2.1 Quaternionic unitary groups

We start by fixing some notation. For more details on quaternion algebras, the reader is referred to [Reference Voight36]. In this work, a quaternion algebra will mean a central simple algebra of dimension 4 over $\mathbb {Q}$ . After selecting a basis, we can write it in the form

$$\begin{align*}\mathbb{B}=\mathbb{Q}\oplus\mathbb{Q}\zeta\oplus\mathbb{Q}\xi\oplus\mathbb{Q}\zeta\xi, \end{align*}$$

where

$$\begin{align*}\zeta^2=\alpha,\xi^2=\beta,\zeta\xi=-\xi\zeta, \end{align*}$$

with $\alpha ,\beta $ nonzero square-free integers. We assume in this paper that $\mathbb {B}$ is definite, i.e., $\alpha ,\beta <0$ . The main involution of $\mathbb {B}$ is given by

$$\begin{align*}\overline{\cdot}:\mathbb{B}\to\mathbb{B}:a+b\zeta+c\xi+d\zeta\xi\mapsto\overline{a+b\zeta+c\xi+d\zeta\xi}=a-b\zeta-c\xi-d\zeta\xi. \end{align*}$$

We warn the reader that we may, by abusing the notation, denote $\overline {\cdot }$ various involution of algebras (for example, complex conjugation on quadratic imaginary fields), but it will be always clear from the context what is meant. The trace and the norm are defined by $\mathrm {tr}(x)=x+\overline {x},N(x)=x\overline {x}$ for $x\in \mathbb {B}$ . As usual, we write $M_n(\mathbb {B})$ for the set of $n\times n$ matrices with entries in $\mathbb {B}$ . We also use the notation $\mathbb {B}_n^m$ for the set of $m\times n$ matrices with entries in $\mathbb {B}$ . For $X\in M_n(\mathbb {B})$ , we write $X^{\ast }={^t\!\overline {X}},\hat {X}=(X^{\ast })^{-1}$ for the conjugate transpose and its inverse (if makes sense).

Identify $\zeta ,\xi $ with $\sqrt {\alpha },\sqrt {\beta }\in \overline {\mathbb {Q}}$ , and let $\mathbb {K}=\mathbb {Q}(\xi )$ . We define the embedding

$$\begin{align*}\mathfrak{i}:\mathbb{B}\to M_2(\mathbb{K}),a+b\zeta+c\xi+d\zeta\xi\mapsto\left[\begin{array}{@{}cc@{}} a+c\xi & \alpha(b-d\xi)\\ b+d\xi & a-c\xi \end{array}\right]. \end{align*}$$

One easily checks that for $x\in \mathbb {B}$ ,

$$\begin{align*}\mathfrak{i}(x)^{\ast}=I^{-1}\mathfrak{i}(x^{\ast})I,\,\,\,\,I:=\left[\begin{array}{@{}cc@{}} -\alpha & 0\\ 0 & 1 \end{array}\right], \end{align*}$$

$$\begin{align*}{^t\!\mathfrak{i}(x)}=J^{-1}\mathfrak{i}(x^{\ast})J,\,\,\,\,\,J:=\left[\begin{array}{@{}cc@{}} 0 & -1\\ 1 & 0 \end{array}\right], \end{align*}$$

and $\mathfrak {i}$ induces an isomorphism

$$\begin{align*}\mathfrak{i}:\mathbb{B}\stackrel{\sim}\longrightarrow\{x\in M_2(\mathbb{K}):\overline{x}IJ=IJx\}. \end{align*}$$

We extend this map to an embedding $\mathfrak {i}:M_n(\mathbb {B})\to M_{2n}(\mathbb {K})$ by sending $x=(x_{ij})$ to $(\mathfrak {i}(x_{ij}))$ . Denote $I_n'=\mathrm {diag}[I,{\ldots },I],J_n'=\mathrm {diag}[J,{\ldots },J]$ with n copies. Then, for ${x\in M_n(\mathbb {B})}$ ,

$$\begin{align*}\mathfrak{i}(x)^{\ast}=I_n^{\prime-1}\mathfrak{i}(x^{\ast})I_n',\,\,\,\,\,{^t\!\mathfrak{i}(x)}=J_n^{\prime-1}\mathfrak{i}(x^{\ast})J_n', \end{align*}$$

and $\mathfrak {i}$ induces an isomorphism

$$\begin{align*}\mathfrak{i}:M_n(\mathbb{B})\stackrel{\sim}\longrightarrow\{x\in M_{2n}(\mathbb{K}):\overline{x}I_n'J_n'=I_n'J_n'x\}. \end{align*}$$

