2.1 Lattices and local–global principles

Throughout this section, let R be a principal ideal domain, K its field of fractions, and L a finitely generated free R-module.

1. (i) if $r \in R$ and $x \in L$ , then $q(rx) = r^2 q(x)$ ;

2. (ii) the map $(x, y) := q(x + y) - q(x) - q(y)$ for $x, y \in L$ is a symmetric bilinear form.

The pair (L, q) is called a lattice over R. If $R = K$ is a field, we call it quadratic space.

In this paper we will consider only lattices over $\mathbb{Z}$ and $\mathbb{Z}_{p}$ . If we do not specify the ring R of a lattice, we implicitly assume that $R=\mathbb{Z}$ .

For $x \in L$ , we write $x^\perp := \{y \in L : (x, y) = 0 \}$ for the orthogonal complement of x. We call q non-degenerate if $x^\perp \neq L$ for every $x \in L \setminus \{0\}$ . From now on we will assume that every lattice is non-degenerate.

It is well known that every quadratic space V over the real numbers $\mathbb{R}$ of dimension n is isometric to a quadratic space $\mathbb{R}^n$ with quadratic form

$$q(x) = x_1^2 + \cdots + x_{b^+}^2 - x_{b^+ + 1}^2 - \cdots - x_{n}^2.$$

We denote this quadratic space by $\mathbb{R}^{b^+, b^-}$ , where $ b^- = n -b^+$ . The tuple $(b^+, b^-)$ is uniquely determined and is called the signature of V.

Given a lattice (L, q) over a ring R and a ring extension $R \to R'$ , one obtains a lattice $(L \otimes R', q)$ . Here, $q(x \otimes r') = r'^2 q(x)$ for every $x \in L$ and $r' \in R'$ . In particular, if V is a rational quadratic space, i.e. a quadratic space over $\mathbb{Q}$ , we define its signature to be the signature of $V \otimes \mathbb{R}$ .

For an R-lattice L, we define its dual lattice L’ as

$$L' := \{ x \in L \otimes K : (x, y) \in R \text{ for all } y \in L \}.$$

If L is integral, then $L \subseteq L'$ . The quotient group $D_{L} := L' / L$ is called the discriminant group of L. If L is even, then the quadratic form on L induces a map $D_{L} \to K / R$ , which we also denote by q. Given two discriminant groups $D_{L}$ and $D_{M}$ , a bijective map $g : D_{L} \to D_{M}$ is called an isometry if $q(g(x)) = q(x)$ for all $x \in D_{L}$ .

An even lattice is called unimodular if $D_{L}$ is trivial, and we call L maximal if $D_{L}$ is anisotropic, i.e. if $q(x) = 0$ for $x \in D_{L}$ , then $x = 0 \in D_{L}$ .

Let L be a $\mathbb{Z}$ -lattice and v a place of $\mathbb{Q}$ , i.e. either v is (a non-archimedean place associated to) some prime number or (the archimedean place) $v = \infty$ . We define $L_v := L \otimes \mathbb{Z}_{p}$ if ${v = p}$ , and ${L_v := L \otimes \mathbb{R}}$ if ${v = \infty}$ .

Isometric lattices are obviously in the same genus and we write $\operatorname{gen}(L)$ for the set of isometry classes of lattices in the same genus of L. This set is always finite; see [Reference KneserKne02, Satz 21.3]. If L represents another lattice M, then L also represents M locally, i.e. $L_v$ represents $M_v$ for every place v. The converse fails in general, but the local–global principle asserts that if L represents M locally, then there exists a lattice $\tilde{L} \in \operatorname{gen}{(L)}$ such that $\tilde{L}$ represents M; see [Reference KneserKne02, Satz 30.9] for a slightly weaker statement. The following lemma is a slight refinement of this.

Proof. We have $L = U^r \oplus \tilde{L}$ , where U is a hyperbolic plane and $\tilde{L}$ is an even lattice. We write ${f_i, f_i'}$ for a standard basis of the ith hyperbolic plane with $(f_i, f_i') = 1$ and $q(f_i) = q(f_i') = 0$ . Since $D_L = D_{\tilde{L}}$ , we may assume $\gamma \in \tilde{L}'$ and define

$$\tilde{\gamma}_i := \gamma_i + f_i + \frac{1}{2}\sum_{j = 1}^{r} ((e_i, e_j) - (\gamma_i, \gamma_j)) f_j' \in \gamma_i + L.$$

We obtain the isometric embedding g by mapping the basis element $e_i$ to $\tilde{\gamma}_i$ for ${i = 1, \ldots, r}$ .

Proof. The positive-definite symmetric matrix $T \in q(\gamma)$ corresponds to a positive-definite lattice M with basis $(e_i)_{i = 1, \ldots, r}$ such that $q((e_i)_{i = 1, \ldots, r}) = T$ . By Lemma 2.6, the lattice $L^+$ represents M locally and thus, by Lemma 2.5, there exist an even lattice $\tilde{L}^+ \in \operatorname{gen}(L^+)$ , an isometric embedding $g \colon M \to \tilde{L}^{+'}$ , and an isometry $g' : D_{L^{+}} \to D_{\tilde{L}^{+}}$ with $(g(e_i))_{i = 1, \ldots, r}$ in $g'(\gamma) + (\tilde{L}^+)^r$ . By Remark 2.7 we have an isometry

$$U \oplus \tilde{L}^+ \to U \oplus L^+ = L,$$

so that we can assume $U \oplus \tilde{L}^+ = L$ and $g' \colon D_L \to D_L$ . Again by Remark 2.7, the map $O(L) \to O(D_L)$ is surjective and thus there exists some $\tilde{g}' \in O(L)$ mapping to g’. Hence, replacing g by $\tilde{g}'^{-1} \circ g$ and $\tilde{L}^+$ by $\tilde{g}'^{-1}(\tilde{L}^+)$ , we obtain

$$g((e_i)_{i = 1, \ldots, r}) \in \gamma + L^r$$

and $g((e_i)_{i = 1, \ldots, r}) \in (\tilde{L}^{+'})^r$ . Moreover, $\lambda = g((e_i)_{i = 1, \ldots, r})$ satisfies $q(\lambda) = T$ .

2.2 Orthogonal Shimura varieties

Let L be a (non-degenerate) even lattice of signature (b,2), for some $b \gt 0$ . The Grassmannian associated to L is the set of negative-definite planes in $V=L\otimes\mathbb{R}$ , namely,

\begin{equation*} \mathrm{Gr}(L)=\{z\subset V : \text{$\dim z=2$ and $(\cdot{,}\cdot)|_z<0$}\}. \end{equation*}

The Hermitian symmetric space $\mathcal{D}$ attached to V may be identified with $\mathrm{Gr}(L)$ ; see [Reference Bruinier, van der Geer, Harder and ZagierBvdG⁺08, Part 2, § $2.4$ ]. From now on, we write $\mathcal{D}$ and $\mathrm{Gr}(L)$ interchangeably.

Let $\widetilde{\mathrm{SO}}(L)$ be the discriminant kernel of $\mathrm{SO}(L)$ , namely the kernel of the natural homomorphism $\mathrm{SO}(L)\to\text{Aut}(D_{L})$ . An orthogonal Shimura variety is a quotient of the form $X=\Gamma\backslash\mathcal{D}$ for some finite-index subgroup $\Gamma\subseteq\widetilde{\mathrm{SO}}(L)$ .

By the theorem of Baily and Borel, the locally symmetric space X admits a unique algebraic structure that makes it a quasi-projective variety of dimension b. This variety may possibly be compact only if $b\le 2$ .

3.1 Very homogeneous polynomials and differential operators

The Siegel theta series we will employ to prove the injectivity of the Kudla–Millson theta lift are special theta series arising from certain homogeneous polynomials on $(\mathbb{R}^{b,2})^2$ . To illustrate such notion of homogeneity, we need to introduce another piece of notation. Let $(g_0(e_j))_j$ be the standard basis of the quadratic space $\mathbb{R}^{b,2}$ . For every vector ${x=\sum_{j=1}^{b+2}x_jg_0(e_j)\in \mathbb{R}^{b,2}}$ , we define ${x^+=\sum_{j=1}^bx_jg_0(e_j)}$ and ${x^-=\sum_{j=b+1}^{b+2}x_jg_0(e_j)}$ . For every $\boldsymbol{x}=(x_1,x_2)\in (\mathbb{R}^{b,2})^2$ , we define ${\boldsymbol{x}^+=(x_1^+,x_2^+)}$ and $\boldsymbol{x}^-=(x_1^-,x_2^-)$ .

Definition 3.1. We say that a polynomial $\mathcal{P}\colon(\mathbb{R}^{b,2})^2\to\mathbb{C}$ is very homogeneous of degree $(m^+,m^-)$ if it splits as a product of two polynomials $\mathcal{P}(\boldsymbol{x})=\mathcal{P}_b(\boldsymbol{x}^+)\mathcal{P}_2(\boldsymbol{x}^-)$ such that

\begin{equation*} \mathcal{P}_b(\boldsymbol{x}^+N)=(\det N)^{m^+}\mathcal{P}_b(\boldsymbol{x}^+)\quad\text{and}\quad\mathcal{P}_2(\boldsymbol{x}^-N)=(\det N)^{m^-}\mathcal{P}_2(\boldsymbol{x}^-), \end{equation*}

for every $N\in\mathbb{C}^{2\times2}$ .

This homogeneity property is the same as the one introduced in [Reference RoehrigRoe21]. Very homogeneous polynomials are a (not necessarily harmonic) generalization of the ‘harmonic forms’ defined by Freitag [Reference FreitagFre83, Definition 3.5] and Maass [Reference MaassMaa59] to indefinite quadratic spaces. To avoid confusion with the harmonic differential forms on the Hermitian domain $\mathcal{D}$ , we decided to refer to such polynomials with a different terminology.

An example of a very homogeneous harmonic polynomial of degree (1,0) for $b\geq 2$ is $\mathcal{P}(\boldsymbol{x})=\det(\begin{smallmatrix} x_{1,1} & x_{1,2} \\ x_{2,1} & x_{2,2} \end{smallmatrix}{\kern-1pt})$ . Large powers of it provide examples of very homogeneous non-harmonic polynomials; see Lemma 6.6 and [Reference RoehrigRoe21, Remark 3.3] for further examples.

Remark 3.2. Let $\mathcal{P}$ be a very homogeneous polynomial on $(\mathbb{R}^{b,2})^2$ of degree $(m^+,m^-)$ and let $N=(\begin{smallmatrix} \lambda & 0\\ 0 & \lambda\end{smallmatrix})$ , for some $\lambda\in\mathbb{R}\setminus\{0\}$ . For every $\boldsymbol{x}=(x_1,x_2)\in(\mathbb{R}^{b,2})^2$ , we have

\begin{equation*} \mathcal{P}_b(\lambda \boldsymbol{x}^+)=\mathcal{P}_b(\boldsymbol{x}^+N) = \det N^{m^+}\mathcal{P}_b(\boldsymbol{x}^+)=\lambda^{2m^+}\mathcal{P}_b(\boldsymbol{x}). \end{equation*}

The case of $\mathcal{P}_2(\boldsymbol{x}^-)$ is analogous. We have just shown that the polynomials $\mathcal{P}_b$ and $\mathcal{P}_2$ are homogeneous of even degree in the classical sense, if considered as polynomials on ${(\mathbb{R}^{b,0})^2\cong\mathbb{R}^{2b}}$ and ${(\mathbb{R}^{0,2})^2\cong\mathbb{R}^4}$ , respectively. An analogous procedure shows that $\mathcal{P}_b$ is homogeneous of degree $m^+$ on each copy of $\mathbb{R}^{b,2}$ in $(\mathbb{R}^{b,2})^2$ . A similar statement holds for $\mathcal{P}_2$ as well.

Let $\Delta$ be the standard Laplacian on $(\mathbb{R}^{b,2})^2$ defined as

(3.2) \begin{equation} \Delta=\bigg(\frac{\partial}{\partial\boldsymbol{x}}\bigg)^{\!\!t}\cdot\frac{\partial}{\partial\boldsymbol{x}}\quad\text{where }\frac{\partial}{\partial\boldsymbol{x}}=\bigg( \frac{\partial}{\partial x_{i,j}} \bigg)_{\!\!1\le i\le b+2,\, 1\le j\le 2}. \end{equation}

We consider

(3.3) \begin{equation} \mathrm{tr}\Delta=\sum_{i=1}^{b+2}\sum_{j=1}^{2}\frac{\partial^2}{\partial x_{i,j}^2}\quad\text{and}\quad\!\exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)=\sum_{m=0}^\infty\frac{1}{m!}\bigg(\frac{-\mathrm{tr} (\Delta y^{-1})}{8\pi}\bigg)^{\!\!m}, \end{equation}

for any symmetric positive-definite matrix $y\in\mathbb{R}^{2\times 2}$ , as operators acting on the space of smooth $\mathbb{C}$ -valued functions on $(\mathbb{R}^{b,2})^2$ ; see (4.12) for an explicit expression of $\mathrm{tr} (\Delta y^{-1})$ . We say that a smooth function $f\colon(\mathbb{R}^{b,2})^2\to\mathbb{R}$ is harmonic if $\mathrm{tr}\Delta f=0$ .

Remark 3.3. If a very homogeneous polynomial $\mathcal{P}$ is harmonic, then $\Delta\mathcal{P}=0$ , namely,

\begin{equation*} \sum_{j=1}^{b+2}\frac{\partial^2 \mathcal{P}}{\partial x_{j,\eta} \partial x_{j,\xi}}=0\quad\text{for every~$1\le\eta$, $\xi\le 2$}; \end{equation*}

see e.g. [Reference FreitagFre83, Bemerkung 3.3]. This implies that

\begin{equation*} \exp \bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1}) \bigg)(\mathcal{P})=\mathcal{P}. \end{equation*}

This is analogous to the case of homogeneous harmonic polynomials in the genus-1 case; see [Reference ZuffettiZuf24b, Remark 3.2].

3.2 The Weil representation and theta functions

We recall here the Schrödinger model $\omega_{\infty,2}$ of the genus-2 Weil representation on the space of Schwartz functions on a quadratic space of general signature; see [Reference Funke and MillsonFM02, § 4] and [Reference Funke and MillsonFM06, § 7] for further details. We then use it to construct Siegel theta functions of genus 2.

Let V be a real quadratic space of signature $(b^+,b^-)$ . Only at a later point will we restrict to the signature of interest for the present paper, i.e. $b^-=2$ . We write $\mathrm{Mp}_4(\mathbb{R})$ for the metaplectic group of order 4, namely, the double cover of $\mathrm{Sp}_4(\mathbb{R})$ realized by the two choices of holomorphic square roots of $\tau\mapsto\det(C\tau+D)$ for every $(\begin{smallmatrix} A & B\\ C & D \end{smallmatrix})\in\mathrm{Sp}_4(\mathbb{R})$ . Hence, it consists of elements of the form

\begin{align*} \bigg(\bigg(\begin{matrix} A & B\\[3pt] C & D \end{matrix}\bigg), \pm\det(C\tau + D)^{1/2} \bigg)\quad \text{where~$\bigg(\begin{matrix} A & B\\[1pt] C & D \end{matrix}\bigg)\in\mathrm{Sp}_4(\mathbb{R})$.} \end{align*}

Occasionally, we just write $(\begin{smallmatrix} A & B\\ C & D \end{smallmatrix})$ for $((\begin{smallmatrix} A & B\\ C & D \end{smallmatrix}),\det(C\tau + D)^{1/2})$ if it is clear from the context that we are working with the metaplectic cover. The group operation in $\mathrm{Mp}_4(\mathbb{R})$ is given by $(M,f(\cdot))\cdot (N,g(\cdot)) =(MN , f(N(\cdot))g(\cdot))$ ; see [Reference ZhangZha09, § $2.1$ ] for further information. The group $\mathrm{Mp}_4(\mathbb{Z})$ is the inverse image of $\mathrm{Sp}_4(\mathbb{Z})$ under the covering map. For simplicity, we denote by S the element $S=((\begin{smallmatrix} 0 & -I_2\\ I_2 & 0 \end{smallmatrix}),\det\tau^{1/2})$ of $\mathrm{Mp}_4(\mathbb{Z})$ . Note that $S^2=Z$ , where $Z=({-}I_4,-1)$ lies in the center of $\mathrm{Mp}_4(\mathbb{Z})$ .

Definition 3.4. The Schrödinger model $\omega_{\infty,2}$ of the Weil representation provides an action of $\mathrm{Mp}_4(\mathbb{R})\times\mathrm{O}(V)$ on the space $\mathcal{S}(V^2)$ of Schwartz functions on $V^2$ as follows. The action of $\mathrm{O}(V)$ is given by

\begin{align*} \omega_{\infty,2}(g)\varphi(\boldsymbol{v})=\varphi(g^{-1}(\boldsymbol{v})), \end{align*}

for every $\varphi\in\mathcal{S}(V^2)$ and $g\in\mathrm{O}(V)$ . The action of $\mathrm{Mp}_4(\mathbb{R})$ is given by

(3.4)

where $\widehat{\varphi}$ is the Fourier transform of $\varphi$ normalized as in Appendix A.1.

Let $\boldsymbol{\delta},\boldsymbol{\nu}\in V^2$ . For any $\varphi \in \mathcal{S}(V^2)$ , we define a new Schwartz function $\varphi'$ with characteristics $(\boldsymbol{\delta},\boldsymbol{\nu})$ by

\begin{align*} \varphi'(\boldsymbol{v}, \boldsymbol{\delta},\boldsymbol{\nu}) := \varphi(\boldsymbol{v} + \boldsymbol{\nu}) e(({-}\boldsymbol{v} + \boldsymbol{\nu}/2, \boldsymbol{\delta})). \end{align*}

Then a straightforward calculation using Lemma A.1 yields

(3.5) \begin{align} \omega_{\infty,2} (\widetilde{M})(\varphi')(\boldsymbol{v},\boldsymbol{\delta},\boldsymbol{\nu})= (\omega_{\infty,2}(\widetilde{M})\varphi)'(\boldsymbol{v},(\boldsymbol{\delta},\boldsymbol{\nu})\widetilde{M}^{t}) \end{align}

for all $\widetilde{M}=(M,\phi)\in\mathrm{Mp}_4(\mathbb{R})$ , where we simply denote by $(\boldsymbol{\delta},\boldsymbol{\nu})\widetilde{M}^{t}$ the tuple $(\boldsymbol{\delta},\boldsymbol{\nu})M^t$ . From now on, we will write $M \in \mathrm{Mp}_4(\mathbb{R})$ instead of $\widetilde{M}$ .

We now construct theta series with characteristics $(\boldsymbol{\delta},\boldsymbol{\nu}) \in V^2$ attached to lattice cosets of $L^2$ , where L is an even lattice such that $V=L\otimes\mathbb{R}$ .

Definition 3.5. The Siegel theta function of index $\sigma\in D_{L}^2$ and characteristics $(\boldsymbol{\delta},\boldsymbol{\nu})\in V^2$ is defined as

\begin{align*} \theta_{L, 2}^\sigma(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi) := \sum_{\boldsymbol{\lambda} \in L^2 + \sigma} (\omega_{\infty,2}(M) \varphi')(g(\boldsymbol{\lambda}, (\boldsymbol{\delta},\boldsymbol{\nu})M^{-t})) \end{align*}

for every $M \in \mathrm{Mp}_4(\mathbb{R})$ , $ g \in G$ , and $\varphi \in \mathcal{S}(V^2)$ .

The real analytic functions $\theta_{L, 2}^\sigma$ behave with respect to the action of the generators of $\mathrm{Mp}_4(\mathbb{Z})$ as follows. If $A\in\mathrm{SL}_2(\mathbb{Z})$ , then

\begin{align*} \theta_{L, 2}^\sigma\bigg(\bigg(\begin{matrix} A & 0\\ 0 & {A}^{-t} \end{matrix}\bigg)M, (\boldsymbol{\delta}, \boldsymbol{\nu})\bigg(\begin{matrix} A & 0\\ 0 & {A}^{-t} \end{matrix}\bigg)^{\!\!t}, g, \varphi\bigg) &= \theta_{L, 2}^{\sigma A}(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi). \end{align*}

If $B\in\mathrm{Sym}_2(\mathbb{Z})$ , then

\begin{align*} \theta_{L, 2}^\sigma\bigg(\bigg(\begin{matrix} I_2 & B\\ 0 & I_2 \end{matrix}\bigg)M, (\boldsymbol{\delta}, \boldsymbol{\nu})\bigg(\begin{matrix} I_2 & B\\ 0 & I_2 \end{matrix}\bigg)^{\!\!t}, g, \varphi\bigg) = e(q(\sigma) B) \cdot \theta_{L, 2}^\sigma(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi). \end{align*}

Furthermore, by the Poisson summation formula we have that

\begin{align*} \theta_{L, 2}^\sigma\bigg(\bigg(\begin{matrix} 0 & -I_2\\ I_2 & 0 \end{matrix}\bigg)M, (\boldsymbol{\delta}, \boldsymbol{\nu}) \bigg(\begin{matrix} 0 & -I_2\\ I_2 & 0 \end{matrix}\bigg)^{\!\!t}, g, \varphi\bigg) &= \frac{i^{b^- - b^+}}{\lvert D_L \rvert} \sum_{\sigma' \in D_L^2} e({-}\sigma', \sigma) \cdot \theta_{L, 2}^{\sigma'}(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi). \end{align*}

These equations define a representation $\rho_{L,2}$ of $\mathrm{Mp}_4(\mathbb{Z})$ on the group algebra $\mathbb{C}[D_L^2]$ . Let $(\mathfrak{e}_{\sigma})_{{\sigma}\in D_{L}^2}$ be the standard basis of the group algebra $\mathbb{C}[D_{L}^2]$ , and let $\langle\cdot{,}\cdot\rangle$ be the standard Hermitian scalar product on $\mathbb{C}[D_{L}^2]$ defined as

\[ \bigg\langle \sum_{{\sigma}\in D_{L}^2}\lambda_{\sigma}\mathfrak{e}_{\sigma},\sum_{{\sigma}\in D_{L}^2}\mu_{\sigma}\mathfrak{e}_\sigma \bigg\rangle = \sum_{{\sigma}\in D_{L}^2}\lambda_{\sigma}\overline{\mu_{\sigma}}. \]

The Weil representation $\rho_{L,2}$ is unitary with respect to such scalar product; see [Reference ZhangZha09] and [Reference Bruinier and ZuffettiBZ24] for further details on $\rho_{L,2}$ and generalizations.

