219–233 (1999)MathSciNetView ArticleMATHGoogle Scholar
  23. Borcherds, R.E.: Automorphic forms with singularities on Grassmannians. Invent. Math. 132, 491–562 (1998)MathSciNetView ArticleMATHGoogle Scholar
  24. Berndt, B.C.: Number theory in the spirit of Ramanujan. Student Mathematical Library, 34. American Mathematical Society, Providence, RI (2006)Google Scholar

Copyright

© SpringerNature 2018

Advertisement