For a matrix with entries in quaternion algebra, the determinant $\det $ and the trace $\mathrm {tr}$ will mean the reduced norm and the reduced trace. That is taking the determinant and the trace for its image under $\mathfrak {i}$ . It is well known that the definition of reduced norm and trace is indeed independent of the choice of such embedding and the field $\mathbb {K}$ . Denote

$$\begin{align*}\mathrm{GL}_n(\mathbb{B})=\{g\in M_n(\mathbb{B}):\det(g)\neq 0\},\mathrm{SL}_n(\mathbb{B})=\{g\in M_n(\mathbb{B}):\det(g)=1\}. \end{align*}$$

Let $\mathbb {A}$ be the adele ring of $\mathbb {Q}$ . By a place v, we mean either a finite place corresponding to a prime or the archimedean place $\infty $ . The set of finite places is denoted by $\mathbf {h}$ . We write $\mathbb {A}=\mathbb {A}_{\mathbf {h}}\mathbb {R}$ with finite adeles $\mathbb {A}_{\mathbf {h}}$ and $x=x_{\mathbf {h}}x_{\infty }$ with $x\in \mathbb {A},x_{\mathbf {h}}\in \mathbb {A}_{\mathbf {h}},x_{\infty }\in \mathbb {R}$ . Fix embeddings $\overline {\mathbb {Q}}\to \overline {\mathbb {Q}}_v$ and set $\mathbb {B}_v=\mathbb {B}\otimes _{\mathbb {Q}}\mathbb {Q}_v$ . The previous definition of trace, norm, and determinant naturally extends locally or adelically. Fix a maximal order $\mathcal {O}$ of $\mathbb {B}$ and set $\mathcal {O}_v=\mathcal {O}\otimes _{\mathbb {Z}}\mathbb {Z}_v$ . For a place v, we say v splits if $\mathbb {B}_v\cong M_2(\mathbb {Q}_v)$ . If this is the case, we fix an isomorphism $\mathfrak {i}_v:\mathbb {B}_v\stackrel {\sim }\longrightarrow M_2(\mathbb {Q}_v)$ and assume $\mathfrak {i}_v(\mathcal {O}_v)=M_2(\mathbb {Z}_v)$ for finite place. $\mathbb {B}$ is called indefinite if $B_v$ is split for $v=\infty $ and definite otherwise. In particular, $\mathbb {B}$ is definite if $\alpha ,\beta <0$ , and indefinite otherwise. That is, for the infinite place, by our assumption, $\mathbb {B}_{\infty }$ is the Hamilton quaternion

$$\begin{align*}\mathbb{H}=\mathbb{R}\oplus\mathbb{R}\mathbf{i}\oplus\mathbb{R}\mathbf{j}\oplus\mathbb{R}\mathbf{ij},\,\,\,\,\,\mathbf{i}^2=\mathbf{j}^2=-1,\mathbf{ij}=-\mathbf{ji}, \end{align*}$$

and the map $\mathfrak {i}$ above induces an isomorphism

$$\begin{align*}\mathfrak{i}:M_n(\mathbb{H})\stackrel{\sim}\longrightarrow \{x\in M_{2n}(\mathbb{C}):\overline{x}J_n'=J_n'x\}. \end{align*}$$

In this paper, we consider the following algebraic groups:

$$\begin{align*}G=G_n(\mathbb{Q})=\{g\in\mathrm{SL}_n(\mathbb{B}):g^{\ast}\phi g=\phi\},\phi=\left[\begin{array}{@{}ccc@{}} 0 & 0 & -1_m\\ 0 & \zeta\cdot1_r & 0\\ 1_m & 0 & 0 \end{array}\right]. \end{align*}$$