With the notation introduced above, we may rephrase the behavior of the functions $\theta^\sigma_{L,2}$ with respect to $\mathrm{Mp}_4(\mathbb{Z})$ by saying that the vector-valued theta function

$$ \Theta_{L, 2}(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi) := \sum_{\sigma \in D_L^2} \theta_{L, 2}^\sigma(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi) \mathfrak{e}_\sigma $$

satisfies

(3.6) \begin{equation} \Theta_{L,2}(\gamma M, (\boldsymbol{\delta}, \boldsymbol{\nu})\gamma^t, g, \varphi) = \rho_{L,2}(\gamma) \Theta_{L, 2}(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi) \end{equation}

for all $\gamma \in \mathrm{Mp}_4(\mathbb{Z})$ .

3.3 Siegel modular forms of genus 2

In this section we employ the representation $\rho_{L,2}$ of $\mathrm{Mp}_4(\mathbb{Z})$ on $\mathbb{C}[D_{L}^2]$ to construct vector-valued holomorphic Siegel modular forms of genus 2.

Let $k\in {\frac{1}{2}}\mathbb{Z}$ . A weight-k Siegel modular form of genus 2 with respect to the Weil representation $\rho_{L,2}$ is a holomorphic function $f\colon\mathbb{H}_2\to\mathbb{C}[D_{L}^2]$ such that

\[ f(\gamma\cdot\tau)=\phi(\tau)^{2k}\rho_{L,2}(\gamma)f(\tau)\quad\text{for all~$\gamma=\bigg(\bigg(\begin{matrix} A & B\\ C & D \end{matrix}\bigg),\phi(\tau)\bigg)\in\mathrm{Mp}_4(\mathbb{Z})$}, \]

where $\gamma\cdot\tau=(A\tau+B)(C\tau+D)^{-1}$ . We denote the scalar components of f by $f_{\sigma}$ , so that $f=\sum_{{\sigma}\in D_{L}^2} f_{\sigma} \mathfrak{e}_{\sigma}$ .

The invariance of f with respect to translations in $\mathrm{Mp}_4(\mathbb{Z})$ implies that the functions ${e({-} q({\sigma})\tau)f_{\sigma}(\tau)}$ are periodic with respect to integral symmetric $2\times 2$ matrices, for every ${\sigma}\in D_{L}^2$ . Hence, f admits a Fourier expansion of the form

(3.7) \begin{equation} f(\tau)=\sum_{{\sigma}\in D_{L}^2}\sum_{\substack{T\in\Lambda_2+q({\sigma})\\ T\ge 0}} a_f(\sigma,T) \cdot \mathfrak{e}_{\sigma}(T\tau), \end{equation}

where $\Lambda_2$ is the set of symmetric half-integral $2\times 2$ matrices and $\mathfrak{e}_\sigma(t):= e(t)\mathfrak{e}_\sigma$ . If it is clear that the Fourier coefficients are referred to a given modular form f, then we drop the index f and denote the coefficients simply by $a(\sigma,T)$ .

We say that a Siegel modular form is a cusp form if all its Fourier coefficients indexed over degenerate matrices vanish. We define $M^k_{2,L}$ , respectively $S^k_{2,L}$ , to be the space of weight-k and genus-2 Siegel modular forms, respectively cusp forms, with respect to the Weil representation $\rho_{L,2}$ .

3.4 Siegel theta functions of genus 2 à la Borcherds

Let ${K_\infty'} \subseteq \mathrm{Mp}_4(\mathbb{R})$ be the preimage of $\mathrm{U}(2) := := \{ (\begin{smallmatrix} A & B \\ -B & A \end{smallmatrix}) : AA^t + BB^t = 1 \}$ under the metaplectic cover. We define a character $\chi_{1/2}$ on ${K_\infty'}$ as

\begin{align*}\chi_{1/2}\bigg(\bigg(\begin{matrix} A & B \\ -B & A \end{matrix}\bigg), \pm \det({-}B\tau + A)^{1/2}\bigg) = \pm \det({-}Bi + A)^{-1/2}.\end{align*}

Let $k\in{\frac{1}{2}}\mathbb{Z}$ . If a Schwartz function $\varphi \in \mathcal{S}(V^2)$ satisfies $\omega_{\infty,2}(M') \varphi = \chi_{1/2}^{2k}(M') \varphi$ for all $M' \in {K_\infty'}$ , then

\begin{align*}\Theta_{L,2}(M M', (\boldsymbol{\delta}, \boldsymbol{\nu}), g, \varphi) = \chi_{1/2}^{2k}(M') \Theta_{L, 2}(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi) \quad \text{for all $M' \in {K_\infty'}$.}\end{align*}

In this case, we may associate to $\Theta_{L, 2}(M, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi)$ a smooth function on $\mathbb{H}_2$ that behaves similarly to the Siegel modular forms constructed in § 3.3, with the following procedure. For every $\tau \in \mathbb{H}_2$ , we denote by $M_\tau:=(\begin{smallmatrix} I_2 & x\\ 0 & I_2 \end{smallmatrix})(\begin{smallmatrix} y^{1/2} & 0\\ 0 & (y^{1/2})^{-t} \end{smallmatrix})$ the standard element of $\mathrm{Sp}_4(\mathbb{R})$ mapping $iI_2$ to $\tau$ . Then, the theta function

\begin{align*}\Theta_{L, 2}(\tau, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi) := \det y^{-k/2} \Theta_{L, 2}(M_\tau, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi)\end{align*}

satisfies

\begin{align*}\Theta_{L,2}(\gamma \cdot \tau, (\boldsymbol{\delta}, \boldsymbol{\nu})\gamma^t, g, \varphi) = \phi(\tau)^{2k} \rho_{L,2}(\gamma) \Theta_{L, 2}(\tau, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi)\end{align*}

for all $\gamma \in \mathrm{Mp}_4(\mathbb{Z})$ .

Example 3.6. Let $(b^+,b^-)=(b,2)$ . The standard Gaussian $\varphi_{0,2}$ of $(\mathbb{R}^{b,2})^2$ is defined as

\begin{equation*} \varphi_{0,2}(\boldsymbol{x})=\exp\bigg({-}\pi\sum_{i=1}^{b+2}\sum_{j=1}^2 x_{i,j}^2 \bigg)\quad\text{for every~$\boldsymbol{x}=(x_1,x_2)\in(\mathbb{R}^{b,2})^2$}, \end{equation*}

where $x_j=(x_{1,j},\dots,x_{b+2,j})^t\in \mathbb{R}^{b,2}$ . The standard Gaussian of $V^2$ is the composition of ${\varphi_{0,2}}$ with $g_0$ , where the latter is as in (3.1). For simplicity, we will denote the standard Gaussian of $V^2$ with $\varphi_{0,2}$ in place of $\varphi_{0,2}\circ g_0$ . It is well known that $\varphi_{0,2}$ is an eigenfunction under the action of ${K_\infty'}$ of weight $k=b/2-1$ , in the sense that $\omega_{\infty,2}(M')\varphi_{0,2} = \chi_{1/2}^{b-2}(M')\varphi_{0,2}$ ; see e.g. [Reference Borel and WallachBW00, Chapter VIII, § 1].

From now on, we assume that the signature of V is (b,2). Let $\mathcal{P}$ be a very homogeneous polynomial of degree $(m^+,m^-)$ on $(\mathbb{R}^{b,2})^2$ . For simplicity, we write $\mathcal{P}(\boldsymbol{v})$ for the value $\mathcal{P}(g_0(\boldsymbol{v}))$ where $\boldsymbol{v}\in V^2$ , i.e. we consider $\mathcal{P}$ as a polynomial in the coordinates of $\boldsymbol{v}$ with respect to the basis $(e_j)_j$ of V. It is easy to see, using Lemma A.2, that the Schwartz function

(3.8) \begin{equation} \varphi(\boldsymbol{v}) = \exp({-}\operatorname{tr}\Delta / 8 \pi)(\mathcal{P})(\boldsymbol{v}) \varphi_{0,2}(\boldsymbol{v}) \end{equation}

is an eigenfunction of ${K_\infty'}$ of weight $b/2-1+m^+-m^-$ , namely,

(3.9) \begin{equation} \omega(M')\varphi = \chi_{1/2}^{b-2+2m^+-2m^-}(M') \varphi \quad\text{for all~$M'\in {K_\infty'}$.} \end{equation}

This will be relevant in § 6 only for special cases of $\mathcal{P}$ , for which (3.9) follows immediately from the similar behavior of the Kudla–Millson Schwartz function; see [Reference Kudla and MillsonKM86, Theorem 3.1, (ii)] and § 6.1 for further details.

The theta functions arising from $\varphi$ as in (3.8) are generalizations in genus 2 of Borcherds’ Siegel theta functions [Reference BorcherdsBor98, § 4]. They can also be considered as vector-valued analogues of the theta functions introduced by Roehrig in [Reference RoehrigRoe21]. These considerations can easily be checked using the following explicit rewriting of $\theta_{L,2}^{\sigma}$ à la Borcherds. Recall that for fixed $g\in G$ , we define $z=g^{-1}(z_0)\in\mathrm{Gr}(L)$ to be the negative-definite plane mapping to $z_0$ under g.

Lemma 3.7. If $\varphi(\boldsymbol{v}) = \exp({-}\operatorname{tr}\Delta / 8 \pi)(\mathcal{P})(\boldsymbol{v}) \varphi_{0,2}(\boldsymbol{v})$ for some very homogeneous polynomial $\mathcal{P}$ of degree $(m^+,m^-)$ on $(\mathbb{R}^{b,2})^2$ , then

(3.10) \begin{align} \theta_{L,2}^{\sigma}(\tau,\boldsymbol{\delta},\boldsymbol{\nu},g,\varphi) &= \det y^{1 + m^-} \sum_{\boldsymbol{\lambda}\in L^2+{\sigma}}\exp \bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1}) \bigg)(\mathcal{P})(g(\boldsymbol{\lambda}+\boldsymbol{\nu}))\nonumber \\ &\quad\times e( q((\boldsymbol{\lambda}+\boldsymbol{\nu})_{z^\perp})\tau+ q((\boldsymbol{\lambda}+\boldsymbol{\nu})_z)\bar{\tau}- (\boldsymbol{\lambda}+\boldsymbol{\nu}/2,\boldsymbol{\delta})). \end{align}

The theta functions considered in Lemma 3.7 play a central role in the present paper. For this reason, we reserve a special notation for them, given as follows.

Definition 3.8. Let $\mathcal{P}$ be a very homogeneous polynomial on $(\mathbb{R}^{b,2})^2$ . We denote by $\theta_{L,2}^{\sigma}(\tau,\boldsymbol{\delta},\boldsymbol{\nu},g,\mathcal{P})$ the theta function arising from $\varphi=\exp({-}\operatorname{tr}\Delta / 8 \pi)(\mathcal{P})(\boldsymbol{v}) \varphi_{0,2}(\boldsymbol{v})$ ; see (3.10) for an explicit formulation. The $\mathbb{C}[D_{L}^2]$ -valued Siegel theta function associated to L and the polynomial $\mathcal{P}$ is

\begin{align*} \Theta_{L,2}(\tau,\boldsymbol{\delta},\boldsymbol{\nu},g,\mathcal{P}) := \sum_{{\sigma}\in D_{L}^2}\theta_{L,2}^\sigma(\tau,\boldsymbol{\delta},\boldsymbol{\nu},g,\mathcal{P}) \mathfrak{e}_{\sigma}. \end{align*}

If $\boldsymbol{\delta},\boldsymbol{\nu}=0$ , we usually drop them from the notation and simply write $\Theta_{L,2}(\tau,g,\mathcal{P})$ .

Proof of Lemma 3.7. By (3.5), we may rewrite

\begin{align*} \theta_{L,2}^\sigma(\tau, \boldsymbol{\delta}, \boldsymbol{\nu}, g, \varphi) &= \det y^{({-}b/2+1-m^++m^-)/2} \sum_{\boldsymbol{\lambda}\in L^2+\sigma} (\omega_{\infty,2}(M_\tau)\varphi)'(g(\boldsymbol{\lambda},\boldsymbol{\delta},\boldsymbol{\nu})). \end{align*}

For any $\boldsymbol{v}\in V^2$ , we may compute

\begin{align*} (\omega_{\infty,2}(M_\tau)\varphi)(\boldsymbol{v}) &= \det y^{(b+2)/4} \cdot e( x q(\boldsymbol{v}))\cdot\varphi(\boldsymbol{v}y^{1/2}), \end{align*}

where $\varphi(\boldsymbol{v}y^{1/2}) = \exp({-}\operatorname{tr}\Delta / 8 \pi)(\mathcal{P})(\boldsymbol{v}y^{1/2}) \varphi_{0,2}(\boldsymbol{v}y^{1/2})$ . By [Reference RoehrigRoe21, Lemma 4.4, ( $4.5$ )] and the very homogeneity of $\mathcal{P}$ , we deduce that

\begin{align*} \exp({-}\operatorname{tr}\Delta/8\pi)(\mathcal{P})(\boldsymbol{v}y^{1/2}) = \det y^{(m^++m^-)/2}\exp({-}\operatorname{tr}(\Delta y^{-1})/8\pi)(\mathcal{P})(\boldsymbol{v}). \end{align*}

This and the rewriting $e( xq(\boldsymbol{v}))\cdot\varphi_{0,2}(g(\boldsymbol{v}y^{1/2})) = e( \mathrm{tr} (q(\boldsymbol{v}_{z^\perp})\tau)+\mathrm{tr}(q(\boldsymbol{v}_z)\bar{\tau}))$ imply the claimed formula for $\theta_{L,2}^{\sigma}(\tau,\boldsymbol{\delta},\boldsymbol{\nu},g,\varphi)$ .

Remark 3.9. The Siegel theta functions of Definition 3.8 satisfy

(3.11) \begin{equation} \Theta_{L,2}(\gamma\cdot \tau,(\boldsymbol{\delta},\boldsymbol{\nu})\gamma^t,g,\mathcal{P}) = \phi(\tau)^{b-2+2(m^++m^-)} \rho_{L,2}(\gamma) \Theta_{L,2}(\tau,\boldsymbol{\delta},\boldsymbol{\nu},g,\mathcal{P}) \end{equation}

for all $\gamma\in\mathrm{Mp}_4(\mathbb{Z})$ .

3.5 Jacobi forms and Jacobi Siegel theta functions

Let $\mathcal{H}(\mathbb{R}) := \mathbb{R}^3$ be the Heisenberg group, where the multiplication is given by

\begin{align*}(r, s, t) \cdot (r', s', t') := (r + r', s + s', t + t' + r s' - r's)\end{align*}

and define the metaplectic Jacobi group by $\mathcal{J}(\mathbb{R}) := \mathcal{H}(\mathbb{R}) \rtimes \mathrm{Mp}_2(\mathbb{R})$ , where the action of $\mathrm{Mp}_2(\mathbb{R})$ on $\mathcal{H}(\mathbb{R})$ is given by

\begin{align*}(r, s, t) \cdot \bigg(\begin{matrix} a & b \\ c & d\end{matrix}\bigg) = (a r + c s, b r + d s, t).\end{align*}

The Jacobi group $\mathcal{J}(\mathbb{R})$ acts on $\mathbb{H} \times \mathbb{C}$ under

\begin{align*}(r, s, t) (\tau_1, \tau_2) := (\tau_1, \tau_2 + r + s \tau_1) \quad\text{and}\quad \bigg(\bigg(\begin{matrix} a & b \\ c & d\end{matrix}\bigg), \phi\bigg) (\tau_1, \tau_2) := \bigg(\frac{a \tau_1 + b}{c \tau_1 + d}, \frac{\tau_2}{c \tau_1 + d}\bigg).\end{align*}

The metaplectic Jacobi group $\mathcal{J}(\mathbb{R})$ can be embedded into $\mathrm{Mp}_4(\mathbb{R})$ as

\begin{align*} \bigg(r, s, t, \bigg(\begin{matrix} a & b \\ c & d\end{matrix}\bigg)\bigg) \mapsto \begin{pmatrix} a & 0 & b & a s - b r \\ r & 1 & s & t \\ c & 0 & d & c s - d r \\ 0 & 0 & 0 & 1 \end{pmatrix} \end{align*}

and by mapping the square root $\phi(\tau_1)$ of $c \tau_1 + d$ to $\tilde{\phi}((\begin{smallmatrix} \tau_1 & \tau_2 \\ \tau_2 & \tau_3 \end{smallmatrix})) = \phi(\tau_1)$ . Its image is the Klingen parabolic $C_{1,2}(\mathbb{R})$ , which is the stabilizer of the standard 1-dimensional boundary component of $\mathbb{H}_2$ . Moreover, the action of $\mathrm{Mp}_4(\mathbb{R})$ on $(\begin{smallmatrix} \tau_1 & \tau_2 \\ \tau_2 & \tau_3 \end{smallmatrix}) \in \mathbb{H}_2$ restricts to the above action of $\mathcal{J}(\mathbb{R})$ on $(\tau_1, \tau_2) \in \mathbb{H} \times \mathbb{C}$ .

For $M \in \mathcal{J}(\mathbb{R}), \eta \in L', g \in G$ , and a Schwartz function $\varphi \in \mathcal{S}(V^2)$ , we define the Jacobi theta function with characteristics $\delta, \nu \in V$ by

\begin{align*}\Theta_{L, \eta}^\sigma(M, \delta, \nu, g, \varphi) = \sum_{\lambda \in L + \sigma} (\omega_{\infty,2}(M) \varphi')(g(\lambda, \eta, ((\delta, 0),(\nu, 0))M^{-t})).\end{align*}

Then, for $b \in \mathbb{Z}$ , we have

\begin{align*}\Theta_{L, \eta}^\sigma\bigg(\bigg(\begin{matrix} 1 & b\\ 0 & 1 \end{matrix}\bigg)M, (\delta, \nu)\bigg(\begin{matrix} 1 & b\\ 0 & 1 \end{matrix}\bigg)^{\!\!t}, g, \varphi\bigg) = e(q(\sigma) b) \Theta_{L, \eta}^\sigma(M, \delta, \nu, g, \varphi)\end{align*}

and

\begin{align*}\Theta_{L, \eta}^\sigma\bigg(\bigg(\begin{matrix} 0 & -1\\ 1 & 0 \end{matrix}\bigg)M, (\delta, \nu) \bigg(\begin{matrix} 0 & -1\\ 1 & 0 \end{matrix}\bigg)^{\!\!t}, g, \varphi\bigg) = \frac{\sqrt{i}^{b^- - b^+}}{\sqrt{\lvert D_L \rvert}} \sum_{\sigma' \in D_L} e({-}\sigma', \sigma) \Theta_{L,\eta}^{\sigma'}(M, \delta, \nu, g, \varphi).\end{align*}

Both equations can be seen by observing that the restriction of the Schrödinger model of the Weil representation of $\mathrm{Mp}_4(\mathbb{R})$ to $\mathrm{Mp}_2(\mathbb{R}) \subseteq \mathcal{J}(\mathbb{R})$ yields the Schrödinger model of the Weil representation of $\mathrm{Mp}_2(\mathbb{R})$ on $\mathcal{S}(V)$ . Moreover, a direct calculation yields, for every $(r, s, t) \in \mathcal{H}(\mathbb{Z})$ ,

\begin{align*} \Theta_{L, \eta}^\sigma((r, s, t) M, \delta, \nu, g, \varphi) = e(r(\sigma, \eta) + (t - rs) q(\eta)) \Theta_{L,\eta}^{\sigma - s \eta}(M, \delta, \nu, g, \varphi). \end{align*}

These equations define a representation $\rho_{L, \eta}$ of $\mathcal{J}(\mathbb{Z})$ on the group ring $\mathbb{C}[D_L]$ such that

\begin{align*}\rho_{L, 2}(M, \tilde{\phi}) \mathfrak{e}_{(\sigma, \eta)} = (\rho_{L, \eta}(M, \phi) \mathfrak{e}_\sigma) \otimes \mathfrak{e}_\eta\end{align*}

for all $(M, \phi) \in \mathcal{J}(\mathbb{Z})$ , where we identify $\mathbb{C}[D_L] \otimes \mathbb{C}[D_L]$ with $\mathbb{C}[D_L^2]$ using the isomorphism

(3.12) \begin{equation} \mathbb{C}[D_L] \otimes \mathbb{C}[D_L] \to \mathbb{C}[D_L^2], \quad \mathfrak{e}_{\sigma_1} \otimes \mathfrak{e}_{\sigma_2} \mapsto \mathfrak{e}_{(\sigma_1, \sigma_2)}. \end{equation}

Definition 3.10. A vector-valued Jacobi form of weight k and index N with respect to the Weil representation $\rho_{L, \sigma_2}$ is a holomorphic function $\phi : \mathbb{H} \times \mathbb{C} \to \mathbb{C}[D_L]$ with

\begin{align*}\phi(\tau_1, \tau_2 + r + s \tau_1) = e({-}N s^2 \tau_1 - 2 N s \tau_2 - r s N - t N) \rho_{L, \sigma_2}(r, s, t) \phi(\tau_1, \tau_2)\end{align*}

for $(r, s, t) \in \mathcal{H}(\mathbb{Z})$ and

\begin{align*}\phi(\gamma (\tau_1, \tau_2)) = (c \tau_1 + d)^{k} e\bigg(\frac{N c \tau_2^2}{c \tau_1 + d}\bigg) \rho_{L, \sigma_2}(\gamma) \phi(\tau_1, \tau_2)\end{align*}

for $\gamma = ((\begin{smallmatrix}a & b \\ c & d\end{smallmatrix}), \phi) \in \mathrm{Mp}_2(\mathbb{Z})$ . If $N = q(\eta)$ for $\eta \in \sigma_2 + L$ , we say that f is a vector-valued Jacobi form of weight k and index $\eta$ .