Here, $n=2m+r$ and we assume that $m>1$ is the global Witt index of the group G. Such a group is usually called a quaternionic unitary group, and has $\mathbb {Q}$ -rank equal to m. For a split place v, $G(\mathbb {Q}_v)$ is isomorphic to an orthogonal group. That is,

(2.1) $$ \begin{align} G(\mathbb{Q}_v)\cong\left\{g\in\mathrm{SL}_{2n}(\mathbb{Q}_v):{^t\!g}\left[\begin{array}{@{}ccc@{}} 0 & 0 & 1_{m_v}\\ 0 & \theta & 0\\ 1_{m_v} & 0 & 0 \end{array}\right]g=\left[\begin{array}{@{}ccc@{}} 0 & 0 & 1_{m_v}\\ 0 & \theta & 0\\ 1_{m_v} & 0 & 0 \end{array}\right]\right\}, \end{align} $$

with some anisotropic matrix $\theta ={^t\!\theta }\in \mathrm {SL}_{r_v}(\mathbb {Q}_v)$ (that is, the corresponding quadratic form does not represent zero), and $2n=2m_v+r_v$ for some positive integers $m_v,r_v$ with $m_v \geq 2m$ and $r_v \leq 2r$ . In particular, $G(\mathbb {Q}_v)$ is totally isotropic if $r=0$ or (using [Reference Shimura33, Lemma 1.7]) if $\alpha \in \mathbb {Q}_v^{\times 2}$ . In these two cases, we have $m_v=n, r_v=0$ , and

$$\begin{align*}G(\mathbb{Q}_v)\cong\left\{g\in\mathrm{SL}_{2n}(\mathbb{Q}_v):{^t\!g}\left[\begin{array}{@{}ccc@{}} 0 & 1_{n}\\ 1_{n} & 0 \end{array}\right]g=\left[\begin{array}{@{}ccc@{}} 0 & 1_{n}\\ 1_{n} & 0 \end{array}\right]\right\}. \end{align*}$$

We remark here that the condition on m being the Witt index of our group G implies that $r \leq 3$ . Indeed, using a result in [Reference Hijikata13], we know that for $r \geq 4$ , $\zeta \cdot 1_r$ is isotropic if and only if it is locally isotropic for all finite places v and infinity. But the latter (i.e., locally isotropic for all finite places and infinity) is always the case for $r \geq 4$ . Indeed, for v split, this follows from [Reference Shimura33, Theorem 7.6] and for v nonsplit (including $\infty $ ) from [Reference Tsukamoto34].

We will discuss the local archimedean group $G(\mathbb {R})$ and the associated symmetric spaces in the next subsection.

We fix an integral two-sided ideal $\mathfrak {n}=(\mathcal {N})$ of $\mathcal {O}$ generated by $\mathcal {N}=\prod _vp_v^{n_v}\in \mathbb {Z}$ . We define an open compact subgroup $K_1(\mathfrak {n})\subset G(\mathbb {A}_{\mathbf {h}})$ by $K_1(\mathfrak {n})=\prod _vK_v$ , where

$$\begin{align*}K_v=\left\{\gamma=\left[\begin{array}{@{}ccc@{}} a & b & c\\ g & e & f\\ h & l & d \end{array}\right]\in G(\mathcal{O}_v):\gamma\equiv\left[\begin{array}{@{}ccc@{}} 1_m & \ast & \ast\\ 0 & 1_r & \ast\\ 0 & 0 & 1_m \end{array}\right]\text{ mod }p_v^{n_v}\right\}. \end{align*}$$

It is well known (see, for example, [Reference Platonov and Rapinchuk22, p. 251]) that we have a finite decomposition

$$\begin{align*}G(\mathbb{A})=\bigcup_{j}G(\mathbb{Q})t_jK_1(\mathfrak{n})G(\mathbb{R}). \end{align*}$$