Example 3.11. Let $\varphi \in \mathcal{S}(V)$ be a Schwartz function such that $\omega(M') \varphi = \chi_{1/2}^{2k}(M') \varphi$ for all $M' \in \mathrm{Mp}_2(\mathbb{R}) \cap {K_\infty'}$ and some $k \in {\frac{1}{2}}\mathbb{Z}$ . Let $M_{\tau_1, \tau_2} \in \mathcal{J}(\mathbb{R})$ be given by $M_{\tau_1} \cdot ({x_2}/{\sqrt{y_1}}, {y_2}/{\sqrt{y_1}}, 0)$ , where $M_{\tau_1} :=(\begin{smallmatrix} 1 & x_1 \\ 0 & 1 \end{smallmatrix})(\begin{smallmatrix} y_1^{1/2} & 0\\ 0 & y_1^{-1/2} \end{smallmatrix})$ , so that $M_{\tau_1, \tau_2} (i, 0) = (\tau_1, \tau_2)$ . Then

\begin{align*}\Theta_{L, \eta}(\tau_1, \tau_2, \delta, \nu, g, \varphi) := y_1^{k/2} \Theta_{L, \eta}(M_{\tau_1, \tau_2}, \delta, \nu, g, \varphi)\end{align*}

satisfies

\begin{align*} &\Theta_{L, \eta}(\tau_1, \tau_2 + r + s \tau_1, \delta, \nu, g, \varphi) \\ &\quad= e({-}q(\eta) s^2 \tau_1 - 2q(\eta) s \tau_2 - rs q(\eta) - tq(\eta)) \rho_{L, \eta}(r, s, t) \Theta_{L, \eta}(\tau_1, \tau_2, \delta, \nu, g, \varphi) \end{align*}

and

\begin{align*}\Theta_{L, \eta}(M(\tau_1, \tau_2), (\delta, \nu)M^t, g, \varphi) = \phi(\tau)^{2k} e\bigg(\frac{q(\eta) c \tau_2^2}{c \tau_1 + d}\bigg) \rho_{L, \eta}(M) \Theta_{L, \eta}(\tau_1, \tau_2, \delta, \nu, g, \varphi).\end{align*}

Assume that L has signature $(b^+, b^-)$ and fix an isometry $g_0 : L \otimes \mathbb{R} \to \mathbb{R}^{b^+, b^-}$ . Let

\begin{align*}\Delta = \sum_{i = 1}^{b^+ + b^-} \frac{\partial^2}{\partial x_i^2}\end{align*}

be the Laplace operator on $\mathbb{R}^{b^+ + b^-}$ . If $\mathcal{P} : \mathbb{R}^{b^+, b^-} \to \mathbb{C}$ is a homogeneous polynomial, then

\begin{align*}\varphi(\lambda) = \exp({-} \Delta / 8 \pi)(\mathcal{P})(g_0(\lambda)) \varphi_0(\lambda)\end{align*}

is a Schwartz function that satisfies

\begin{align*}\omega(M') \varphi = \chi_{1/2}^{2k}(M') \varphi\end{align*}

for all $M' \in \mathrm{Mp}_2(\mathbb{R}) \cap K'$ , where $k = b^+/2 + m^+ - b^- / 2$ . Write $z_0^\perp$ , respectively $z_0$ , for the preimage of $\mathbb{R}^{b^+, 0}$ , respectively $\mathbb{R}^{0, b^-}$ , under $g_0$ . For an isometry $g \colon L \otimes \mathbb{R} \to L \otimes \mathbb{R}$ , we define $z^\perp$ , respectively z, to be the preimage of $z_0^\perp$ , respectively $z_0$ , under g. Then $z_0$ and z are negative-definite subspaces of dimension $b^-$ in $L \otimes \mathbb{R}$ , so they lie in the Grassmannian $\mathrm{Gr}(L)$ . If we denote by $\lambda_{z^\perp}$ and $\lambda_z$ the projections of $\lambda$ to the subspace $z^\perp$ and z respectively, for $\lambda \in L \otimes \mathbb{R}$ , then

\begin{align*} &\theta^\sigma_{L, \eta}(\tau_1, \tau_2, \delta, \nu, g, \mathcal{P}) := \theta^\sigma_{L, \eta}(\tau_1, \tau_2, \delta, \nu, g, \varphi) \\ &\quad= y_1^{b^-/2} \exp(4 \pi q(\eta_{z}) y_2^2 / y_1) \sum_{\lambda \in \sigma + L} \exp({-}\Delta / 8 \pi y_1)(\mathcal{P})(g(\lambda + y_2\eta/y_1 + \nu)) \\ &\qquad\times e(q((\lambda + \nu)_{z^\perp}) \tau_1 + q((\lambda + \nu)_{z}) \overline{\tau_1} + \tau_2 (\lambda + \nu, \eta_{z^\perp}) + \overline{\tau_2} (\lambda + \nu, \eta_{z}) - (\lambda + \nu / 2, \delta)) \end{align*}

and

\begin{align*}\Theta_{L, \eta}(\tau_1, \tau_2, \delta, \nu, g, \mathcal{P}) := \Theta_{L, \eta}(\tau_1, \tau_2, \delta, \nu, g, \varphi) = \sum_{\sigma \in D_L} \theta_{L, \eta}^\sigma(\tau_1, \tau_2, \delta, \nu, g, \mathcal{P}) \mathfrak{e}_\sigma.\end{align*}

If $\phi$ is a vector-valued Jacobi form of weight k and index $\eta \in L'$ , then the transformation properties imply that $\phi$ has a Fourier expansion of the form

\begin{align*}\phi(\tau_1, \tau_2) = \sum_{\substack{\sigma \in D_L \\ r \in \mathbb{Z} + q(\sigma) \\ s \in \mathbb{Z} + (\sigma, \eta)}} a_\phi(\sigma, r, s) \mathfrak{e}_\lambda(r \tau_1 + s \tau_2).\end{align*}

We will drop the index of the Fourier coefficients if the Jacobi form is clear from the context. Since $\phi(\tau_1, \tau_2 + \tau_1) = e({-}q(\eta) \tau_1 - 2 q(\eta) \tau_2) \rho_{L, \eta}(0, 1, 0) \phi(\tau_1, \tau_2)$ , we deduce that

\begin{align*}a(\sigma, r, s) = a(\sigma + \eta, r + s + q(\eta), s + 2 q(\eta)),\end{align*}

hence

\begin{align*} a(\sigma + \eta, q(\sigma + \eta), (\sigma + \eta, \eta)) &= a(\sigma + \eta, q(\sigma) + (\sigma, \eta) + q(\eta), (\sigma, \eta) + 2q(\eta)) \\ &= a(\sigma, q(\sigma), (\sigma, \eta)). \end{align*}

A Jacobi form $\phi$ is said to be a Jacobi cusp form if $a(\sigma, r, s) \neq 0$ implies $4 q(\eta) r \gt s^2$ .

Jacobi forms occur as the Fourier–Jacobi coefficients of Siegel modular forms. That is, if ${f \in M_{2, L}^k}$ , then there exist Jacobi forms $\phi_{\sigma_2, m}(\tau_1, \tau_2)$ of weight k and index m with respect to $\rho_{L, \sigma_2}$ such that

\begin{align*}f(\tau) = \sum_{\sigma_2 \in D_L} \sum_{m \in \mathbb{Z} + q(\sigma_2)} \phi_{\sigma_2, m}(\tau_1, \tau_2) \otimes \mathfrak{e}_{\sigma_2}(m \tau_3).\end{align*}

If f has Fourier expansion as in (3.7), then the Fourier coefficients $a_{\sigma_2, m}$ of $\phi_{\sigma_2, m}$ are related to those of f as

(3.13) \begin{align} a_{\sigma_2, m}(\sigma_1, r, s) = a_f\bigg((\sigma_1, \sigma_2), \bigg(\begin{matrix} r & s / 2 \\ s / 2 & m\end{matrix}\bigg)\bigg). \end{align}

Let L be an even indefinite lattice of signature (b,2). We illustrate here the Fourier–Jacobi expansion of $\Theta_{L, 2}$ in terms of the Jacobi Siegel theta functions introduced in the current section.

Let $\Theta_{L, 2}(\tau,\boldsymbol{\delta},\boldsymbol{\nu},g,\mathcal{P})$ be the Siegel theta function of genus 2 associated to some very homogeneous polynomial $\mathcal{P}$ of degree (m,0). The polynomial $\mathcal{P}$ is homogeneous of degree m in the first variable; see Remark 3.2. Since $\mathcal{P}((\lambda,\eta)\cdot N)=\det N^m\cdot\mathcal{P}(\lambda,\eta)$ for every $N\in\mathbb{C}^{2\times 2}$ , if we choose $N=(\begin{smallmatrix} 1 & 0\\ C & 1 \end{smallmatrix})$ for some $C\in\mathbb{C}$ , then we have that

\begin{align*} (\lambda,\eta)\cdot N =(\lambda + C\eta, \eta)\quad\text{and}\quad\det N=1. \end{align*}

We then deduce that $\mathcal{P}(\lambda, \eta) = \mathcal{P}(\lambda + C\eta, \eta)$ for every $C\in\mathbb{C}$ . This property simplifies the construction of Jacobi Siegel theta functions arising from polynomials of the form $\mathcal{P}(\cdot,\eta)$ , for some fixed $\eta$ . Moreover, it plays a key role in proving the injectivity result on Jacobi Petersson inner products that we will consider in § 5.2.

For simplicity, we assume here that $\boldsymbol{\nu}=0$ . This is sufficient for the main goal of the present paper, namely, to prove the injectivity of the genus-2 Kudla–Millson lift.

Recall from (3.12) that we may regard $\mathbb{C}[D_{L}^2]$ as the product $\mathbb{C}[D_{L}]\otimes\mathbb{C}[D_{L}]$ . Then the Fourier–Jacobi expansion of $\Theta_{L, 2}(\tau,\boldsymbol{\delta},0,g,\mathcal{P})$ is

\begin{align*} &\Theta_{L,2}(\tau,\boldsymbol{\delta},0,g,\mathcal{P}) \\ &\quad= \bigg(\frac{\det y}{y_1}\bigg)^{\!\!1/2} \sum_{\sigma \in D_L} \sum_{\eta \in \sigma + L} \Theta_{L, \eta}\bigg(\tau_1, \tau_2, \delta_1, 0, g, \exp\bigg({-}\frac{y_1}{\det y} \Delta_2\bigg) \mathcal{P}(\cdot, g(\eta))\bigg) \\ &\qquad\times e({-}2 i q(\eta_w) (\det y) / y_1 - (\eta, \delta_2)) \otimes \mathfrak{e}_{\sigma}(q(\eta) \tau_3) \\ &\quad= \bigg(\frac{\det y}{y_1}\bigg)^{\!\!1/2} \sum_{\sigma \in D_L} \sum_{m \in \mathbb{Z}+ q(\sigma)} \sum_{\substack{\eta \in \sigma + L \\ q(\eta) = m}} \Theta_{L, \eta}\bigg(\tau_1, \tau_2, \delta_1, 0, g, \exp\bigg({-}\frac{y_1}{\det y} \Delta_2\bigg) \mathcal{P}(\cdot, g(\eta))\bigg) \\ &\qquad\times e({-}2 i q(\eta_w) (\det y) / y_1 - (\eta, \delta_2)) \otimes \mathfrak{e}_{\sigma}(m \tau_3), \end{align*}

where $\Delta_2$ is the Laplace operator acting on the second variable of $\mathcal{P}$ .

The following lemma gathers some properties of the elements of $\mathbb{H}\times\mathbb{C}$ . The proof is a straightforward calculation.

1. (i) First, ${\Im(\tau_2 + r + s \tau_1)}/{\Im(\tau_1)} = {\Im(\tau_2)}/{\Im(\tau_1)} + s$ for every $(r, s) \in \mathbb{R}^2$ .

2. (ii) Then, ${\Im(\tau_2/\tau_1)}/{\Im({-}1/\tau_1)} = \tau_1 ({\Im(\tau_2)}/{\Im(\tau_1)} - {\tau_2}/{\tau_1})$ .

3. (iii) If $(r, s) \in \mathbb{R}^2$ and $N \in \mathbb{Z}$ , then

   \begin{align*} \exp(4 \pi N \Im(\tau_2 + s \tau_1 + r)^2 / \Im(\tau_1)) = \exp(4 \pi s^2 y_1 N + 8 \pi s y_2 N) \cdot \exp(4 \pi N y_2^2 / y_1). \end{align*}

4. (iv) For $N \in \mathbb{Z}$ ,

   \begin{align*} \exp(4 \pi N \Im(\tau_2 / \tau_1)^2 / \Im({-} 1 / \tau_1)) = e(N \tau_2^2 / \tau_1 - N \overline{\tau_2}^2 / \overline{\tau_1}) \cdot \exp(4 \pi N y_2^2 / y_1). \end{align*}

4.1 On auxiliary polynomials defined on subspaces

We begin here by illustrating how to rewrite very homogeneous polynomials as combinations of polynomials defined on certain subspaces of $(\mathbb{R}^{b,2})^2$ , in analogy with [Reference BorcherdsBor98, § 5], and study their homogeneity. Examples of such decomposition are collected in the appendix of the present paper, namely § A.2.

In fact, we will see that those polynomials on subspaces are not always very homogeneous, in contrast with the case of genus 1; see Lemma A.4. This implies that the associated genus-2 theta functions are not always modular with respect to $\mathrm{Mp}_4(\mathbb{Z})$ . Such unexpected behavior will be investigated further in § 7 using Lemma A.5, which provides the Fourier transforms of the general summands of such theta functions.

For simplicity, we restrict to the case of L splitting off (orthogonally) a hyperbolic plane U, hence $L={{K}}\oplus U$ for some Lorentzian sublattice K. We denote by u,u’ the standard hyperbolic basis of U such that

\begin{align*} u^2=u'^2=0 \quad\text{and}\quad (u,u')=1. \end{align*}

The orthogonal projection from L to K induces a projection $p\colon D_{L}\to D_{{{K}}}$ , which is an isomorphism. We denote also by p the componentwise projection induced on $D_{L}^2$ .

Without loss of generality, we may choose the orthogonal basis $(e_j)_j$ of $L\otimes\mathbb{R}$ in such a way that

(4.1) \begin{align}&K\otimes\mathbb{R}=\langle e_1,\ldots,e_{b-1},e_{b+1}\rangle_{\mathbb{R}}, \quad U\otimes\mathbb{R}=\langle e_b, e_{b+2}\rangle_{\mathbb{R}},\nonumber\\& u=\frac{e_b+e_{b+2}}{\sqrt{2}}\quad\text{and}\quad u'=\frac{e_b-e_{b+2}}{\sqrt{2}}.\end{align}

This assumption will simplify several of the following computations.

The following definitions recall some notions introduced in [Reference BorcherdsBor98, § 5].

We now define certain polynomials on subspaces of $(\mathbb{R}^{b,2})^2$ , to be considered as the analogue in genus 2 of the polynomials defined by Borcherds in [Reference BorcherdsBor98, § 5, p. 508]. Since the very homogeneous polynomials on $(\mathbb{R}^{b,2})^2$ we will work with in the next sections, namely $\mathcal{P}_{\boldsymbol{\alpha}}$ defined in (6.8), are of degree (2,0), we restrict our attention to very homogeneous polynomials of degree $(m^+,0)$ .

Definition 4.3. Let $z\in\mathrm{Gr}(L)$ , and let $g\in G$ be such that g maps z to $z_0$ . For every very homogeneous polynomial $\mathcal{P}$ of degree $(m^+,0)$ on $(\mathbb{R}^{b,2})^2$ , we define the polynomials $\mathcal{P}_{g_{{{K}}},h_1,h_2}$ on $g_0\circ g_{{{K}}} (L\otimes\mathbb{R})^2\cong (\mathbb{R}^{b-1,1})^2$ by

(4.2) \begin{equation} \mathcal{P}( g(\boldsymbol{v}))=\sum_{h_1,h_2}(v_1,u_{z^\perp})^{h_1}(v_2,u_{z^\perp})^{h_2} \mathcal{P}_{g_{{{K}}},h_1,h_2}(g_{{{K}}} (\boldsymbol{v})) \end{equation}

for every $\boldsymbol{v}=(v_1,v_2)\in(\mathbb{R}^{b,2})^2$ , where as usual we drop the isometry $g_0$ from the notation, and write $\mathcal{P}( g(\boldsymbol{v}))$ and $\mathcal{P}_{g_{{{K}}},h_1,h_2}(g_{{{K}}} (\boldsymbol{v}))$ in place of, respectively, $\mathcal{P}( g_0\circ g(\boldsymbol{v}))$ and $\mathcal{P}_{g_{{{K}}},h_1,h_2}(g_0\circ g_{{{K}}} (\boldsymbol{v}))$ . Furthermore, we define

\begin{align*} \mathcal{P}_{g_{{{K}}}, h}(g_{{{K}}} (\boldsymbol{v}), y_2 / y_1) := \sum_{h_1} \bigg({-}\frac{y_2}{y_1}\bigg)^{\!\!h_1} \mathcal{P}_{g_{{{K}}}, h_1, h - h_1}(g_{{{K}}} (\boldsymbol{v})). \end{align*}

Although $\mathcal{P}$ is very homogeneous, the auxiliary polynomials $\mathcal{P}_{g_{{{K}}},h_1,h_2}$ are not necessarily very homogeneous. This will be shown in the case of $\mathcal{P}=\mathcal{P}_{\boldsymbol{\alpha}}$ in Lemma A.4.

Lemma 4.4. We have

\begin{align*} \mathcal{P}_{g_{{{K}}}, h}(g_{{{K}}} (\boldsymbol{v}), y_2 / y_1) = \mathcal{P}_{g_{{{K}}}, 0, h}(g_{{{K}}} (v_1 + y_2v_2/y_1 , v_2)). \end{align*}

Proof. Let $a, c \in \mathbb{R}$ . Then we have

\begin{align*} & a^{m^+} \sum_{h_1, h_2} (v_1,u_{z^\perp})^{h_1}(v_2,u_{z^\perp})^{h_2} \mathcal{P}_{g_{{{K}}}, h_1, h_2}(g_{{{K}}} (\boldsymbol{v})) = a^{m^+} \mathcal{P}( g(\boldsymbol{v})) = \mathcal{P}(g(a v_1 + c v_2, v_2)) \\ &\quad= \sum_{h_1, h_2} (a v_1 + c v_2,u_{z^\perp})^{h_1}(v_2,u_{z^\perp})^{h_2} \mathcal{P}_{g_{{{K}}}, h_1, h_2}( g_{{{K}}}(a v_1 + c v_2, v_2)) \\ &\quad= \sum_{h_1, h_2} \sum_{j} \binom{h_1}{j} a^{j} (v_1, u_{z^\perp})^{j} c^{h_1 - j} (v_2, u_{z^\perp})^{h_1 + h_2 - j} \mathcal{P}_{g_{{{K}}}, h_1, h_2}( g_{{{K}}}(a v_1 + c v_2, v_2)) \\ &\quad= \sum_{h_1, h_2} \sum_{j} \binom{j}{h_1} a^{h_1} (v_1, u_{z^\perp})^{h_1} c^{j - h_1} (v_2, u_{z^\perp})^{j + h_2 - h_1} \mathcal{P}_{g_{{{K}}}, j, h_2}(g_{{{K}}}(a v_1 + c v_2, v_2)) \\ &\quad= \sum_{h_1, h_2} (v_1, u_{z^\perp})^{h_1} (v_2, u_{z^\perp})^{h_2} a^{h_1} \sum_{j} \binom{j}{h_1} c^{j - h_1} \mathcal{P}_{g_{{{K}}}, j, h_1 + h_2 - j}(g_{{{K}}}(a v_1 + c v_2, v_2)). \end{align*}

Hence, we have

\begin{align*} a^{m^+} \mathcal{P}_{g_{{{K}}}, h_1, h_2}(g_{{{K}}}(v)) = a^{h_1} \sum_{j} \binom{j}{h_1} c^{j - h_1} \mathcal{P}_{g_{{{K}}}, j, h_1 + h_2 - j}(g_{{{K}}}(a v_1 + c v_2, v_2)). \end{align*}

Apply this with $a = 1, c = {y_2}/{y_1}$ to obtain

\begin{align*} &\mathcal{P}_{g_{{{K}}}, h}\bigg(g_{{{K}}}(v_1, v_2), \frac{y_2 }{ y_1}\bigg) = \sum_{h_1} \bigg({-}\frac{y_2}{y_1}\bigg)^{\!\!h_1} \mathcal{P}_{g_{{{K}}}, h_1, h - h_1}(g_{{{K}}}(v_1, v_2)) \\ &\quad= \sum_{h_1} \bigg({-}\frac{y_2}{y_1}\bigg)^{\!\!h_1} \sum_{j} \binom{j}{h_1} \bigg(\frac{y_2}{y_1}\bigg)^{\!\!j - h_1} \mathcal{P}_{g_{{{K}}}, j, h - j}\bigg(g_{{{K}}}\bigg(v_1 + \frac{y_2}{y_1} v_2, v_2\bigg)\bigg) \\ &\quad= \sum_{j} \sum_{h_1} \binom{j}{h_1} \bigg({-}\frac{y_2}{y_1}\bigg)^{\!\!h_1} \bigg(\frac{y_2}{y_1}\bigg)^{\!\!j - h_1} \mathcal{P}_{g_{{{K}}}, j, h - j}\bigg(g_{{{K}}}\bigg(v_1 + \frac{y_2}{y_1} v_2, v_2\bigg)\bigg) \\ &\quad= \mathcal{P}_{g_{{{K}}}, 0, h}\bigg(g_{{{K}}}\bigg(v_1 + \frac{y_2}{y_1} v_2, v_2\bigg)\bigg). \end{align*}

In what follows, we will be interested in the auxiliary polynomials $\mathcal{P}_{\boldsymbol{\alpha},g_{{{K}}},h_1,h_2}$ arising as in Definition 4.3 with $\mathcal{P}=\mathcal{P}_{\boldsymbol{\alpha}}$ . Since the related proofs are rather technical, we collect in § A.2 the explicit formulas of these polynomials, together with some of their properties. We remark here only that $\mathcal{P}_{\boldsymbol{\alpha},g_{{{K}}},h_1,h_2}$ are in general not very homogeneous; see Lemma A.4 for details.

4.2 Reduction of the Siegel theta series ${\Theta_{L,2}}$ to smaller lattices

In this section, we explain how to rewrite the theta function $\Theta_{L,2}$ , introduced in § 3.4, with respect to theta functions associated to the Lorentzian sublattice K. Recall that we assume L splits off a hyperbolic plane and that we chose u, u’ and a basis of $L\otimes\mathbb{R}$ as in (4.1).

Lemma 4.5. Let $\mathcal{P}$ be a very homogeneous polynomial of degree $(m^+,0)$ on $(\mathbb{R}^{b,2})^2$ , and let $\sigma\in D_{L}^2$ . We have

(4.3) \begin{align} &\Theta_{L,2}^{\sigma}(\tau,g,\mathcal{P})\nonumber \\ &\quad=\frac{\sqrt{\det y}}{2 u_{z^\perp}^2}\sum_{\boldsymbol{\lambda}\in \sigma + (L/\mathbb{Z}u)^2}\sum_{n\in\mathbb{Z}^{1\times 2}} \sum_{h_1,h_2}\frac{[(n+(u,\boldsymbol{\lambda})\bar{\tau})y^{-1}]_1^{h_1}[(n+(u,\boldsymbol{\lambda})\bar{\tau})y^{-1}]_2^{h_2}}{({-}2i)^{h_1+h_2}}\nonumber \\ &\qquad\times\exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)(\mathcal{P}_{g_{{{K}}},h_1,h_2})(g_{{{K}}}(\boldsymbol{\lambda})) \cdot e( q(\boldsymbol{\lambda}_{w^\perp})\tau+ q(\boldsymbol{\lambda}_w)\bar{\tau})\nonumber \\ &\qquad\times \exp\bigg({-}\frac{\pi}{2 u_{z^\perp}^2}\mathrm{tr} (n+(u,\boldsymbol{\lambda})\tau)^t(n+(u,\boldsymbol{\lambda})\bar{\tau})y^{-1}-\frac{\pi i}{u_{z^\perp}^2}\mathrm{tr}((\boldsymbol{\lambda},u_{z^\perp}-u_z)n)\bigg), \end{align}

where we denote by $[\,\cdot\,]_j$ the extraction of the jth entry of the given tuple.

We may rewrite any element of $L^2+\sigma$ as $\boldsymbol{\lambda}+nu$ , for some $\boldsymbol{\lambda}\in \sigma + (L/\mathbb{Z}u)^2$ and some row-vector $n\in\mathbb{Z}^{1\times 2}$ . To simplify the notation, we write $q(\boldsymbol{\lambda}+nu)_z$ instead of $q((\boldsymbol{\lambda}+nu)_z)$ , and the same for $z^\perp$ in place of z. We define the auxiliary function $f(\boldsymbol{\lambda},g;n)$ as

(4.4) \begin{align} f(\boldsymbol{\lambda},g;n) &= \exp \bigg( {-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1}) \bigg)(\mathcal{P})(g(\boldsymbol{\lambda}+nu))\nonumber \\ &\quad\times e( q(\boldsymbol{\lambda}+nu)_{z^\perp}\tau + q(\boldsymbol{\lambda}+nu)_z\bar{\tau} ), \end{align}

for every $\boldsymbol{\lambda}\in \sigma + (L/\mathbb{Z}u)^2$ , $g\in G$ , and $n\in\mathbb{R}^{1\times 2}$ , where $z=g^{-1}(z_0)$ . We may then rewrite $\Theta_{L,2}^\sigma$ using the Poisson summation formula as

(4.5) \begin{equation} \Theta_{L,2}^\sigma(\tau,g,\mathcal{P})=\sqrt{\det y}\sum_{\boldsymbol{\lambda}\in \sigma + (L/\mathbb{Z}u)^2}\sum_{n\in\mathbb{Z}^{1\times 2}}f(\boldsymbol{\lambda},g;n)=\sqrt{\det y}\sum_{\boldsymbol{\lambda}\in \sigma + (L/\mathbb{Z} u)^2}\sum_{n\in\mathbb{Z}^{1\times 2}}\widehat{f}(\boldsymbol{\lambda},g;n), \end{equation}

where $\widehat{f}(\boldsymbol{\lambda},g;n)$ is the Fourier transform of f with respect to the vector n.

Let $\boldsymbol{\lambda}=(\lambda_1,\lambda_2)\in \sigma + (L/\mathbb{Z}u)^2$ , $n=(n_1,n_2)\in\mathbb{R}^{1\times 2}$ , and $\tau=(\begin{smallmatrix} \tau_1 & \tau_2\\ \tau_2 & \tau_3 \end{smallmatrix})\in\mathbb{H}_2$ , with analogous notation for the real part x and the imaginary part y of $\tau$ . It is easy to see that

\begin{equation*} q(\boldsymbol{\lambda}+nu)_z=q(\boldsymbol{\lambda}_z)+({\boldsymbol{\lambda}_z},nu_z)+q(nu_z)\quad\text{and}\quad q(\boldsymbol{\lambda}_z)=q(\boldsymbol{\lambda}_w)+(\boldsymbol{\lambda},u_z)(\boldsymbol{\lambda},u_z)^t/2u_z^2, \end{equation*}

the same with $z^\perp$ in place of z. We use such relations to rewrite $f(\boldsymbol{\lambda},g;n)$ as

(4.6) \begin{align} f(\boldsymbol{\lambda},g;n) &= \exp \bigg( {-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1}) \bigg)(\mathcal{P})(g(\boldsymbol{\lambda}+nu)) \cdot e( q(\boldsymbol{\lambda}_{z^\perp})\tau+g(q(\boldsymbol{\lambda}_z)\overline{\tau} )\nonumber \\ &\quad\times e( (\boldsymbol{\lambda}_{z^\perp},nu_{z^\perp})\tau+q(nu_{z^\perp})\tau+(\boldsymbol{\lambda}_z,nu_z)\overline{\tau}+q(nu_z)\overline{\tau} ). \end{align}

The second factor on the right-hand side of (4.6) may be computed as

\begin{align*} e( q(\boldsymbol{\lambda}_{z^\perp})\tau+q(\boldsymbol{\lambda}_z)\overline{\tau} ) &= e( q(\boldsymbol{\lambda}_{w^\perp})\tau + q(\boldsymbol{\lambda}_w)\overline{\tau} ) \\ &\quad\times e\bigg( \frac{(\boldsymbol{\lambda},u_{z^\perp})(\boldsymbol{\lambda},u_{z^\perp})^t\tau}{2u_{z^\perp}^2} + \frac{(\boldsymbol{\lambda},u_z)(\boldsymbol{\lambda},u_z)^t\overline{\tau}}{2u_z^2} \bigg). \end{align*}

Let $h(\boldsymbol{\lambda},g;n)$ be the auxiliary function defined as the product between the first and the last factor on the right-hand side of (4.6), that is, the part of $f(\boldsymbol{\lambda},g;n)$ which depends on the entries $n_1$ and $n_2$ of n. Using the relation $q(u_{z^\perp})+q(u_z)=0$ and $n^t\cdot n=(\begin{smallmatrix} n_1^2 & n_1n_2\\ n_1n_2 & n_2^2 \end{smallmatrix})$ , it is easy to see that

(4.7) \begin{align} h(\boldsymbol{\lambda},g;n) &= \exp \bigg( {-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1}) \bigg)(\mathcal{P})(g(\boldsymbol{\lambda}+nu)) \nonumber\\ &\quad \times e( [x(\boldsymbol{\lambda},u) + i y(\boldsymbol{\lambda},u_{z^\perp}-u_z)]n+iu_{z^\perp}^2yn^t n ). \end{align}

We then rewrite (4.5) as

(4.8) \begin{align} \Theta_{L,2}^\sigma(\tau,g,\mathcal{P}) &= \sqrt{\det y} \sum_{\boldsymbol{\lambda}\in \sigma + (L/\mathbb{Z}u)^2} e\bigg( \frac{(\boldsymbol{\lambda},u_{z^\perp})(\boldsymbol{\lambda},u_{z^\perp})^t\tau}{2u_{z^\perp}^2} - \frac{(\boldsymbol{\lambda},u_z)(\boldsymbol{\lambda},u_z)^t\overline{\tau}}{2u_{z^\perp}^2} \bigg)\nonumber \\ &\quad\times e( q(\boldsymbol{\lambda}_{w^\perp})\tau + q(\boldsymbol{\lambda}_w)\overline{\tau} ) \sum_{n\in\mathbb{Z}^{1\times 2}}\widehat{h}(\boldsymbol{\lambda},g;n). \end{align}

The remaining part of the proof is devoted to the computation of $\widehat{h}(\boldsymbol{\lambda},g;n)$ . To simplify the notation, we define

(4.9) \begin{align} A=iu_{z^\perp}^2y\quad\text{and}\quad B=x(\boldsymbol{\lambda},u) + i y(\boldsymbol{\lambda},u_{z^\perp}-u_z)=\tau (\boldsymbol{\lambda},u_{z^\perp})+\bar{\tau}(\boldsymbol{\lambda},u_z), \end{align}

so that we may rewrite h as

(4.10) \begin{align} h(\boldsymbol{\lambda},g;n)= \exp \bigg( {-}\frac{1}{8\pi}\mathrm{tr} (\Delta y^{-1}) \bigg)(\mathcal{P})(g(\boldsymbol{\lambda}+un)) \cdot e( An^tn+ Bn ). \end{align}

We want to make the dependence of $\exp ( -{1}/({8\pi}) \mathrm{tr} (\Delta y^{-1}) )(\mathcal{P})(g(\boldsymbol{\lambda}+un))$ with respect to the variables $n_1$ and $n_2$ explicit. Recall that we split the polynomial $\mathcal{P}$ as

(4.11) \begin{equation} \mathcal{P}(g(\boldsymbol{v}))=\sum_{h_1,h_2}(v_1,u_{z^\perp})^{h_1}\cdot (v_2,u_{z^\perp})^{h_2}\cdot \mathcal{P}_{g_{{{K}}},h_1,h_2}(g_{{{K}}}(\boldsymbol{v})), \end{equation}

and that the operator $-\mathrm{tr}(\Delta y^{-1})/8\pi$ may be rewritten as

(4.12) \begin{equation} -\frac{\mathrm{tr}(\Delta y^{-1})}{8\pi} = -\frac{1}{8\pi \det y}\bigg( y_3\sum_{j=1}^{b+2}\frac{\partial ^2}{\partial x_{j,1}^2}-2y_2\sum_{j=1}^{b+2}\frac{\partial}{\partial x_{j,1}}\frac{\partial}{\partial x_{j,2}} + y_1\sum_{j=1}^{b+2}\frac{\partial^2}{\partial x_{j,2}^2} \bigg). \end{equation}

Since the three factors appearing as the summand on the right-hand side of (4.11) are defined on linearly independent subspaces of $(\mathbb{R}^{b,2})^2$ , we deduceFootnote ¹ that

(4.13) \begin{align}& \exp\bigg(-\frac{1}{8\pi}\mathrm{tr} (\Delta y^{-1})\bigg)(\mathcal{P})(g(\boldsymbol{\lambda}+nu))\nonumber \\ &\quad= \sum_{h_1,h_2}\exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)(\mathcal{P}_{g_{{{K}}},h_1,h_2})(g_{{{K}}} (\boldsymbol{\lambda})) \nonumber\\ &\qquad\times\exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)( (\lambda_1+n_1u,u_{z^\perp})^{h_1}(\lambda_2+n_2u,u_{z^\perp})^{h_2} ). \end{align}

By (4.12), we may rewrite the summands on the right-hand side of (4.13) as

\begin{align*}& \exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)(\mathcal{P}_{g_{{{K}}},h_1,h_2})(g_{{{K}}} (\boldsymbol{\lambda})) \\ &\quad\times \exp\bigg({-}\frac{1}{8\pi u_{z^\perp}^2 \det y}\bigg( y_{2,2}\frac{\partial ^2}{\partial n_1^2}-2y_{1,2}\frac{\partial}{\partial n_1}\frac{\partial}{\partial n_2} + y_{1,1}\frac{\partial^2}{\partial n_2^2} \bigg)\bigg) \\ & \quad\times ( (\lambda_1+n_1u,u_{z^\perp})^{h_1}(\lambda_2+n_2u,u_{z^\perp})^{h_2} ). \end{align*}

We may then rewrite h as

\begin{align*} h(\boldsymbol{\lambda},g,n) &= \sum_{h_1,h_2} \exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)(\mathcal{P}_{g_{{{K}}},h_1,h_2})(g_{{{K}}} (\boldsymbol{\lambda}))\cdot e( An^tn+Bn ) \\ &\quad\times \exp\bigg({-}\frac{1}{8\pi u_{z^\perp}^2 \det y}\bigg( y_{2,2}\frac{\partial ^2}{\partial n_1^2}-2y_{1,2}\frac{\partial}{\partial n_1}\frac{\partial}{\partial n_2} + y_{1,1}\frac{\partial^2}{\partial n_2^2} \bigg)\bigg) \\ &\quad\times( (\lambda_1+n_1u,u_{z^\perp})^{h_1}(\lambda_2+n_2u,u_{z^\perp})^{h_2} ). \end{align*}

We compute the Fourier transform of h, as a function of n, via Lemma A.2(iii). In fact, if we denote by $N_j$ the jth entry of $({-}n-B^t)A^{-1}/2$ , we may compute

\begin{align*} \widehat{h}(\boldsymbol{\lambda},g,n) &= \det({-}2iA)^{-1/2}\sum_{h_1,h_2}\exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)(\mathcal{P}_{g_{{{K}}},h_1,h_2})(g_{{{K}}}(\boldsymbol{\lambda}))\nonumber \\ &\quad\times \exp\bigg(\frac{1}{8\pi u_{z^\perp}^2\det y}\bigg( y_{2,2}\frac{\partial ^2}{\partial N_1^2}-2y_{1,2}\frac{\partial}{\partial N_1}\frac{\partial}{\partial N_2} + y_{1,1}\frac{\partial^2}{\partial N_2^2} \bigg)\bigg)\nonumber \\ &\quad\times \exp\bigg({-}\frac{1}{8\pi u_{z^\perp}^2\det y}\bigg( y_{2,2}\frac{\partial ^2}{\partial N_1^2}-2y_{1,2}\frac{\partial}{\partial N_1}\frac{\partial}{\partial N_2} + y_{1,1}\frac{\partial^2}{\partial N_2^2} \bigg)\bigg) \nonumber\\ &\quad\times ( (\lambda_1+N_1 u,u_{z^\perp})^{h_1}(\lambda_2+N_2 u, u_{z^\perp})^{h_2} )\nonumber \\ &\quad\times e\bigg({-}\frac{1}{4}nn^tA^{-1} -\frac{1}{2} BnA^{-1} -\frac{1}{4} BB^tA^{-1} \bigg) \nonumber\end{align*}

(4.14) \begin{align} &=\det({-}2iA)^{-1/2} \sum_{h_1,h_2}\exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)(\mathcal{P}_{g_{{{K}}},h_1,h_2})(g_{{{K}}}(\boldsymbol{\lambda}))\nonumber \\ &\quad\times (\lambda_1+N_1 u,u_{z^\perp})^{h_1} (\lambda_2+N_2 u, u_{z^\perp})^{h_2} \nonumber \\ &\quad\times e\bigg( {-}\frac{1}{4} nn^tA^{-1} -\frac{1}{2} BnA^{-1} -\frac{1}{4} BB^tA^{-1} \bigg). \end{align}

We may compute $(\lambda_j+N_j u, u_{z^\perp})$ , for $j=1,2$ , as the jth entry of the vector

\begin{align*} (u_{z^\perp},\boldsymbol{\lambda}) + \bigg(\begin{matrix}N_1\\ N_2\end{matrix}\bigg)u_{z^\perp}^2 &= (u_{z^\perp},\boldsymbol{\lambda}) -\frac{1}{2i}(n+(u_{z^\perp},\boldsymbol{\lambda})\tau+(u_z,\boldsymbol{\lambda})\bar{\tau})y^{-1} \\ &=-\frac{1}{2i}( n+(u,\boldsymbol{\lambda})x-i(u,\boldsymbol{\lambda})y)y^{-1})= -\frac{1}{2i}(n+(u,\boldsymbol{\lambda})\bar{\tau})y^{-1}. \end{align*}

We replace the values of A and B in (4.14) using that ${\det A=-u_{z^\perp}^4\det y}$ , and then replace $\widehat{h}(\boldsymbol{\lambda},g;n)$ in (4.8). The resulting formula may be further simplified, rewriting

\begin{align*}& \mathrm{tr}( (n^t ny^{-1})+2 (\boldsymbol{\lambda},u)nxy^{-1} + (\boldsymbol{\lambda},u)(\boldsymbol{\lambda},u)^tx^2y^{-1} + (\boldsymbol{\lambda},u)(\boldsymbol{\lambda},u)^ty ) \\&\quad = \mathrm{tr} (n+(u,\boldsymbol{\lambda})\tau)^t (n+(u,\boldsymbol{\lambda})\bar{\tau})y^{-1}, \end{align*}

to eventually obtain (4.3).

Theorem 4.6. Let $\mu\in({{K}}\otimes\mathbb{R})\oplus\mathbb{R} u$ be the vector

(4.15) \begin{equation} \mu=-u'+u_{z^\perp}/2u_{z^\perp}^2 + u_z/2u_z^2. \end{equation}

The theta function $\Theta_{L,2}(\tau,g,\mathcal{P})$ satisfies

(4.16) \begin{align} \Theta_{L,2}(\tau,g,\mathcal{P}) &= \frac{1}{2 u_{z^\perp}^2} \sum_{c,d\in\mathbb{Z}^{1\times 2}} \sum_{h_1,h_2} \exp\bigg({-}\frac{\pi}{2 u_{z^\perp}^2}\mathrm{tr} (c\tau + d)^t(c\bar{\tau}+d)y^{-1}\bigg) \nonumber\\ &\quad\times \frac{[(c\bar{\tau} + d)y^{-1}]_1^{h_1}[(c\bar{\tau}+d)y^{-1}]_2^{h_2}}{({-}2i)^{h_1+h_2}} \Theta_{{{K}},2}(\tau,\mu d,-\mu c,g_{{{K}}},\mathcal{P}_{g_{{{K}}},h_1,h_2}). \end{align}

In Theorem 4.6, the theta functions $\Theta_{{{K}},2}$ are attached to a lattice M of signature $(b - 1, 1)$ and some polynomials $\mathcal{P}_{g_{{{K}}},h_1,h_2}$ , which may be non-very homogeneous. Such theta functions are constructed as in Definition 3.8. They are absolutely convergent, as illustrated in [Reference RoehrigRoe21, p. 2].

Remark 4.7. The isomorphic projection $p\colon D_{L}^2\to D_{{{K}}}^2$ induces an isomorphism $\mathbb{C}[D_{L}^2]\to\mathbb{C}[D_{{{K}}}^2]$ . The latter is implicitly used in (4.16) to identify values of theta functions on isomorphic group algebras. Moreover, among the entries of $\Theta_{{{K}},2}$ in Theorem 4.6, we should write $\mu_{{{K}}}$ as argument, namely the projection of $\mu$ to ${{K}}\otimes\mathbb{R}$ , instead of $\mu$ . However, since $\mu_{{{K}}}=\mu-(\mu,u')u$ , we have

\begin{align*} \mu_w&=(\mu_{{{K}}})_w=-u'_w,\\ \mu_{w^\perp}&=(\mu_{{{K}}})_{w^\perp}=-u'_{w^\perp},\\ (\mu,u)&=(\mu_{{{K}}},u). \end{align*}

This explains why we may use such an abuse of notation. Note also that the orthogonal projection $L\otimes\mathbb{R}\to{{K}}\otimes\mathbb{R}$ induces an isometric isomorphism $w^\perp\oplus w\to w_{{{K}}}^\perp\oplus w_{{{K}}}={{K}}\otimes\mathbb{R}$ . This implies that we may identify w with $w_{{K}}$ and consider w as an element of $\mathrm{Gr}({{K}})$ ; see [Reference BruinierBru02, p. 42]. Analogously, we may regard $g_{{{K}}}|_{{{K}}\otimes\mathbb{R}}$ as an element of $\mathrm{SO}({{K}}\otimes\mathbb{R})$ .