Moreover, thanks to the weak approximation which is valid for our group (see [Reference Platonov and Rapinchuk22, Proposition 7.11]), we can take $t_j$ such that $(t_j)_v=1$ for $v|\mathfrak {n}$ (compare with [Reference Shimura31, Lemma 8.12]). For finite places v not in the support of $\mathfrak {n}$ , the Iwasawa decomposition is valid, and hence we can take $t_j$ to be upper triangular. Let $\Gamma _1^j=t_jK_1(\mathfrak {n})t_j^{-1}\cap G(\mathbb {Q})$ . We can take $t_0=1$ so that

$$\begin{align*}\Gamma_1^0=\Gamma_1(\mathcal{N})=\left\{\gamma=\left[\begin{array}{@{}ccc@{}} a & b & c\\ g & e & f\\ h & l & d \end{array}\right]\in G(\mathcal{O}):\gamma\equiv\left[\begin{array}{@{}ccc@{}} 1_m & \ast & \ast\\ 0 & 1_r & \ast\\ 0 & 0 & 1_m \end{array}\right]\text{ mod } \mathcal{N}\right\}. \end{align*}$$

2.2 Symmetric spaces

2.2.1 Abstract symmetric spaces

To motivate our definition of symmetric spaces, we start with a rather general and abstract setting before giving explicit realizations of our symmetric spaces. Let $\mathfrak {i}$ be any embedding $M_n(\mathbb {H})\to M_{2n}(\mathbb {C})$ . Then, by the Skolem–Noether theorem, there exists $\alpha \in M_{2n}(\mathbb {C})$ with $\alpha \alpha ^{\ast }=1$ such that ${^t\!\mathfrak {i}}(x)=\alpha \mathfrak {i}(x^{\ast })\alpha ^{-1}$ . Let $\Phi \in \mathrm {GL}_n(\mathbb {B})$ be a skew-hermitian form similar to $\phi $ above, that is, $\Phi = \gamma ^* \phi \gamma $ for some $\gamma \in \mathrm {GL}_n(\mathbb {B})$ . Then the group $G(\mathbb {R})$ is isomorphic to

$$\begin{align*}\mathcal{G}=\{g\in\mathrm{GL}_{2n}(\mathbb{C}):g^{\ast}Hg=H,{^t\!g}Kg=K\}, \end{align*}$$

with $H=\mathfrak {i}(\Phi ),K=\alpha ^{-1}\mathfrak {i}(\Phi )$ . We call it a realization of $G(\mathbb {R})$ . Suppose that we are given two such data $(\mathfrak {i}_1,\Phi _1,H_1,K_1,\mathcal {G}_1)$ and $(\mathfrak {i}_2,\Phi _2,H_2,K_2,\mathcal {G}_2)$ with $\Phi _1=S^{\ast }\Phi _2 S$ . Again, by Skolem–Noether, there exists $\beta $ with $\beta \beta ^{\ast }=1$ such that $\mathfrak {i}_1(x)=\beta ^{-1}\mathfrak {i}_2(x)\beta $ . Put $R=\mathfrak {i}_2(S)\beta $ , then $H_1=R^{\ast }H_2R,K_1={^t\!R}K_2R$ . Therefore, $g\mapsto RgR^{-1}$ gives isomorphism $\mathcal {G}_1\cong \mathcal {G}_2$ .

Following [Reference Pyateskii-Shapiro23], we will define the associated symmetric space via its Borel embedding into its compact dual symmetric space. In our case, we have that the semisimple compact dual of our group is the group $\mathrm {SO}(2n)$ (see [Reference Helgason12, p. 330]), and the corresponding dual symmetric space is $\mathrm {SO}(2n)/\mathrm {U}(n)$ . This space may be identified (see, for example, [Reference Shimura28, p. 6]) with the space $V = L/ GL_{n}(\mathbb {C})$ where

$$\begin{align*}L = \{ U \in\mathbb{C}_n^{2n}: \,\,\, {^t\!U}KU=0\}. \end{align*}$$

We set

$$\begin{align*}\Omega=\{U\in\mathbb{C}^{2n}_{n}:-iU^{\ast}HU>0,{^t\!U}KU=0\}\subset L, \end{align*}$$

with the action of $\mathrm {GL}_n(\mathbb {C})$ by right multiplication and $\mathcal {G}$ by left multiplication. The symmetric space $\mathcal {H}$ is defined as

$$\begin{align*}\mathcal{H}=\{z\in\mathbb{C}_n^{2n}:U(z)\in\Omega\}, \,\,\,\,U(z):=\left[\begin{array}{@{}c@{}} z\\ u_0 \end{array}\right], \end{align*}$$

for some fixed suitable $u_0$ , which we make explicit later. The following lemma is a direct consequence of our definition for $\mathcal {H}$ .