Proof of Theorem 4.6. Recall that $p\colon D_{L}^2\to D_{{{K}}}^2$ is the isomorphism induced by the orthogonal projection $L\to{{K}}$ . Every $\boldsymbol{\lambda}\in\sigma + (L/\mathbb{Z} u)^2$ can be rewritten as ${\boldsymbol{\lambda}=\boldsymbol{\lambda}_{{K}}+cu'}$ in a unique way, where $\boldsymbol{\lambda}_{{K}}\in p(\sigma) + {{K}}^2$ and $c\in\mathbb{Z}^{1\times 2}$ . Since u and u’ are orthogonal to K, we may rewrite the formula provided by Lemma 4.5 as

\begin{align*} \Theta_{L,2}^{\sigma}(\tau,g, \mathcal{P}) &=\frac{\sqrt{\det y}}{2 u_{z^\perp}^2} \sum_{c,d\in\mathbb{Z}^{1\times 2}} \sum_{h_1,h_2} \sum_{\boldsymbol{\lambda}_{{K}}\in p(\sigma)+{{K}}^2} \exp\bigg({-}\frac{\pi}{2 u_{z^\perp}^2}\mathrm{tr} (c\tau + d)^t(c\bar{\tau}+d)y^{-1}\bigg) \\ &\quad\times \frac{[(c\bar{\tau} + d)y^{-1}]_1^{h_1}[(c\bar{\tau}+d)y^{-1}]_2^{h_2}}{({-}2i)^{h_1+h_2}} \cdot \exp\bigg( {-}\frac{\pi i}{u_{z^\perp}^2}\mathrm{tr}((\boldsymbol{\lambda}_{{K}}+cu',u_{z^\perp}-u_z)d) \bigg) \\ &\quad\times\exp\bigg({-}\frac{1}{8\pi}\mathrm{tr}(\Delta y^{-1})\bigg)(\mathcal{P}_{g_{{{K}}},h_1,h_2})(g_{{{K}}}(\boldsymbol{\lambda}_{{K}}-c\mu)) \\ &\quad\times e( q(\boldsymbol{\lambda}_{{K}}-c\mu)_{w^\perp}\tau+ q(\boldsymbol{\lambda}_{{K}}-c\mu)_w\bar{\tau} ) , \end{align*}

where we denote by $[\,\cdot\,]_j$ the extraction of the jth entry, and we write $q(\boldsymbol{v})_w$ instead of $q((\boldsymbol{v})_w)$ , for every $\boldsymbol{v}\in (L\otimes\mathbb{R})^2$ , the same for $w^\perp$ in place of w. To conclude the proof, it is enough to check that

\begin{align*} \exp\bigg( {-}\frac{\pi i}{u_{z^\perp}^2}\mathrm{tr}((\boldsymbol{\lambda}_{{K}}+cu',u_{z^\perp}-u_z)d) \bigg)=e({-}(\boldsymbol{\lambda}_{{K}}-\mu c/2,\mu d)). \end{align*}

This may be proved as in [Reference BorcherdsBor98, end of the proof of Theorem 5.2].

The following results illustrate how to rewrite the formula provided by Theorem 4.6 in terms of vectors in $\mathbb{Z}^{1\times2}$ with coprime entries, as well as in terms of the Klingen parabolic subgroup of $\mathrm{Sp}_4(\mathbb{Z})$ .

Corollary 4.8. The vector-valued theta function $\Theta_{L,2}(\tau,g,\mathcal{P})$ satisfies

(4.17) \begin{align} \Theta_{L,2}(\tau,g,\mathcal{P})&= \frac{1}{2 u_{z^\perp}^2}\Theta_{{{K}},2}(\tau,g_{{{K}}},\mathcal{P}_{g_{{{K}}},0,0})\nonumber \\ &\quad + \frac{1}{2 u_{z^\perp}^2} \sum_{\substack{c,d\in\mathbb{Z}^{1\times 2}\\ \gcd(c,d)=1}} \sum_{r\ge 1} \sum_{h_1,h_2} \bigg({-}\frac{r}{2i}\bigg)^{\!\!h_1+h_2}[(c\bar{\tau} + d)y^{-1}]_1^{h_1}[(c\bar{\tau}+d)y^{-1}]_2^{h_2}\nonumber \\ &\quad\times \exp\!\bigg(\!{-}\frac{\pi r^2}{2 u_{z^\perp}^2}\mathrm{tr} (c\tau + d)^t(c\bar{\tau}+d)y^{-1}\!\bigg) \Theta_{{{K}},2}(\tau,r\mu d,-r\mu c,g_{{{K}}},\mathcal{P}_{g_{{{K}}},h_1,h_2}). \end{align}

Definition 4.9. The Klingen parabolic subgroup $\mathrm{C}_{2,1}$ is the subgroup of matrices in $\mathrm{Sp}_4(\mathbb{Z})$ whose last row equals $(\begin{matrix} 0 & 0 & 0 & 1 \end{matrix})$ , namely,

\begin{equation*} \mathrm{C}_{2,1}=\bigg\{ \bigg(\begin{matrix} * & *\\ 0_{1,3} & {1} \end{matrix}\bigg)\in\mathrm{Sp}_4(\mathbb{Z}) \bigg\}. \end{equation*}

Corollary 4.10. The vector-valued theta function $\Theta_{L,2}(\tau,g,\mathcal{P})$ satisfies

(4.18) \begin{align} \Theta_{L,2}(\tau,g,\mathcal{P})& = \frac{1}{2 u_{z^\perp}^2}\Theta_{{{K}},2}(\tau,g_{{{K}}},\mathcal{P}_{g_{{{K}}},0,0})\nonumber \\ &\quad+ \frac{1}{2 u_{z^\perp}^2} \sum_{(\begin{smallmatrix}* & *\\ c & d\end{smallmatrix})\in\mathrm{C}_{2,1}\backslash\mathrm{Sp}_4(\mathbb{Z})} \sum_{r\ge 1} \sum_{h_1,h_2} \bigg({-}\frac{r}{2i}\bigg)^{\!\!h_1+h_2}[(c\bar{\tau} + d)y^{-1}]_1^{h_1}[(c\bar{\tau}+d)y^{-1}]_2^{h_2}\nonumber \\ &\quad\times \exp\!\bigg(\!{-}\frac{\pi r^2}{2 u_{z^\perp}^2}\mathrm{tr} (c\tau + d)^t(c\bar{\tau}+d)y^{-1}\!\bigg) \Theta_{{{K}},2}(\tau,r\mu d,-r\mu c,g_{{{K}}},\mathcal{P}_{g_{{{K}}},h_1,h_2}), \end{align}

where $(c\,\, d)$ is the last row of $(\begin{smallmatrix}* & *\\[1pt] c & d\end{smallmatrix})\in\mathrm{C}_{2,1}\backslash\mathrm{Sp}_4(\mathbb{Z})$ .

5.1 Reduction to smaller lattices

In this section, we will derive a formula for the Jacobi theta function as a Poincaré series. Therefore, let $u \in L$ be primitive isotropic of level $N_u$ , i.e. $(u, L) = N_u \mathbb{Z}$ , and $u' \in L'$ with $(u, u') = 1$ . We denote by K the orthogonal complement of the span of u and u’.

The idea is the same as in [Reference BorcherdsBor98, § 5]. We first make a partial Poisson summation along $\mathbb{Z} u$ . The sum over the primitive elements of the sublattice $\mathbb{Z} u \oplus \mathbb{Z} u'$ can be identified with a sum over the cosets $\Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})$ , where $\Gamma_\infty$ is the standard parabolic in $\mathrm{SL}_2(\mathbb{Z})$ , which leads to a Poincaré series. We start by calculating the partial Fourier transform along the sublattice $\mathbb{Z} u$ .

Let $\mathcal{P}$ be a homogeneous polynomial of degree m. We define the polynomials $\mathcal{P}_{{g_{{{K}}}}, h}$ of degree $m- h$ , for every $h \in \mathbb{N}$ , by

\begin{align*}\mathcal{P}(g(\lambda)) = \sum_{h} (\lambda, u_{z^\perp})^{h} \mathcal{P}_{{g_{{{K}}}}, h}({g_{{{K}}}}(\lambda)).\end{align*}

Here, ${g_{{{K}}}}\colon L \otimes \mathbb{R} \to L \otimes \mathbb{R}$ is the linear map defined as $\lambda \mapsto g(\lambda_{{w^\perp}} + \lambda_{w})$ , where ${w^\perp}$ and w are the orthogonal complements of $u_{z^\perp}$ and $u_{z}$ in, respectively, $z^\perp$ and z. This is the analogue of its homonym in § 4.1.

Let

(5.1) \begin{equation}\mu = - u' + \frac{u_{z^\perp}}{2 u_{z^\perp}^2} + \frac{u_{z}}{2u_{z}^2}.\end{equation}

The next lemma calculates the partial Fourier transform along $\mathbb{Z} u$ .

Lemma 5.1. Let $\sigma \in L'$ . For $\lambda \in \sigma + L / u$ and $n \in \mathbb{Z}$ , let $g(\lambda, n)$ be defined as

\begin{align*} & e(q(\lambda_{z^\perp}) \tau_1 + q(\lambda_{z}) \overline{\tau_1} + \tau_2 (\lambda , \eta_{z^\perp}) + \overline{\tau_2} (\lambda, \eta_{z})) \cdot \exp({-}\Delta / 8 \pi y_1)(\mathcal{P})\bigg(g\bigg(\lambda + \frac{y_2}{y_1} \eta + n u\bigg)\bigg) \\ &\quad\times e(n^2 (\tau_1 q(u_{z^\perp}) + \overline{\tau_1} q(u_{z})) + n (\tau_1 (\lambda, u_{z^\perp}) + \overline{\tau_1} (\lambda, u_{z}) + \tau_2 (\eta, u_{z^\perp}) + \overline{\tau_2} (\eta, u_{z}))). \end{align*}

Then, for $\lambda \in K + \sigma - c u'$ and $c \in \mathbb{Z}$ with $c \equiv (\sigma, u) \pmod{N_u}$ , we have

\begin{align*} \hat{g}(\lambda + c u', n)&= \frac{1}{\sqrt{2 y_1 u_{z^\perp}^2}} \sum_{h} \frac{(c \overline{\tau_1} + n)^{h}}{({-}2 i y_1)^{h}} e\bigg({-}i\frac{(y_2 (\eta, u_{z^\perp}))^2}{y_1 u_{z^\perp}^2} - \frac{n y_2 (\eta, u_{z^\perp})}{y_1 u_{z^\perp}^2} \bigg) \\ &\quad \times \exp({-} \Delta / 8 \pi y_1)(\mathcal{P}_{{g_{{{K}}}}, h})\bigg({g_{{{K}}}}\bigg(\lambda + \frac{y_2}{y_1} \eta\bigg)\bigg) e({-} n (\sigma, u') + cn q(u')) \\ &\quad\times e\bigg({-}\frac{\lvert n + c \tau_1 \rvert^2 + 2 c (\tau_1 \overline{\tau_2} (\eta, u_{z^\perp}) + \overline{\tau_1} \tau_2 (\eta, u_{z}))}{4 i y_1 u_{z^\perp}^2}\bigg) \cdot e(\tau_1 q((\lambda - c \mu)_{{w^\perp}}) \\ &\quad+ \overline{\tau_1} q((\lambda - c \mu)_{w}) + \tau_2 (\lambda - c \mu, \eta_{{w^\perp}}) + \overline{\tau_2} (\lambda - c \mu, \eta_{w}) - (\lambda - c \mu / 2, n \mu)). \end{align*}

Proof. Let

\begin{align*} A &= \tau_1 q(u_{z^\perp}) + \overline{\tau_1} q(u_{z}), \\ B &= \tau_1 (\lambda, u_{z^\perp}) + \overline{\tau_1} (\lambda, u_{z}) + \tau_2 (\eta, u_{z^\perp}) + \overline{\tau_2} (\eta, u_{z}). \end{align*}

We remark that $(\eta, u) = 0$ implies that $B = \tau_1 (\lambda, u_{z^\perp}) + \overline{\tau_1} (\lambda, u_{z}) + 2iy_2 (\eta, u_{z^\perp})$ . Using

\begin{align*} \exp({-} \Delta / 8 \pi y_1)(\mathcal{P})(g(\lambda + n u)) &= \sum_{h} \exp\bigg({-} \frac{1}{8 \pi y_1 u_{z^\perp}^2} \frac{d^2}{dn^2}\bigg)((\lambda + n u, u_{z^\perp})^{h}) \\ &\quad\times \exp({-} \Delta / 8 \pi y_1)(\mathcal{P}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda)) \end{align*}

and [Reference BorcherdsBor98, Corollary 3.3], we see that the Fourier transform of g with respect to n is given by

(5.2) \begin{align} & \frac{1}{\sqrt{2 y_1 u_{z^\perp}^2}} \sum_{h} \exp({-} \Delta / 8 \pi y_1)(\mathcal{P}_{{g_{{{K}}}}, h}) \bigg({g_{{{K}}}}\bigg(\lambda + \frac{y_2}{y_1} \eta\bigg)\bigg) \nonumber\\ &\quad\times \exp\bigg(\frac{1}{8 \pi y_1 u_{z^\perp}^2} \frac{d^2}{d N^2}\bigg) \bigg(\exp\bigg({-} \frac{1}{8 \pi y_1 u_{z^\perp}^2} \frac{d^2}{dN^2}\bigg) \bigg(\bigg(\lambda + \frac{y_2}{y_1} \eta + N u, u_{z^\perp}\bigg)^{\!\!h}\bigg)\bigg) \nonumber\\ &\quad\times e\bigg({-} \frac{(n + \tau_1(\lambda, u_{z^\perp}) + \overline{\tau_1} (\lambda, u_{z}) + 2 i y_2 (\eta, u_{z^\perp}))^2}{4 i y_1 u_{z^\perp}^2}\bigg) \nonumber\\ &\quad\times e(q(\lambda_{z^\perp}) \tau_1 + q(\lambda_{z}) \overline{\tau_1} + \tau_2 (\lambda , \eta_{z^\perp}) + \overline{\tau_2} (\lambda, \eta_{z})), \end{align}

where $N = -({n + B})/{2 A}$ . Since

\begin{align*}N = - \frac{n + \tau_1 (\lambda, u_{z^\perp}) + \overline{\tau_1} (\lambda, u_{z}) + 2i y_2 (\eta, u_{z^\perp})}{2 i y_1 u_{z^\perp}^2},\end{align*}

we deduce that

\begin{align*} & \exp\bigg(\frac{1}{8 \pi y_1 u_{z^\perp}^2} \frac{d^2}{d N^2}\bigg) \bigg(\exp\bigg({-} \frac{1}{8 \pi y_1 u_{z^\perp}^2} \frac{d^2}{dN^2}\bigg)\bigg(\bigg(\lambda + \frac{y_2}{y_1} \eta + N u, u_{z^\perp}\bigg)^{\!\!h}\bigg)\bigg) \\ &\quad = \bigg(\lambda + \frac{y_2}{y_1} \eta - \frac{n + \tau_1 (\lambda, u_{z^\perp}) + \overline{\tau_1} (\lambda, u_{z}) + 2i y_2 (\eta, u_{z^\perp})}{2 i y_1 u_{z^\perp}^2} u, u_{z^\perp}\bigg)^{\!\!h}. \end{align*}

A simple computation shows that this equals $({-}((\lambda, u) \overline{\tau_1} + n)/2i y_1)^{h}$ . On the other hand, the exponential term in (5.2) may be rewritten as

\begin{align*} & e\bigg({-} \frac{(n + \tau_1(\lambda, u_{z^\perp}) + \overline{\tau_1} (\lambda, u_{z}) + 2 i y_2 (\eta, u_{z^\perp}))^2}{4 i y_1 u_{z^\perp}^2}\bigg) \\ &\quad\times e(q(\lambda_{z^\perp}) \tau_1 + q(\lambda_{z}) \overline{\tau_1} + \tau_2 (\lambda , \eta_{z^\perp}) + \overline{\tau_2} (\lambda, \eta_{z})) \\ &= e\bigg({-}i\frac{(y_2 (\eta, u_{z^\perp}))^2}{y_1 u_{z^\perp}^2} - \frac{n y_2 (\eta, u_{z^\perp})}{y_1 u_{z^\perp}^2} \bigg) \\ &\quad\times e(\tau_1 q(\lambda_{{w^\perp}}) + \overline{\tau_1} q(\lambda_{w}) + \tau_2 (\lambda, \eta_{{w^\perp}}) + \overline{\tau_2} (\lambda, \eta_{w})) \\ &\quad\times e\bigg({-}\frac{\lvert n + (\lambda, u) \tau_1 \rvert^2 + 2 (\lambda, u) (\tau_1 \overline{\tau_2} (\eta, u_{z^\perp}) + \overline{\tau_1} \tau_2 (\eta, u_{z}))}{4 i y_1 u_{z^\perp}^2} - \frac{n (\lambda, u_{z^\perp} - u_{z})}{2 u_{z^\perp}^2}\bigg). \end{align*}

Every element in $\sigma + L / u$ can be written as $\lambda + c u'$ for some $\lambda \in {{K}} + \sigma - cu'$ and $c \in \mathbb{Z}$ such that $c \equiv (\sigma, u)$ mod $N_u$ . We may then compute

\begin{align*} \hat{g}(\lambda + c u', n) &= \frac{1}{\sqrt{2 y_1 u_{z^\perp}^2}} \sum_{h} \exp({-} \Delta / 8 \pi y_1)(\mathcal{P}_{{g_{{{K}}}}, h}) \bigg({g_{{{K}}}}\bigg(\lambda + \frac{y_2}{y_1} \eta\bigg)\bigg) \\ &\quad\times \frac{(c \overline{\tau_1} + n)^{h}}{({-}2i y_1)^{h}} e\bigg({-}i\frac{(y_2 (\eta, u_{z^\perp}))^2}{y_1 u_{z^\perp}^2} - \frac{n y_2 (\eta, u_{z^\perp})}{y_1 u_{z^\perp}^2} \bigg) \\ &\quad\times e(\tau_1 q((\lambda + c u')_{{w^\perp}}) + \overline{\tau_1} q((\lambda + c u')_{w}) + \tau_2 (\lambda + c u', \eta_{{w^\perp}}) + \overline{\tau_2} (\lambda + c u', \eta_{w})) \\ &\quad\times e\bigg({-}\frac{\lvert n + c \tau_1 \rvert^2 + 2 c (\tau_1 \overline{\tau_2} (\eta, u_{z^\perp}) + \overline{\tau_1} \tau_2 (\eta, u_{z}))}{4 i y_1 u_{z^\perp}^2} - \frac{n (\lambda + c u', u_{z^\perp} - u_{z})}{2 u_{z^\perp}^2}\bigg) \\ &= \frac{1}{\sqrt{2 y_1 u_{z^\perp}^2}} \sum_{h} \frac{(c \overline{\tau_1} + n)^{h}}{({-}2 i y_1)^{h}} e\bigg({-}i\frac{(y_2 (\eta, u_{z^\perp}))^2}{y_1 u_{z^\perp}^2} - \frac{n y_2 (\eta, u_{z^\perp})}{y_1 u_{z^\perp}^2} \bigg) \\ &\quad \times \exp({-} \Delta / 8 \pi y_1)(\mathcal{P}_{{g_{{{K}}}}, h})\bigg({g_{{{K}}}}\bigg(\lambda + \frac{y_2}{y_1} \eta\bigg)\bigg) e({-} n (\sigma, u') + cn q(u')) \\ &\quad\times e\bigg({-}\frac{\lvert n + c \tau_1 \rvert^2 + 2 c (\tau_1 \overline{\tau_2} (\eta, u_{z^\perp}) + \overline{\tau_1} \tau_2 (\eta, u_{z}))}{4 i y_1 u_{z^\perp}^2}\bigg) \cdot e(\tau_1 q((\lambda - c \mu)_{{w^\perp}}) \\ &\quad+ \overline{\tau_1} q((\lambda - c \mu)_{w}) + \tau_2 (\lambda - c \mu, \eta_{{w^\perp}}) + \overline{\tau_2} (\lambda - c \mu, \eta_{w}) - (\lambda - c \mu / 2, n \mu)), \end{align*}

where we used that $(\lambda, u') = (\lambda + c u' - c u', u') \equiv (\sigma - c u', u')$ mod 1 and that

\begin{align*} \frac{(\lambda + c u', u_{z^\perp} - u_{z})}{2 u_{z^\perp}^2} = (\lambda - c \mu / 2, \mu) + c q(u') + (\lambda, u').\\[-43pt] \end{align*}

We will need the following result.

Lemma 5.2 [Reference KieferKie22, Lemma 6.3]. Let $\sigma \in D_{{K}}, \gamma = ((\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}), \phi) \in \mathrm{Mp}_2(\mathbb{Z})$ and $n \in \mathbb{Z}$ . Then

\begin{align*}\rho_L(\gamma)\sum_{m \in \mathbb{Z} / N_u \mathbb{Z}}\mathfrak{e}_{\sigma + \frac{m u}{N_{u}}}\bigg({-}\frac{mn}{N_{u}}\bigg)=(\rho_{{K}}(\gamma) \mathfrak{e}_{\sigma})\sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{\frac{mu}{N_{u}} - nc u'}\bigg({-}\frac{amn}{N_{u}} + q(u') ac n^2\bigg).\end{align*}

Let $\Gamma_\infty = \{(\begin{smallmatrix}1 & n \\ 0 & 1\end{smallmatrix}) : n \in \mathbb{Z}\}$ be the standard parabolic subgroup in $\mathrm{SL}_2(\mathbb{Z})$ . For every ${\gamma\in \mathrm{SL}_2(\mathbb{Z})}$ , we denote by $\tilde{\gamma}\in \mathrm{Mp}_2(\mathbb{Z})$ the standard preimage of $\gamma$ under the metaplectic double cover. The following result is an analogue of [Reference BorcherdsBor98, Theorem 5.2] for the Jacobi Siegel theta functions introduced in Example 3.11. It provides a way to rewrite $\Theta_{L, \eta}(\tau,g,\mathcal{P})$ as a Poincaré series in terms of the theta functions associated to K.

Theorem 5.3. If $(\eta, u) = 0$ , then $\Theta_{L, \eta}(\tau,g,\mathcal{P})$ may be rewritten as

\begin{align*} &\frac{1}{\sqrt{2 u_{z^\perp}^2}} \Theta_{{{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{\frac{m u}{N_u}} \\ &\quad + \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = (\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} e\bigg({-}\frac{q(\eta) c \tau_2^2}{c \tau_1 + d}\bigg) \\ &\quad\times \frac{n^{h}(c \tau_1 + d)^{ - m + (b^- - b^+)/2}}{({-}2 i \Im(\gamma \tau_1))^{h}} e\bigg({-}\frac{n^2 + 4 i n \Im(\gamma \tau_2) (\eta, u_{z^\perp})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\quad\times \rho_{L, \eta}^{-1}(\tilde{\gamma}) \bigg(\Theta_{{{K}}, \eta}(\gamma\tau_1, \gamma\tau_2, n\mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{\frac{m u}{N_u}}\bigg({-} \frac{mn}{N_u}\bigg) \bigg). \end{align*}

Proof. Let $\sigma \in L'$ . We write the elements in $\sigma + L$ as $\lambda + n u$ for some $\lambda \in \sigma + L / u$ and $n \in \mathbb{Z}$ , so that we may compute

\begin{align*}& \theta_{\sigma + L, \eta} (\tau_1, \tau_2, g, \mathcal{P})\\ &\quad = y_1^{-b^+/2} \exp({-}4 \pi q(\eta_{z}) y_2^2 / y_1) \sum_{\lambda \in \sigma + L / u} \sum_{n \in \mathbb{Z}} \exp({-}\Delta / 8 \pi y_1)(\mathcal{P}) \bigg(g\bigg(\lambda + \frac{y_2}{y_1} \eta + n u\bigg)\bigg) \\ &\qquad\times e(q((\lambda + n u)_{z^\perp}) \tau_1 + q((\lambda + n u)_{z}) \overline{\tau_1} + \tau_2 (\lambda + n u, \eta_{z^\perp}) + \overline{\tau_2} (\lambda + n u, \eta_{z})) \\ &\quad= y_1^{-b^+/2} \exp({-}4 \pi q(\eta_{z}) y_2^2 / y_1) \sum_{\lambda \in \sigma + L / u} \sum_{n \in \mathbb{Z}} g(\lambda, n). \end{align*}

The factor in front of the theta function can be rewritten as

\begin{align*}y_1^{b^-/2} \exp(4 \pi q(\eta_{z}) y_2^2 / y_1) = \sqrt{y_1} \cdot e\bigg({-}i \frac{(\eta, u_{z})^2 y_2^2}{y_1 u_{z}^2}\bigg) y_1^{(b^- - 1)/2} \exp(4 \pi q(\eta_{w}) y_2^2 / y_1).\end{align*}

Applying a partial Poisson summation in n and using Lemma 5.1, we may rewrite the theta function as

\begin{align*}& \theta_{\sigma + L, \eta}(\tau_1, \tau_2, g, \mathcal{P})\\ &\quad= \frac{1}{\sqrt{2 u_{z^\perp}^2}} e\bigg({-}i\frac{(y_2 (\eta, u_{z^\perp}))^2}{y_1 u_{z^\perp}^2} \bigg) \cdot e\bigg({-} i \frac{(\eta, u_{z})^2 y_2^2}{y_1 u_{z}^2}\bigg) \\ &\qquad\times \sum_{\substack{c, d \in \mathbb{Z} \\ c \equiv (\sigma, u) \pmod N}} \sum_{h} \frac{(c \overline{\tau_1} + d)^{h}}{({-}2 i y_1)^{h}} e\bigg({-} \frac{dy_2 (\eta, u_{z^\perp})}{y_1 u_{z^\perp}^2} \bigg) \cdot e({-} d (\sigma, u') + cd q(u')) \\ &\qquad\times e\bigg({-}\frac{\lvert c \tau_1 + d \rvert^2 + 2 c (\tau_1 \overline{\tau_2} (\eta, u_{z^\perp}) + \overline{\tau_1} \tau_2 (\eta, u_{z}))}{4 i y_1 u_{z^\perp}^2}\bigg) \\ &\qquad\times \theta_{\sigma - cu' + {{K}}, \eta}(\tau_1, \tau_2, d \mu, -c \mu, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \\ &\quad= \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\substack{c, d \in \mathbb{Z} \\ c \equiv (\sigma, u) \pmod N}} \sum_{h} \frac{(c \overline{\tau_1} + d)^{h}}{({-}2 i y_1)^{h}} e({-} d (\sigma, u') + cd q(u')) \\ &\qquad\times e\bigg({-}\frac{\lvert c \tau_1 + d \rvert^2 + 2 (c\tau_1 + d) \overline{\tau_2} (\eta, u_{z^\perp}) + 2 (c\overline{\tau_1} + d) \tau_2 (\eta, u_{z})}{4 i y_1 u_{z^\perp}^2}\bigg) \\ &\qquad\times\theta_{\sigma - cu' + {{K}}, \eta}(\tau_1, \tau_2, d \mu, -c \mu, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}). \end{align*}

Summing over $\sigma \in D_L$ and using the correspondence between coprime integers (c, d) and elements in $\Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})$ yields

\begin{align*} &\Theta_{L, \eta}(\tau_1, \tau_2, g, \mathcal{P}) \\ &\quad= \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\substack{\sigma \in D_L \\ (\sigma, u) \equiv 0 \pmod{N_u}}} \theta_{\sigma + {{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \mathfrak{e}_{\sigma} \\ &\qquad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\substack{c, d \in \mathbb{Z} \\ (c, d) = 1}}\; \sum_{\substack{\sigma \in D_L \\ c \equiv (\sigma, u) \pmod{N_u}}} \sum_{n = 1}^\infty \sum_{h} \frac{n^{h}(c \overline{\tau_1} + d)^{h}}{({-}2 i y_1)^{h}} e({-} nd (\sigma, u') + n^2 cd q(u')) \\ &\qquad\times e\bigg({-}\frac{n^2 \lvert c \tau_1 + d \rvert^2 + 2 n (c\tau_1 + d) \overline{\tau_2} (\eta, u_{z^\perp}) + 2n (c\overline{\tau_1} + d) \tau_2 (\eta, u_{z})}{4 i y_1 u_{z^\perp}^2}\bigg) \\ &\qquad\times \theta_{\sigma - ncu' + {{K}}, \eta}(\tau_1, \tau_2, nd \mu, -nc \mu, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \mathfrak{e}_{\sigma} \end{align*}

\begin{align*} &\quad= \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\substack{\sigma \in D_L \\ (\sigma, u) \equiv 0 \pmod{N_u}}} \theta_{\sigma + {{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \mathfrak{e}_{\sigma} \\ &\qquad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = (\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} \frac{n^{h}(c \overline{\tau_1} + d)^{h}}{({-}2 i y_1)^{h}} \\ &\qquad\times e\bigg({-}\frac{n^2 + 2 n \overline{\gamma \tau_2} (\eta, u_{z^\perp}) + 2n \gamma \tau_2 (\eta, u_{z})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\qquad\times \sum_{\substack{\sigma \in D_L \\ c \equiv (\sigma, u) \pmod{N_u}}} \theta_{\sigma - ncu' + {{K}}, \eta}(\tau_1, \tau_2, n d \mu, -n c \mu, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \mathfrak{e}_{\sigma}({-} nd (\sigma, u') + n^2 cd q(u')) \\ &\quad= \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\substack{\sigma \in D_L \\ (\sigma, u) \equiv 0 \pmod{N_u}}} \theta_{\sigma + {{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \mathfrak{e}_{\sigma} \\ &\qquad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = (\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} \frac{n^{h}(c \tau_1 + d)^{-h}}{({-}2 i \Im(\gamma \tau_1))^{h}} e\bigg({-}\frac{n^2 + 4 i n \Im(\gamma \tau_2) (\eta, u_{z^\perp})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\qquad\times \sum_{\substack{\sigma \in D_L \\ c \equiv (\sigma, u) \pmod{N_u}}} \theta_{\sigma - ncu' + {{K}}, \eta}(\tau_1, \tau_2, n d \mu, -n c \mu, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \mathfrak{e}_{\sigma}({-} nd (\sigma, u') + n^2 cd q(u')). \end{align*}

We apply the change of variables $\sigma \mapsto \sigma + ncu'$ and obtain

\begin{align*} &\frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\substack{\sigma \in D_L \\ (\sigma, u) \equiv 0 \pmod{N_u}}} \theta_{\sigma + {{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \mathfrak{e}_{\sigma} \\ &\quad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = \big(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\big) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} \frac{n^{h}(c \tau_1 + d)^{-h}}{({-}2 i \Im(\gamma \tau_1))^{h}} e\bigg({-}\frac{n^2 + 4 i n \Im(\gamma \tau_2) (\eta, u_{z^\perp})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\quad\times \sum_{\substack{\sigma \in D_L \\ (\sigma, u) \equiv 0 \pmod{N_u}}} \theta_{\sigma + {{K}}, \eta}(\tau_1, \tau_2, n d \mu, -n c \mu, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \\ &\quad\times \mathfrak{e}_{\sigma + ncu'}({-} nd (\sigma, u') - n^2 cd q(u')). \end{align*}

The elements $\sigma \in D_L$ with $(\sigma, u) \equiv 0 \pmod{N_u}$ are represented by $\sigma + \frac{m u}{N_u}$ with $\sigma \in D_{{K}}$ and $m \in \mathbb{Z} / N_u \mathbb{Z}$ . We may then rewrite $\Theta_{{{K}}, \eta}$ as

\begin{align*} &\frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\sigma \in D_{{K}}} \theta_{\sigma + {{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \mathfrak{e}_{\sigma} \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}{/}{N_u}} \\ &\quad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = \big(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\big) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} \frac{n^{h}(c \tau_1 + d)^{-h}}{({-}2 i \Im(\gamma \tau_1))^{h}} e\bigg({-}\frac{n^2 + 4 i n \Im(\gamma \tau_2) (\eta, u_{z^\perp})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\quad\times \sum_{\sigma \in D_{{K}}} \theta_{\sigma + {{K}}, \eta}(\tau_1, \tau_2, n d \mu, -n c \mu, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \mathfrak{e}_{\sigma} \\ &\quad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{\frac{m u}{N_u} + ncu'}\bigg({-} \frac{mnd}{N_u} - n^2 cd q(u')\bigg) \end{align*}

\begin{align*} &= \frac{1}{\sqrt{2 u_{z^\perp}^2}} \Theta_{{{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{\frac{m u}{N_u}} \\ &\quad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = \big(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\big) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} \frac{n^{h}(c \tau_1 + d)^{-h}}{({-}2 i \Im(\gamma \tau_1))^{h}} e\bigg({-}\frac{n^2 + 4 i n \Im(\gamma \tau_2) (\eta, u_{z^\perp})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\quad\times \Theta_{{{K}}, \eta}(\tau_1, \tau_2, n d \mu, -n c \mu, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{\frac{m u}{N_u} + ncu'}\bigg({-} \frac{mnd}{N_u} - n^2 cd q(u')\bigg). \end{align*}

The transformation formula of $\Theta_{{{K}}, \eta}$ provided by Example 3.11 implies that the formula above equals

\begin{align*} & \frac{1}{\sqrt{2 u_{z^\perp}^2}} \Theta_{{{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}} \\ &\quad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = \big(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\big) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} e\bigg({-}\frac{q(\eta) c \tau_2^2}{c \tau_1 + d}\bigg) \\ &\quad\times \frac{n^{h}(c \tau_1 + d)^{{b^-}/{2} - {b^+}/{2} - m}}{({-}2 i \Im(\gamma \tau_1))^{h}} e\bigg({-}\frac{n^2 + 4 i n \Im(\gamma \tau_2) (\eta, u_{z^\perp})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\quad\times (\rho_{{{K}}, \eta}^{-1}(\gamma) \Theta_{{{K}}, \eta}(\gamma\tau_1, \gamma\tau_2, n\mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h})) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u} + ncu'}\bigg({-} \frac{mnd}{N_u} - n^2 cd q(u')\bigg) \\ &= \frac{1}{\sqrt{2 u_{z^\perp}^2}} \Theta_{{{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}} \\ &\quad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = \big(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\big) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} e\bigg({-}\frac{q(\eta) c \tau_2^2}{c \tau_1 + d}\bigg) \\ &\quad\times \frac{n^{h}(c \tau_1 + d)^{{b^-}/{2} - {b^+}/{2} - m}}{({-}2 i \Im(\gamma \tau_1))^{h}} e\bigg({-}\frac{n^2 + 4 i n \Im(\gamma \tau_2) (\eta, u_{z^\perp})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\quad\times \rho_{L, \eta}^{-1} (\tilde{\gamma}) \bigg(\Theta_{{{K}}, \eta}(\gamma\tau_1, \gamma\tau_2, n\mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}}\bigg({-} \frac{mn}{N_u}\bigg) \bigg), \end{align*}

where in the last equality we used Lemma 5.2.

5.2 Evaluating Petersson inner products

Let $\mathcal{P} : \mathbb{R}^{b^+, 0} \to \mathbb{C}$ be a homogeneous polynomial of degree m and let $\eta \in L'$ . Let $\phi \colon \mathbb{H} \times \mathbb{C} \to \mathbb{C}[D_L]$ be a Jacobi form of weight ${k = m + (b^+ - b^-)/2}$ and index $\eta \in L'$ . Let $\langle\cdot{,}\cdot\rangle$ be the standard Hermitian product of $\mathbb{C}[D_L]$ . We define the Jacobi Petersson inner product of $\phi$ and $\Theta_{L, \eta}$ as

\begin{align*}\langle \phi, \Theta_{L, \eta}(\cdot, \cdot, g, \mathcal{P}) \rangle_{\operatorname{Pet}}= \int_{\mathcal{J} {{\backslash}} \mathbb{H} \times \mathbb{C}} \langle\phi(\tau_1, \tau_2), \Theta_{L, \eta}(\tau_1, \tau_2, g, \mathcal{P}) \rangle y_1^k \exp({-}4 \pi q(\eta) y_2^2 / y_1) \frac{d\tau_2\, d\tau_1}{y_1^3}.\end{align*}

In this section, we deduce from the rewriting of $\Theta_{L, \eta}$ as a Poincaré series given by Theorem 5.3 an unfolding of the Jacobi Petersson inner products defined above. To do so, we need to recall that a fundamental domain for the action of $\mathcal{H}(\mathbb{Z}) \rtimes \Gamma_\infty$ on $\mathbb{H} \times \mathbb{C}$ is given by

\begin{align*}\{ (\tau_1, \tau_2) \in \mathbb{H} \times \mathbb{C} \mid -1/2 \leq \Re(\tau_1) \leq 1/2, \tau_2 \in T(\tau_1) \},\end{align*}

where we denote by $T(\tau_1)$ the torus

\begin{align*}T(\tau_1) = \{\tau_2 \in \mathbb{C} \mid 0 \leq y_2 \leq y_1,\, y_2 x_1 / y_1 \leq x_2 \leq 1 + y_2 x_1 / y_1 \}.\end{align*}

Let $u \in L$ be primitive isotropic of level $N_u$ , and let $u' \in L'$ be such that $(u, u') = 1$ . The notation K, ${g_{{{K}}}}$ , $\mathcal{P}_{{g_{{{K}}}}, h}$ , $\mu$ is as in § 5.1.

Lemma 5.4. Let $\eta \in u^\perp$ and consider the linear map

\begin{align*}{{K}}' \to (\eta^\perp \cap {{K}})', \quad \lambda \mapsto \lambda_{\eta} = \lambda - \frac{(\lambda, \eta)}{\eta^2} \eta_{{{K}}}.\end{align*}

Then

\begin{align*}&\langle \phi, \Theta_{L, \eta}(\cdot, \cdot, g, \mathcal{P})\rangle_{\operatorname{Pet}} \\&\quad= \frac{1}{\sqrt{2 u_{z^\perp}^2}} \langle \phi, \Theta_{{{K}}, \eta}(\cdot, \cdot, g_K, \mathcal{P}_{g_K, 0}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}} \rangle_{\operatorname{Pet}} + \frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i}\bigg)^{\!\!h} \\ &\qquad\times \int_0^\infty y_1^{b^- + k - h - 5/2} \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \sum_{\tilde{\lambda} \in (\eta^\perp \cap {{K}})'} \sum_{\substack{\lambda \in {{K}}' / \eta_{{{K}}} \\ \lambda_{\eta} = \tilde{\lambda}}} \int_{-\infty}^{\infty} e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \\ &\qquad\times \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda_{\eta} + y_2 \eta)) \cdot \exp({-}4 \pi q((\lambda_{\eta} + y_2 \eta)_{{w^\perp}}) y_1) dy_2\, dy_1 \\ &\qquad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + (n \lambda_{\eta}, \mu) + \frac{(n\lambda, \eta)(\eta, u')}{\eta^2}\bigg) \cdot a\bigg(\lambda + \frac{m u}{N_{u}}, q(\lambda), (\lambda, \eta)\bigg), \end{align*}

where $a(\cdot, \cdot, \cdot)$ are the Fourier coefficients of $\phi$ .

Proof. We write $\Theta_{L, \eta}$ as

\begin{align*} &\frac{1}{\sqrt{2 u_{z^\perp}^2}} \Theta_{{{K}}, \eta}(\tau_1, \tau_2, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, 0}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}} \\ &\quad+ \frac{1}{\sqrt{2 u_{z^\perp}^2}} \sum_{\gamma = \big(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\big) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})} \sum_{n = 1}^\infty \sum_{h} e\bigg({-}\frac{q(\eta) c \tau_2^2}{c \tau_1 + d}\bigg) \\ &\quad\times \frac{n^{h} (c \tau_1 + d)^{(b^--b^+)/2 - m}}{({-}2 i \Im(\gamma \tau_1))^{h}} e\bigg({-}\frac{n^2 + 4 i n \Im(\gamma \tau_2) (\eta, u_{z^\perp})}{4 i \Im(\gamma \tau_1) u_{z^\perp}^2}\bigg) \\ &\quad\times \rho_{L, \eta}^{-1}(\tilde{\gamma}) \bigg(\Theta_{{{K}}, \eta} (\gamma\tau_1, \gamma\tau_2, n\mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}}\bigg({-} \frac{mn}{N_u}\bigg) \bigg) \end{align*}

and unfold the integral against this Poincaré series to obtain

(5.3) \begin{align} & \frac{1}{\sqrt{2 u_{z^\perp}^2}} \bigg\langle \phi, \Theta_{{{K}}, \eta}(\cdot, \cdot, g_K, \mathcal{P}_{g_K, 0}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}} \bigg\rangle_{\operatorname{Pet}} + \frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i} \bigg)^{\!\!h} \nonumber\\ &\quad\times \int_0^\infty \exp\bigg({-}\frac{\pi n^2}{2y_1 u_{z^\perp}^2}\bigg) y_1^{k - h} \int_{-1/2}^{1/2} \int_{\tau_2 \in T(\tau_1)} e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{y_1 u_{z^\perp}^2}\bigg) \nonumber\\ &\quad\times \bigg\langle \phi(\tau_1, \tau_2), \Theta_{{{K}}, \eta}(\tau_1, \tau_2, n\mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}}\bigg({-} \frac{mn}{N_u}\bigg) \bigg\rangle \nonumber\\ &\quad\times \exp({-}4 \pi q(\eta) y_2^2 / y_1) d\tau_2 \,\frac{d\tau_1}{y_1^3}. \end{align}

The factor of 2 in the second summand of the formula above comes from the trivial action of $(\begin{smallmatrix}-1 & 0 \\ 0 & -1\end{smallmatrix}) \in \Gamma_\infty {{\backslash}} \mathrm{SL}_2(\mathbb{Z})$ . The first summand has the correct form so that we will only consider the second summand of (5.3). We apply the change of variables

\begin{align*}(x_2, y_2) \mapsto (x_2 + y_2 x_1, y_1 y_2)\end{align*}

which transforms the rectangle $[0, 1]^2$ to the torus $T(\tau_1)$ . Hence, the second summand of (5.3) is given by

(5.4) \begin{align} & \frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i}\bigg)^{\!\!h} \int_0^\infty \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) y_1^{k - h} \int_{-1/2}^{1/2} \int_{0}^1 \int_0^1 e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \nonumber\\ &\quad\times \bigg\langle \phi(\tau_1, x_2 + y_2 \tau_1), \Theta_{{{K}}, \eta}(\tau_1, x_2 + y_2 \tau_1, n\mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}}\bigg({-} \frac{mn}{N_u}\bigg) \bigg\rangle \nonumber \\ &\quad\times \exp({-}4 \pi q(\eta) y_2^2 y_1) dx_2 \,dy_2 \,\frac{dx_1 \,dy_1}{y_1^2} \nonumber \\[6pt] &= \frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \frac{n^{h}}{(2i)^{h}} \int_0^\infty y_1^{k - h - 2} \int_{0}^1 \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \nonumber \\ &\quad\times \exp({-}4 \pi q(\eta) y_2^2 y_1) \cdot e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \int_{-1/2}^{1/2} \int_0^1 \bigg\langle \phi(\tau_1, x_2 + y_2 \tau_1), \nonumber \\ &\qquad \Theta_{{{K}}, \eta}(\tau_1, x_2 + y_2 \tau_1, n\mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}}\bigg({-} \frac{mn}{N_u}\bigg) \bigg\rangle\, dx_2 \, dx_1 \, dy_2 \, dy_1. \end{align}