Lemma 2.1 There is a bijection $\mathcal {H}\times \mathrm {GL}_n(\mathbb {C})\to \Omega $ given by $z\times \lambda =U(z)\lambda $ .

Note that $\mathcal {G}$ acts on $\Omega $ by left multiplication. By the above lemma, it follows that for any element $\alpha \in \mathcal {G}$ , we can find a $z'\in \mathcal {H}$ and an $\lambda (\alpha ,z)\in \mathrm {GL}_n(\mathbb {C})$ such that

$$\begin{align*}\alpha U(z)=U(z')\lambda(\alpha,z). \end{align*}$$

We then define the action of $G(\mathbb {R})$ on $\mathcal {H}$ by $\alpha .z:=\alpha z:=z'$ and $\lambda (\alpha ,z)$ satisfies the cocycle relation

$$\begin{align*}\lambda(\alpha_1\alpha_2,z)=\lambda(\alpha_1,\alpha_2 z)\lambda(\alpha_2,z)\text{ for }\alpha_1,\alpha_2\in \mathcal{G},z\in\mathcal{H}. \end{align*}$$

We set $j(\alpha ,z):=\det (\lambda (\alpha ,z))\in \mathbb {C}^{\times }$ . We call $\lambda (\alpha ,z)$ or $j(\alpha ,z)$ automorphy factors. More explicitly, write $\alpha =\left [\begin {array}{@{}cc@{}} a & b\\ c & d \end {array}\right ]$ ,

$$\begin{align*}\alpha U(z)=\left[\begin{array}{@{}c@{}} az+bu_0\\ cz+du_0 \end{array}\right]=\left[\begin{array}{@{}c@{}} (az+bu_0)(cz+du_0)^{-1}u_0\\ u_0 \end{array}\right]u_0^{-1}(cz+du_0). \end{align*}$$

That is, $\alpha z=(az+bu_0)(cz+du_0)^{-1}u_0$ ,and $\lambda (\alpha ,z)=u_0^{-1}(cz+du_0)$ .

For $z_1,z_2\in \mathcal {H}$ , we set

$$\begin{align*}&\eta(z_1,z_2):=iU(z_1)^{\ast}HU(z_2),\\ &\delta(z_1,z_2):=\det(\eta(z_1,z_2))\,\, \text{and} \,\,\eta(z):=\eta(z,z),\delta(z):=\delta(z,z). \end{align*}$$

We now note that

$$\begin{align*}U(z_1)^{\ast}HU(z_2)=\lambda(\alpha,z_1)^{\ast}U(\alpha z_1)^{\ast}HU(\alpha z_2)\lambda(\alpha,z_2) \end{align*}$$

and

$$\begin{align*}iU(\alpha z_1)^{\ast}HU(\alpha z_2)=\left[\begin{array}{@{}cc@{}}\eta(\alpha z_1,\alpha z_2) & \ast\\ \ast & \ast \end{array}\right],iU(z_1)^{\ast}HU( z_2)=\left[\begin{array}{@{}cc@{}}\eta(z_1,z_2) & \ast\\ \ast & \ast \end{array}\right]. \end{align*}$$

In particular, we obtain that

$$\begin{align*}\lambda(\alpha,z_1)^{\ast}\eta(\alpha z_1,\alpha z_2)\lambda(\alpha,z_2)=\eta(z_1,z_2), \end{align*}$$

and after taking the determinant, we have

$$\begin{align*}\overline{j(\alpha,z_1)}\delta(\alpha z_1,\alpha z_2)j(\alpha,z_2)=\delta(z_1,z_2). \end{align*}$$

In particular,

$$\begin{align*}\lambda(\alpha,z)^{\ast}\eta(\alpha z)\lambda(\alpha,z)=\eta(z),\,\,\,\delta(\alpha z)=|j(\alpha,z)|^{-2}\delta(z). \end{align*}$$