We insert the Fourier expansion of the Jacobi theta function

\begin{align*} &\Theta_{{{K}}, \eta}(\tau_1, x_1 + y_2 \tau_1, n \mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \\[6pt] &\quad= y_1^{{b^{-}- 1}/{2}} \exp(4 \pi q(\eta_{w}) y_2^2 y_1) \sum_{\lambda \in {{K}}'} \exp({-}\Delta / 8 \pi y_1)(\mathcal{P}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda + y_2 \eta)) \\ &\qquad\times \mathfrak{e}_{\lambda}(q(\lambda) x_1 + q_{{w^\perp}}(\lambda) i y_1 + (x_2 + y_2 x_1) (\lambda, \eta) + i y_2 y_1 (\lambda, \eta)_{{w^\perp}} - (\lambda, n \mu)) \end{align*}

to obtain, for the integrals over $x_1, x_2$ appearing on the right-hand side of (5.4),

\begin{align*} &\int_{-1/2}^{1/2} \int_0^1 \bigg\langle \phi(\tau_1, x_2 + y_2 \tau_1), \Theta_{{{K}}, \eta}(\tau_1, x_2 + y_2 \tau_1, n\mu, 0, {g_{{{K}}}}, \mathcal{P}_{{g_{{{K}}}}, h}) \\ &\quad\times\sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} \mathfrak{e}_{{m u}/{N_u}}\bigg({-} \frac{mn}{N_u}\bigg) \bigg\rangle\, dx_2 \, dx_1 \\[6pt] &= y_1^{({b^- - 1)}/{2}} \exp(4 \pi q(\eta_{w}) y_2^2 y_1) \sum_{\lambda \in {{K}}'} \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda + y_2 \eta)) \\ &\quad\times \exp({-}2\pi(q_{{w^\perp}}(\lambda) y_1 + y_2 y_1 (\lambda, \eta)_{{w^\perp}})) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + (\lambda, n \mu)\bigg) \\ &\quad\times \int_{-1/2}^{1/2} \int_0^1 \phi_{\lambda + {m u}/{N_{u}}}(\tau_1, x_2 + y_2 \tau_1) e({-}q(\lambda) x_1 - (x_2 + y_2 x_1) (\lambda, \eta))\, dx_2\, dx_1 \end{align*}

\begin{align*} &= y_1^{({b^- - 1})/{2}} \exp(2 \pi q(\eta) y_2^2 y_1) \sum_{\lambda \in {{K}}'} \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda + y_2 \eta)) \\[6pt] &\quad\times \exp({-}2 \pi q_{{w^\perp}}(\lambda + y_2 \eta) y_1) \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + (\lambda, n \mu)\bigg) \\ &\quad\times \int_{-1/2}^{1/2} \int_0^1 \phi_{\lambda + {m u}/{N_{u}}}(\tau_1, x_2 + y_2 \tau_1) e({-}q(\lambda) x_1 - (x_2 + y_2 x_1) (\lambda, \eta))\, dx_2\, dx_1. \end{align*}

Next, we insert the Fourier expansion

\begin{align*}\phi_{\lambda + {m u}/{N_{u}}}(\tau_1, x_2 + y_2 \tau_1) = \sum_{\substack{r \in \mathbb{Z} + q(\lambda) \\ s \in \mathbb{Z} + (\lambda + {m u}/{N_{u}}, \eta)}} a\bigg(\lambda + \frac{m u}{N_{u}}, r, s\bigg) \cdot e(r \tau_1 + s (x_2 + y_2 \tau_1)),\end{align*}

calculate the integrals, and obtain

\begin{align*} &y_1^{(b^- - 1)/2} \exp(2 \pi q(\eta) y_2^2 y_1) \sum_{\lambda \in {{K}}'} \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda + y_2 \eta)) \\[6pt] &\quad\times \exp({-}2 \pi q_{{w^\perp}}(\lambda + y_2 \eta) y_1) \exp({-}2 \pi ((\lambda, \eta)y_2 y_1 + q(\lambda) y_1)) \\[6pt] &\quad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + (\lambda, n \mu)\bigg) \cdot a\bigg(\lambda + \frac{m u}{N_{u}}, q(\lambda), (\lambda, \eta)\bigg) \\[6pt] &= y_1^{(b^- - 1)/2} \exp(4 \pi q(\eta) y_2^2 y_1) \sum_{\lambda \in {{K}}'} \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda + y_2 \eta)) \\[6pt] &\quad\times \exp({-}2 \pi q_{{w^\perp}}(\lambda + y_2 \eta) y_1) \cdot \exp({-}2 \pi q(\lambda + y_2 \eta) y_1) \\[6pt] &\quad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + (\lambda, n \mu)\bigg) \cdot a\bigg(\lambda + \frac{m u}{N_{u}}, q(\lambda), (\lambda, \eta)\bigg). \end{align*}

Hence, the second summand of (5.3) is given by

\begin{align*} & \frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i}\bigg)^{\!\!h} \int_0^\infty y_1^{b^-/2 + k - h - 5/2} \int_{0}^1 \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \cdot e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \\[6pt] &\quad\times \sum_{\lambda \in {{K}}'} \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda + y_2 \eta)) \cdot \exp({-}4 \pi q((\lambda + y_2 \eta)_{{w^\perp}}) y_1) dy_2\, dy_1 \\[6pt] &\quad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + (\lambda, n \mu)\bigg) \cdot a\bigg(\lambda + \frac{m u}{N_{u}}, q(\lambda), (\lambda, \eta)\bigg). \end{align*}

By summing over $\lambda + l \eta_{{K}}$ , where $\lambda \in {{K}}' / \eta_{{K}}$ and $l \in \mathbb{Z}$ , and by using that

\begin{align*}a(\lambda + \eta, q(\lambda + \eta), (\lambda + \eta, \eta)) = a(\lambda, q(\lambda), (\lambda, \eta)),\end{align*}

we obtain

\begin{align*} & \frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i}\bigg)^{\!\!h} \int_0^\infty y_1^{b^-/2 + k - h - 5/2} \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \\[4pt] &\quad\times \sum_{\lambda \in {{K}}' / \eta_{{{K}}}} \sum_{l \in \mathbb{Z}} \int_{0}^1 e\bigg(\frac{n (y_2 + l) (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \cdot \exp({-}4 \pi q((\lambda + (y_2 + l) \eta)_{{w^\perp}}) y_1) \\[4pt] &\quad\times \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda + (y_2 + l) \eta)) dy_2 \,dy_1 \\[4pt] &\quad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} - n l (\eta, u') + (\lambda, n \mu)\bigg) \cdot a\bigg(\lambda + \frac{m u}{N_{u}} - l (\eta, u') u, q(\lambda), (\lambda, \eta)\bigg) \\[4pt] &= \frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i}\bigg)^{\!\!h} \int_0^\infty y_1^{b^-/2 + k - h - 5/2} \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \\[4pt] &\quad\times \sum_{\lambda \in {{K}}' / \eta_{{{K}}}} \int_{-\infty}^{\infty} e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \cdot \exp({-}4 \pi q((\lambda + y_2 \eta)_{{w^\perp}}) y_1) \\[4pt] &\quad\times \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda + y_2 \eta)) dy_2 \,dy_1 \\[4pt] &\quad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + (\lambda, n \mu)\bigg) \cdot a\bigg(\lambda + \frac{m u}{N_{u}}, q(\lambda), (\lambda, \eta)\bigg). \end{align*}

We remark that $(\eta, u_{z^\perp})/u_{z^\perp}^2 = (\eta, \mu + u')$ . We use $\lambda_{\eta} = \lambda - {(\lambda, \eta)}/{\eta^2} \eta_{{K}}$ and make the substitution $y_2 \mapsto y_2 - {(\lambda, \eta)}/{\eta^2}$ to rewrite this as

\begin{align*} &\frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i}\bigg)^{\!\!h} \int_0^\infty y_1^{b^-/2 + k - h - 5/2} \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \sum_{\lambda \in {{K}}' / \eta_{{{K}}}} \int_{-\infty}^{\infty} e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \\[4pt] &\quad\times \exp({-}\Delta / 8 \pi y_1)(\overline{\mathcal{P}}_{{g_{{{K}}}}, h})({g_{{{K}}}}(\lambda_{\eta} + y_2 \eta)) \cdot \exp({-}4 \pi q((\lambda_{\eta} + y_2 \eta)_{{w^\perp}}) y_1) dy_2 \,dy_1 \\[4pt] &\quad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + \frac{(n\lambda, \eta)(\eta, u')}{\eta^2}\bigg) \cdot a\bigg(\lambda + \frac{m u}{N_{u}}, q(\lambda), (\lambda, \eta)\bigg) \cdot e(n \lambda_{\eta}, \mu) \\[4pt] &= \frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i}\bigg)^{\!\!h} \int_0^\infty y_1^{b^-/2 + k - h - 5/2} \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \\[4pt] &\quad\times \sum_{\tilde{\lambda} \in (\eta^\perp \cap {{K}})'} \sum_{\substack{\lambda \in {{K}}' / \eta_K \\[4pt] \lambda_{\eta} = \tilde{\lambda}}} \int_{-\infty}^{\infty} e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \\[4pt] &\quad\times \exp({-}\Delta / 8 \pi y_1) (\overline{\mathcal{P}}_{{g_{{{K}}}}, h}) ({g_{{{K}}}}(\tilde{\lambda} + y_2 \eta)) \cdot \exp({-}4 \pi q((\tilde{\lambda} + y_2 \eta)_{{w^\perp}}) y_1) dy_2 \,dy_1 \\[4pt] &\quad\times \sum_{m \in \mathbb{Z} / N_u \mathbb{Z}} e\bigg(\frac{mn}{N_u} + \frac{(n\lambda, \eta)(\eta, u')}{\eta^2}\bigg) \cdot a\bigg(\lambda + \frac{m u}{N_{u}}, q(\lambda), (\lambda, \eta)\bigg) \cdot e(n \tilde{\lambda}, \mu). \end{align*}

This shows the assertion.

Theorem 5.5. Let $\eta \in u^\perp$ and assume $N_{u} = 1$ , so that L is the orthogonal sum of a hyperbolic plane and K. Moreover, assume that $\mathcal{P}_{{g_{{{K}}}}, h}$ is harmonic and satisfies $\mathcal{P}_{{g_{{{K}}}}, h}(\lambda + C \eta) = \mathcal{P}_{{g_{{{K}}}}, h}(\lambda)$ for every $C \in \mathbb{C}$ and $\lambda \in {{K}}$ . Then $\langle \phi, \Theta_{L, \eta}(\cdot, \cdot, g, \mathcal{P}) \rangle_{\operatorname{Pet}}$ , as a function of G, admits a Fourier expansion of the form

\begin{align*}\langle \phi, \Theta_{L, \eta}(\cdot, \cdot, g, \mathcal{P}) \rangle_{\operatorname{Pet}}=\sum_{\tilde{\lambda} \in (\eta^\perp \cap K)'} c_{\tilde{\lambda}}(g) \cdot e(\tilde{\lambda}, \mu).\end{align*}

The constant term of the expansion is the coefficient

\begin{align*}c_0(g) = \frac{1}{\sqrt{2} \lvert u_{z^\perp}\rvert}\langle \phi, \Theta_{{{K}}, \eta}(\cdot, \cdot, g_K, \mathcal{P}_{g_K, 0}) \rangle_{\operatorname{Pet}}.\end{align*}

If $\tilde\lambda$ has positive norm, then the Fourier coefficient associated to $\tilde\lambda$ is

(5.5) \begin{align}c_{\tilde{\lambda}}(g) &= \frac{2}{\lvert u_{z^\perp} \rvert \lvert \eta_{w^\perp} \rvert}\sum_{n \mid \tilde{\lambda}}\sum_{h}\frac{n^{b^- + h + k - 4}}{(2i)^{-h}}\bigg(\frac{\lvert \eta_{z^\perp} \rvert}{2 \lvert u_{z^\perp} \rvert (\tilde{\lambda}_{w^\perp}^2 \eta_{w^\perp}^2 - (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})^2)^{1/2}}\bigg)^{\!\!b^- / 2 + k - h - 2}\nonumber\\&\quad\times\exp\bigg(-\frac{\pi i (\eta, u_{z^\perp}) (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})}{u_{z^\perp}^2 \eta_{w^\perp}^2}\bigg)K_{b^-/2 + k - h - 2}\bigg(2 \pi \frac{\lvert \eta_{z^\perp} \rvert (\tilde{\lambda}_{w^\perp}^2\eta_{w^\perp}^2 - (\tilde{\lambda}_{w^\perp},\eta_{w^\perp})^2)^{1/2}}{\lvert u_{z^\perp} \rvert\eta_{w^\perp}^2}\bigg) \nonumber\\&\quad\times\overline{\mathcal{P}}_{{g_{{{K}}}}, h}({g_{{{K}}}}(\tilde{\lambda})) \sum_{\substack{\lambda \in {{K}}' / \eta_K \\ \lambda_{\eta} = \tilde{\lambda}}} e\bigg(\frac{(\lambda, \eta)(\eta, u')}{\eta^2}\bigg)\cdot a(\lambda / n, q(\lambda) / n^2, (\lambda, \eta) / n),\end{align}

where $K_s(x)$ is the K-Bessel function. In all remaining cases, the Fourier coefficients vanish.

The Eichler transformation associated to the isotropic lattice vector u and $\lambda \in \eta_{{{K}}}^\perp \cap {{K}} \otimes \mathbb{R}$ is defined as

\begin{align*}E(u, \lambda)(v) := v - (v, u) \lambda + (v, \lambda) u- q(\lambda) (v, u) u.\end{align*}

The fact that $\langle \phi, \Theta_{L, \eta} \rangle_{\operatorname{Pet}} \colon G\to\mathbb{C}$ admits a Fourier expansion of the form

\begin{equation*}\langle \phi, \Theta_{L, \eta}(\cdot, \cdot, g, \mathcal{P}) \rangle_{\operatorname{Pet}}=\sum_{\tilde{\lambda} \in (\eta^\perp \cap {{K}})'}c_{\tilde{\lambda}}(g)\cdot e(\tilde{\lambda}, \mu),\end{equation*}

for some Fourier coefficients $c_{\tilde{\lambda}}$ , follows from its invariance under the Eichler transformations $E(u, \lambda)$ , where $\lambda \in \eta^\perp \cap K$ . This invariance can easily be checked using the following result, which is known to experts and whose proof is left to the reader.

1. (i) We have that $E(u,\lambda)(w')=w$ and that $E(u,\lambda)(w'^\perp)=w^\perp$ .

2. (ii) We have that $E(u,\lambda)(\mu')=\mu + \lambda + q(\lambda)u$ .

3. (iii) We have that $g_{{{K}}}(v)=(g\circ E(u,\lambda))_{{{K}}}(v)$ for every $v\in{{K}}\otimes\mathbb{R}$ .

4. (iv) We have $u^2_{z^\perp} = u^2_{E(u,\lambda)(z^\perp)}$ .

5. (v) We have $v^2_{w^\perp} = v^2_{w'^\perp}$ for every $v\in{{K}}\otimes\mathbb{R}$ .

To check that the Fourier coefficients $c_{\tilde{\lambda}}$ are as provided by Lemma 5.4, it is enough to check that $(2 u_{z^\perp}^2)^{-1/2} \langle f, \Theta_{{{K}}, \eta}(\cdot, \cdot, g_K, \mathcal{P}_{g_K, 0}) \rangle_{\operatorname{Pet}}$ and the coefficients $c_{\tilde{\lambda}}$ are invariant with respect to Eichler transformations $E(u, \lambda)$ , where $\lambda \in \eta^\perp \cap K\otimes\mathbb{R}$ . This follows easily from Lemma 5.6

We conclude the section with the proof of Theorem 5.5.

Proof of Theorem 5.5. We use Lemma 5.4 and only consider the second summand. It is given by

(5.6) \begin{align} &\frac{\sqrt{2}}{\lvert u_{z^\perp} \rvert} \sum_{n = 1}^\infty \sum_{h} \bigg(\frac{n}{2i}\bigg)^{\!\!h} \int_0^\infty y_1^{b^-/2 + k - h - 5/2}\nonumber \\ &\quad\times \exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \sum_{\tilde{\lambda} \in (\eta^\perp \cap {{K}})'} \sum_{\substack{\lambda \in {{K}}' / \eta_K \\ \lambda_{\eta} = \tilde{\lambda}}} \overline{\mathcal{P}}_{{g_{{{K}}}}, h}({g_{{{K}}}}(\tilde{\lambda})) \nonumber\\ &\quad\times \int_{-\infty}^{\infty} e\bigg(\frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \cdot \exp({-}4 \pi q((\tilde{\lambda} + y_2 \eta)_{{w^\perp}}) y_1) dy_2\, dy_1 \nonumber\\ &\quad\times e\bigg(\frac{(n\lambda, \eta)(\eta, u')}{\eta^2}\bigg) \cdot a(\lambda, q(\lambda), (\lambda, \eta)) \cdot e(n \tilde{\lambda}, \mu). \end{align}

The inner integral of (5.6) is a Fourier transform of a Gaussian and is given by

\begin{align*} &\exp\bigg({-}\frac{\pi n^2}{2 y_1 u_{z^\perp}^2}\bigg) \int_{-\infty}^{\infty} \exp\bigg(2 \pi i \frac{n y_2 (\eta, u_{z^\perp})}{u_{z^\perp}^2}\bigg) \exp({-}4 \pi q((\tilde{\lambda} + y_2 \eta)_{{w^\perp}}) y_1) dy_2 \\ &\quad= (2 \eta_{w^\perp}^2 y_1)^{-1/2} \exp\bigg({-} \frac{\pi n^2 \eta_{z^\perp}^2}{2 y_1u_{z^\perp}^2 \eta_{w^\perp}^2} - 2 \pi y_1 \frac{\tilde{\lambda}_{w^\perp}^2 \eta_{w^\perp}^2 - (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})^2}{\eta_{w^\perp}^2}\bigg) \\ &\qquad\times \exp\bigg({-}\frac{\pi i n (\eta, u_{z^\perp}) (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})}{u_{z^\perp}^2 \eta_{w^\perp}^2}\bigg). \end{align*}

Hence, the integral over $y_1$ of (5.6) is given by

\begin{align*}\int_0^\infty y_1^{b^-/2 + k - h - 3} \exp\bigg({-} \frac{\pi n^2 \eta_{z^\perp}^2}{2 y_1u_{z^\perp}^2 \eta_{w^\perp}^2} - 2 \pi y_1 \frac{\tilde{\lambda}_{w^\perp}^2 \eta_{w^\perp}^2 - (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})^2}{\eta_{w^\perp}^2}\bigg).\end{align*}

Using the formula [Reference Erdélyi, Magnus, Oberhettinger and TricomiEMO⁺54, p. 313, 6.3(17)]

\begin{align*}\int_{y_1 = 0}^\infty \exp\bigg({-}\alpha y_1 - \frac{\beta}{y_1} \bigg) y_1^\gamma \frac{dy_1}{y_1} = 2 \bigg(\frac{\beta}{\alpha}\bigg)^{\!\!\gamma / 2} K_\gamma(2 (\alpha \beta)^{1/2})\end{align*}

shows that the integral over $y_1$ of (5.6) is equal to

\begin{align*} &2 \Bigg( \frac{n \lvert \eta_{z^\perp} \rvert}{2 \lvert u_{z^\perp} \rvert (\tilde{\lambda}_{w^\perp}^2 \eta_{w^\perp}^2 - (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})^2)^{1/2}}\Bigg)^{\!\!b^- / 2 + k - h - 2} \\ &\quad\times K_{b^-/2 + k - h - 2} \Bigg( 2 \pi n \frac{\lvert \eta_{z^\perp} \rvert (\tilde{\lambda}_{w^\perp}^2 \eta_{w^\perp}^2 - (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})^2)^{1/2}}{\lvert u_{z^\perp} \rvert \eta_{w^\perp}^2} \Bigg). \end{align*}

Hence, (5.6) is given by

\begin{align*} &\frac{2}{\lvert u_{z^\perp} \rvert \lvert \eta_{w^\perp} \rvert} \sum_{n = 1}^\infty n^{b^- /2 + k - 2} \sum_{h} (2i)^{-h} \\ &\quad\times \sum_{\tilde{\lambda} \in (\eta^\perp \cap {{K}})'} \sum_{\substack{\lambda \in {{K}}' / \eta_K \\ \lambda_{\eta} = \tilde{\lambda}}} \overline{\mathcal{P}}_{{g_{{{K}}}}, h}({g_{{{K}}}}(\tilde{\lambda})) \Bigg(\frac{\lvert \eta_{z^\perp} \rvert}{2 \lvert u_{z^\perp} \rvert (\tilde{\lambda}_{w^\perp}^2 \eta_{w^\perp}^2 - (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})^2)^{1/2}}\Bigg)^{\!\!b^- / 2 + k - h - 2} \end{align*}

\begin{align*} &\quad\times \exp\bigg({-}\frac{\pi i n (\eta, u_{z^\perp}) (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})}{u_{z^\perp}^2 \eta_{w^\perp}^2}\bigg) K_{b^-/2 + k - h - 2}\Bigg( 2 \pi n \frac{\lvert \eta_{z^\perp} \rvert (\tilde{\lambda}_{w^\perp}^2 \eta_{w^\perp}^2 - (\tilde{\lambda}_{w^\perp}, \eta_{w^\perp})^2)^{1/2}}{\lvert u_{z^\perp} \rvert \eta_{w^\perp}^2}\Bigg) \\ &\quad\times e\bigg(\frac{(n\lambda, \eta)(\eta, u')}{\eta^2}\bigg) \cdot a(\lambda, q(\lambda), (\lambda, \eta)) \cdot e(n \tilde{\lambda}, \mu). \end{align*}

Rewriting this as a divisor sum yields the result.

6.1 The Kudla–Millson Schwartz function

We keep the same notation introduced in § 4. In particular, we fix once and for all an orthogonal basis $(e_j)_j$ of V such that $(e_\alpha,e_\alpha)=1$ for every $\alpha=1,\dots,b$ , and $(e_\mu,e_\mu)=-1$ for $\mu=b+1,b+2$ . We define the standard majorant $(\cdot{,}\cdot)_z$ of $V^2$ with respect to $z\in\mathrm{Gr}(L)$ as

(6.1) \begin{equation} (\boldsymbol{v},\boldsymbol{v})_z=(\boldsymbol{v}_{z^\perp},\boldsymbol{v}_{z^\perp})-(\boldsymbol{v}_z,\boldsymbol{v}_z)\quad\text{for every~$\boldsymbol{v}\in V^2$.} \end{equation}

Let $\mathfrak{g}$ be the Lie algebra of G, and let $\mathfrak{g}=\mathfrak{p}+\mathfrak{k}$ be its Cartan decomposition. It is well known that $\mathfrak{p}\cong\mathfrak{g}/\mathfrak{k}$ is isomorphic to the tangent space of $\mathcal{D}$ at any base point $z_0$ . We may choose $z_0$ to be the negative-definite plane spanned by $e_{b+1}$ and $e_{b+2}$ .

With respect to the basis of V chosen above, we have

(6.2) \begin{equation} \mathfrak{p}\cong\bigg\{\bigg( \begin{matrix} 0 & X\\ X^t & 0 \end{matrix} \bigg)|X\in\text{Mat}_{b,2}(\mathbb{R})\bigg\}\cong \text{Mat}_{b,2}(\mathbb{R}). \end{equation}

We may assume that the chosen isomorphism is such that the complex structure on $\mathfrak{p}$ is given as the right-multiplication by $J=(\begin{smallmatrix}0 & 1\\ -1 & 0\end{smallmatrix})\in\mathrm{GL}_2(\mathbb{R})$ on $\text{Mat}_{b,2}(\mathbb{R})$ .

The Kudla–Millson Schwartz function $\varphi_{\text{ KM},2}$ is a G-invariant element of $\mathcal{S}(V^2)\otimes\mathcal{A}^4(\mathcal{D})$ , where $\mathcal{A}^4(\mathcal{D})$ is the space of differential 4-forms on $\mathcal{D}$ . Recall that

\begin{equation*} [\mathcal{S}(V^2)\otimes\mathcal{A}^4(\mathcal{D})]^G \cong [ \mathcal{S}(V^2)\otimes{\bigwedge}^4(\mathfrak{p}^*) ]^K, \end{equation*}

where the isomorphism is given by evaluating at the base point $z_0$ of $\mathcal{D}$ , and $K_\infty$ is the standard maximal compact of G stabilizing the base point $z_0\in\mathrm{Gr}(V)$ . We may then define $\varphi_{\text{ KM},2}$ firstly as an element of $[ \mathcal{S}(V^2)\otimes{\bigwedge}^4(\mathfrak{p}^*)]^{{K_\infty}}$ , and then spread it to the whole $\mathcal{D}$ via the action of G.

Definition 6.1. We denote by $X_{\alpha,\mu}$ , with $1\le\alpha\le b$ and $1\le\mu\le 2$ , the basis elements of $\text{Mat}_{b,2}(\mathbb{R})$ given by matrices with 1 at the $(\alpha,\mu)$ th entry and zero otherwise. These elements give a basis of $\mathfrak{p}$ via the isomorphism (6.2). Let $\omega_{\alpha,\mu}$ be the element of the dual basis which extracts the $(\alpha,\mu)$ th coordinate of elements in $\mathfrak{p}$ , and let $A_{\alpha,\mu}$ be the left-multiplication by $\omega_{\alpha,\mu}$ . The function $\varphi_{\text{ KM},2}$ is defined by applying the operator

\begin{equation*} \mathcal{D}^{b,2}_2=\frac{1}{4}\prod_{j,\mu=1}^2 \sum_{\alpha=1}^b\bigg( x_{\alpha,j}-\frac{1}{2\pi}\frac{\partial}{\partial x_{\alpha,j}} \bigg)\otimes A_{\alpha,\mu} \end{equation*}

to the standard Gaussian $\varphi_{0,2}\otimes 1\in[\mathcal{S}(V^2)\otimes{\bigwedge}^4(\mathfrak{p}^*)]^{{K_\infty}}$ , namely,

\begin{equation*} \varphi_{\text{ KM},2}=\mathcal{D}^{b,2}_2(\varphi_{0,2}\otimes 1). \end{equation*}

As a differential form, $\varphi_{\text{ KM},2}$ is closed; see [Reference Kudla and MillsonKM86]. The following result provides an explicit formula for $\varphi_{\text{ KM},2}$ . The idea of the proof is analogous to that in [Reference ZuffettiZuf24b, § 2], where we illustrated how to rewrite the Kudla–Millson Schwartz function of genus 1 in terms of certain polynomials $\mathcal{Q}_{(\alpha,\beta)}$ . Recall that the latter are defined on $\mathbb{R}^{b,2}$ as

(6.3) \begin{equation} \mathcal{Q}_{(\alpha,\beta)}(x):=\begin{cases} \mathcal{P}_{(\alpha,\beta)}(x) & \text{if $\alpha\neq\beta$,}\\[6pt]\displaystyle \mathcal{P}_{(\alpha,\beta)}(x)-\frac{1}{2\pi} & \text{otherwise,} \end{cases} \quad\text{where } \mathcal{P}_{(\alpha,\beta)}(x):= 2x_\alpha x_\beta, \end{equation}

for every $x=(x_1,\dots,x_{b+2})^t\in\mathbb{R}^{b,2}$ .

As in Definition 3.8, for simplicity we consider $\mathcal{Q}_{(\alpha,\beta)}$ also as a polynomial in the coordinates of V with respect to the basis $(e_j)_j$ chosen above. Hence, we drop $g_0$ from the notation and write $\mathcal{Q}_{(\alpha,\beta)}(v)$ instead of $\mathcal{Q}_{(\alpha,\beta)}( g_0(v))$ . This convention will be used for all polynomials defined on V and $V^2$ .

Proposition 6.2. The Kudla–Millson Schwartz function $\varphi_{\text{ KM},2}\in[ \mathcal{S}(V^2)\otimes{\bigwedge}^4(\mathfrak{p}^*) ]^{{K_\infty}}$ may be rewritten as

(6.4) \begin{equation}\begin{aligned} \varphi_{\text{ KM},2}(\boldsymbol{v},z_0) &= \sum_{\substack{\alpha_1,\beta_1=1\\ \alpha_1<\beta_1}}^b \sum_{\substack{\alpha_2,\beta_2=1\\ \alpha_2<\beta_2}}^b (\mathcal{Q}_{(\alpha_1,\alpha_2,\beta_1,\beta_2)}\cdot\varphi_{0,2})(\boldsymbol{v}) \\ &\quad\times\omega_{\alpha_1,1}\wedge\omega_{\alpha_2,2}\wedge\omega_{\beta_1,1}\wedge\omega_{\beta_2,2}, \end{aligned}\end{equation}

where $\mathcal{Q}_{(\alpha_1,\alpha_2,\beta_1,\beta_2)}$ is the polynomial on $V^2$ defined as

\begin{equation*} \mathcal{Q}_{(\alpha_1,\alpha_2,\beta_1,\beta_2)}(\boldsymbol{v})=\sum_{\sigma,\sigma'\in S_2}\mathrm{sgn}(\sigma)\mathrm{sgn}(\sigma') \mathcal{Q}_{(\sigma(\alpha_1),\sigma'(\alpha_2))}(v_1) \cdot \mathcal{Q}_{(\sigma(\beta_1),\sigma'(\beta_2))}(v_2), \end{equation*}

and where we denote by $\sigma$ , respectively $\sigma'$ , a permutation of the indexes $\{\alpha_1,\beta_1\}$ , respectively $\{\alpha_2,\beta_2\}$ .

To simplify the notation, we will frequently replace $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ by a vector of indices $\boldsymbol{\alpha}$ .

Remark 6.3. From (6.3), we may rewrite the product of polynomials appearing in the summand of the defining sum of $\mathcal{Q}_{\boldsymbol{\alpha}}$ more explicitly as

\begin{align*} \mathcal{Q}_{(\alpha_1,\alpha_2)}(v_1)\cdot \mathcal{Q}_{(\beta_1,\beta_2)}(v_2)=\begin{cases} 4 x_{\alpha_1,1} \cdot x_{\alpha_2,1} \cdot x_{\beta_1,2} \cdot x_{\beta_2,2} & \text{if $\alpha_1\neq\alpha_2$ and $\beta_1\neq\beta_2$,}\\[6pt]\displaystyle 2 x_{\alpha_1,1} \cdot x_{\alpha_2,1} \cdot \bigg( 2x_{\beta_1,2}^2-\frac{1}{2\pi}\bigg) & \text{if$\alpha_1\neq\alpha_2$ and $\beta_1=\beta_2$,}\\[12pt]\displaystyle \bigg(2 x_{\alpha_1,1}^2-\frac{1}{2\pi}\bigg) \cdot 2x_{\beta_1,2} \cdot x_{\beta_2,2} & \text{if $\alpha_1=\alpha_2$ and$\beta_1\neq\beta_2$,}\\[12pt]\displaystyle \bigg(2 x_{\alpha_1,1}^2-\frac{1}{2\pi}\bigg)\bigg( 2x_{\beta_1,2}^2-\frac{1}{2\pi}\bigg) & \text{if $\alpha_1=\alpha_2$ and $\beta_1=\beta_2$.} \end{cases} \end{align*}

Moreover, an easy calculation shows that

\begin{align*}\mathcal{Q}_{(\alpha_1,\alpha_2)}(v_1)\cdot \mathcal{Q}_{(\beta_1,\beta_2)}(v_2) = \exp\bigg({-}\frac{\mathrm{tr}\Delta}{8 \pi}\bigg)\mathcal{P}_{(\alpha_1,\alpha_2)}(v_1) \mathcal{P}_{(\beta_1,\beta_2)}(v_2),\end{align*}

where $\Delta$ is the Laplacian on $(\mathbb{R}^{b,2})^2$ defined in (3.2).

Proof of Proposition 6.2. For simplicity, we write $\mathcal{F}_{\alpha,j}=x_{\alpha,j}-{1}/{2\pi}\,{\partial}/({\partial x_{\alpha,j}})$ , for every ${j=1,2}$ and ${\alpha=1,\dots,b}$ . We may use such operators to rewrite

(6.5) \begin{equation}\begin{aligned} \varphi_{\text{ KM},2} &= \sum_{\substack{\alpha_1,\beta_1=1\\ \alpha_1\neq\beta_1}}^b\sum_{\substack{\alpha_2,\beta_2=1\\ \alpha_2\neq\beta_2}}^b \frac{1}{4}\mathcal{F}_{\alpha_1,1}\mathcal{F}_{\alpha_2,1}\mathcal{F}_{\beta_1,2}\mathcal{F}_{\beta_2,2}(\varphi_{0,2}) \cdot \omega_{\alpha_1,1}\wedge\omega_{\alpha_2,2}\wedge \omega_{\beta_1,1}\wedge \omega_{\beta_2,2}, \end{aligned}\end{equation}

where we deleted all summands associated to wedge products containing two functionals which are equal. We may compute

(6.6) \begin{align} \frac{1}{4}\mathcal{F}_{\alpha_1,1}\mathcal{F}_{\alpha_2,1}\mathcal{F}_{\beta_1,2}\mathcal{F}_{\beta_2,2}(\varphi_{0,2}) & = \frac{1}{4}\mathcal{F}_{\alpha_1,1}\mathcal{F}_{\alpha_2,1}\mathcal{F}_{\beta_1,2}(2 x_{\beta_2,2}\cdot\varphi_{0,2}) \nonumber\\ &= \begin{cases} \displaystyle\frac{1}{2}\mathcal{F}_{\alpha_1,1}\mathcal{F}_{\alpha_2,1}(2x_{\beta_1,2}x_{\beta_2,2}\cdot\varphi_{0,2} ) &\text{if $\beta_1\neq\beta_2$,}\\[8pt]\displaystyle\frac{1}{2}\mathcal{F}_{\alpha_1,1}\mathcal{F}_{\alpha_2,1}\bigg(\bigg(2x_{\beta_1,2}^2-\frac{1}{2\pi}\bigg)\varphi_{0,2}\!\bigg) &\text{if $\beta_1=\beta_2$.} \end{cases} \end{align}

Since the entries $x_1$ and $x_2$ of $g_0(\boldsymbol{v})$ are independent of each other, we may repeat an analogous procedure to compute the action of the operator ${\frac{1}{2}}\mathcal{F}_{1,\alpha_1}\mathcal{F}_{1,\alpha_2}$ on the right-hand side of (6.6), to deduce that

(6.7) \begin{align} \varphi_{\text{ KM},2}(\boldsymbol{v},z_0) &= \sum_{\substack{\alpha_1,\beta_1=1\\ \alpha_1\neq\beta_1}}^b\sum_{\substack{\alpha_2,\beta_2=1\\ \alpha_2\neq\beta_2}}^b(\mathcal{Q}_{(\alpha_1,\alpha_2)}(v_1)\cdot \mathcal{Q}_{(\beta_1,\beta_2)}(v_2)\cdot\varphi_{0,2}(\boldsymbol{v})) \nonumber\\ &\quad\times \omega_{\alpha_1,1}\wedge\omega_{\alpha_2,2}\wedge \omega_{\beta_1,1}\wedge \omega_{\beta_2,2}. \end{align}

The wedge products appearing on the right-hand side of (6.7) are linearly dependent in the vector space ${\bigwedge}^4(\mathfrak{p}^*)$ . A set of linearly independent wedge products, with respect to which we can write all those appearing in (6.7), is

\begin{equation*} \{\omega_{\alpha_1,1}\wedge\omega_{\alpha_2,2}\wedge \omega_{\beta_1,1}\wedge \omega_{\beta_2,2}\text{ : such that $\alpha_1<\beta_1$ and $\alpha_2<\beta_2$}\}. \end{equation*}

If we rewrite (6.7) with respect to such set, taking into account permutations of the indexes $\{\alpha_1,\beta_1\}$ and $\{\alpha_2,\beta_2\}$ , then we obtain (6.4).

Corollary 6.4. The extension of $\varphi_{\text{ KM},2}\in[\mathcal{S}(V^2)\otimes{\bigwedge}^4(\mathfrak{p}^*)]^{{K_\infty}}$ to the whole of $\mathcal{D}$ is

\begin{equation*} \varphi_{\text{ KM},2}(\boldsymbol{v},z) = \sum_{\substack{\alpha_1,\beta_1=1\\ \alpha_1<\beta_1}}^b \sum_{\substack{\alpha_2,\beta_2=1\\ \alpha_2<\beta_2}}^b (\mathcal{Q}_{\boldsymbol{\alpha}}\varphi_{0,2})(g(\boldsymbol{v})) \cdot g^*(\omega_{\alpha_1,1}\wedge\omega_{\alpha_2,2}\wedge\omega_{\beta_1,1}\wedge\omega_{\beta_2,2}), \end{equation*}

where $g\in G$ is any isometry that maps $z\in\mathcal{D}$ to the base point $z_0$ .

We define additional auxiliary polynomials $\mathcal{P}_{\boldsymbol{\alpha}}$ on $V^2$ as

(6.8) \begin{align} \mathcal{P}_{\boldsymbol{\alpha}}(\boldsymbol{v}) &{{}:= 4 \det\bigg( \begin{matrix} x_{\alpha_1,1} & x_{\alpha_1,2}\\ x_{\beta_1,1} & x_{\beta_1,2} \end{matrix} \bigg) \det\bigg( \begin{matrix} x_{\alpha_2,1} & x_{\alpha_2,2}\\ x_{\beta_2,1} & x_{\beta_2,2} \end{matrix} \bigg)} \nonumber\\ &= 4 \sum_{\sigma,\sigma'\in S_2} \mathrm{sgn}(\sigma)\mathrm{sgn}(\sigma') x_{\sigma(\alpha_1),1} \cdot x_{\sigma'(\alpha_2),1} \cdot x_{\sigma(\beta_1),2} \cdot x_{\sigma'(\beta_2),2}, \end{align}

for every $\boldsymbol{v}=(\sum_{j}x_{j,1}e_j,\sum_{j}x_{j,2}e_j)\in V^2$ , where $\sigma$ and $\sigma'$ are permutations of, respectively, $\{\alpha_1,\beta_1\}$ and $\{\alpha_2,\beta_2\}$ . Note that $\mathcal{Q}_{\boldsymbol{\alpha}}=\exp({-}\mathrm{tr}\Delta/8 \pi) \mathcal{P}_{\boldsymbol{\alpha}}$ and that if $\alpha_1\neq\alpha_2$ and $\beta_1\neq\beta_2$ , then $\mathcal{P}_{\boldsymbol{\alpha}}$ is harmonic and $\mathcal{Q}_{\boldsymbol{\alpha}}=\mathcal{P}_{\boldsymbol{\alpha}}$ .

Remark 6.5. Let $S(V^2)$ be the polynomial Fock space in $\mathcal{S}(V^2)$ , and let $\iota\colon S(V^2)\to\mathcal{P}(\mathbb{C}^{2(b+2)})$ be the associated intertwining map, where $\mathcal{P}(\mathbb{C}^{2(b+2)})$ is the space of complex polynomials in $2(b+2)$ variables; see [Reference Funke and MillsonFM06, Appendix A] for precise definitions. In place of working with functions in the Fock space, it is sometimes easier to consider their images under $\iota$ . We remark that even if $\mathcal{Q}_{\boldsymbol{\alpha}}\varphi_{0,2}$ lies in $S(V^2)$ , there is no simplification in working with its image $\iota(\mathcal{Q}_{\boldsymbol{\alpha}}\varphi_{0,2})$ . In fact, one can easily check that

\begin{align*} \iota(\mathcal{Q}_{\boldsymbol{\alpha}}\varphi_{0,2})(\boldsymbol{z}) = \frac{1}{2^6}\det\bigg(\begin{matrix} z_{\alpha_1,1} & z_{\alpha_1,2} \\ z_{\beta_1,1} & z_{\beta_1,2} \end{matrix}\bigg) \det\bigg(\begin{matrix} z_{\alpha_2,1} & z_{\alpha_2,2} \\ z_{\beta_2,1} & z_{\beta_2,2} \end{matrix}\bigg), \end{align*}

which is of the same form as (6.8). Here we denoted $\boldsymbol{z}=(z_{1,1},\dots,z_{b+2,1},z_{1,2},\dots,z_{b+2,2})$ .

The polynomials $\mathcal{P}_{\boldsymbol{\alpha}}$ will play a key role in the upcoming sections, thanks to the following result. We suggest that the reader recalls the construction of very homogeneous polynomials from Definition 3.1.

6.2 The Kudla–Millson theta function in terms of Siegel theta functions

Let X be an orthogonal Shimura variety arising from L; see § 2.2. We fix once and for all the weight $k=1+b/2$ .

Definition 6.7. The Kudla–Millson theta form of genus 2 is defined as

(6.9) \begin{equation} \Theta(\tau,z,\varphi_{\text{KM},2})=\det y^{-k/2}\sum_{\sigma\in D_{L}^2}\sum_{\boldsymbol{\lambda}\in \sigma + L^2}(\omega_{\infty,2}({M_\tau})\varphi_{\text{KM},2})(\boldsymbol{\lambda},z) \mathfrak{e}_\sigma, \end{equation}

for every $\tau=x+iy\in\mathbb{H}_2$ and $z\in\mathrm{Gr}(L)$ , where ${M_\tau}=\big(\begin{smallmatrix} 1 & x\\ 0 & 1 \end{smallmatrix}\big)\big(\begin{smallmatrix} y^{1/2} & 0\\ 0 & (y^{1/2})^{-t} \end{smallmatrix}\big)$ is the standard element of $\mathrm{Sp}_4(\mathbb{R})$ mapping ${iI_2\in\mathbb{H}_2}$ to $\tau$ .

Let $A^k_{2,L}$ be the space of $\mathbb{C}[D_{L}^2]$ -valued analytic functions on $\mathbb{H}_2$ satisfying the weight-k modular transformation property with respect to $\mathrm{Mp}_4(\mathbb{Z})$ under the Weil representation $\rho_{L,2}$ . The Kudla–Millson theta function (6.9) is a non-holomorphic modular form with respect to the variable $\tau\in\mathbb{H}_2$ , and a closed 4-form with respect to the variable ${z\in\mathrm{Gr}(L)}$ . In short, $\Theta(\tau,z,\varphi_{\text{ KM},2})\in A^k_{2,L}\otimes\mathcal{Z}^4(\mathcal{D})$ . In fact, the Kudla–Millson theta function is $\widetilde{\mathrm{SO}}(L)$ -invariant, hence it descends to an element of $A^k_2\otimes\mathcal{Z}^4(X)$ .

Recall that $\Lambda_2$ is the set of symmetric half-integral $2\times 2$ matrices. Kudla and Millson [Reference Kudla and MillsonKM90] proved that the cohomology class $[\Theta(\tau,z,\varphi_{\text{ KM},2})]$ is a holomorphic modular form with values in $H^{2,2}(X)$ . Moreover, they proved that the Fourier coefficient of $[\Theta(\tau,z,\varphi_{\text{ KM},2})]$ associated to the indices $\sigma\in D_{L}$ and $T\in q(\sigma) + \Lambda_2$ with $T \gt 0$ is a Poincaré dual form for the special cycle of the same indices in X. The class $[\Theta(\tau,z,\varphi_{\text{ KM},2})]$ is the well-known Kudla generating series of special cycles; see [Reference KudlaKud04, Theorem 3.1].