We now discuss the relation between different realizations of the symmetric space $\mathcal {H}$ . Given $H_1,K_1$ and $H_2,K_2$ as above, we have seen at the beginning of this subsection that we can find an R such that $H_1=R^{\ast }H_2R,K_1={^t\!R}K_2R$ . We then have an isomorphism $\Omega _1\cong \Omega _2$ given by $U\mapsto RU$ which induces isomorphism $\rho :\mathcal {H}_1\cong \mathcal {H}_2$ . Indeed, for $z_1\in \mathcal {H}_1$ , there exists some $z_2\in \mathcal {H}_2,\mu (z_1)\in \mathrm {GL}_n(\mathbb {C})$ such that

(2.2) $$ \begin{align} R\left[\begin{array}{@{}c@{}}z_1\\u_{01}\end{array}\right]=\left[\begin{array}{@{}c@{}}z_2\\u_{02}\end{array}\right]\mu(z_1), \end{align} $$

and the isomorphism can be given by $\rho (z_1)=z_2$ .

In the following lemma, we write $\rho $ also for the isomorphism $\mathcal {G}_1 \rightarrow \mathcal {G}_2$ given by $\rho (g_1):= R g_1 R^{-1}$ .

Lemma 2.2 Let $\rho :\mathcal {G}_1\to \mathcal {G}_2,\rho :\mathcal {H}_1\to \mathcal {H}_2$ given as above. Then:

(1) $\rho (\alpha z)=\rho (\alpha )\rho (z)$ with $\alpha \in \mathcal {G}_1,z\in \mathcal {H}_1$ .

(2) $\lambda (\rho (\alpha ),\rho (z))=\mu (\alpha z)\lambda (\alpha ,z)\mu (z)^{-1}$ .

(3) $\eta (\rho (z_1),\rho (z_2))=\widehat {\mu (z_1)}\eta (z_1,z_2)\mu (z_2)^{-1}$ for $z_1,z_2\in \mathcal {H}_1$ .

Proof (1) It suffices to prove that $\left [\begin {array}{@{}c@{}}\rho (\alpha z)\\u_{02}\end {array}\right ]=\left [\begin {array}{@{}c@{}}\rho (\alpha )\rho (z)\\u_{02}\end {array}\right ]$ . By definition of the isomorphism and action,

$$\begin{align*}\left[\begin{array}{@{}c@{}}\rho(\alpha z)\\u_{02}\end{array}\right]&=R\left[\begin{array}{@{}c@{}} \alpha z\\ u_{01} \end{array}\right]\mu(\alpha z)^{-1}= R\alpha\left[\begin{array}{@{}c@{}} z\\ u_{01} \end{array}\right]\lambda(\alpha,z)^{-1}\mu(\alpha z)^{-1}\\&=\rho(\alpha)\left[\begin{array}{@{}c@{}} \rho(z)\\ u_{02} \end{array}\right]\mu(z)\lambda(\alpha,z)^{-1}\mu(\alpha z)^{-1}\\&=\left[\begin{array}{@{}c@{}} \rho(\alpha)\rho(z)\\ u_{02} \end{array}\right]\lambda(\rho(\alpha),\rho(z))\mu(z)\lambda(\alpha,z)^{-1}\mu(\alpha z)^{-1}. \end{align*}$$

We must have $\lambda (\rho (\alpha ),\rho (z))\mu (z)\lambda (\alpha ,z)^{-1}\mu (\alpha z)^{-1}=1$ , and our desired result follows which we also obtain (2). (3) can be computed similarly by definition of $\eta $ .

2.2.2 The symmetric space $\mathfrak {Z}$

We now apply the above considerations to some explicit realizations of $G(\mathbb {R})$ . We first define a symmetric space $\mathfrak {Z}$ which can be directly obtained from $G(\mathbb {R})$ . This realization is useful in the computations of the doubling map and Lemma 4.6.

Note that the map $\mathfrak {i}$ defined above induces the following isomorphism on $\mathbb {Q}$ -groups:

$$\begin{align*}\mathfrak{i}:G\stackrel{\sim}\longrightarrow\mathfrak{G}=\{g\in\mathrm{GL}_{2n}(\mathbb{K}):g^{\ast}\Phi g=\Phi,{^t\!g}\Psi g=\Psi\}, \end{align*}$$

$$\begin{align*}\Phi=\left[\begin{array}{@{}cccc@{}} 0 & 0 & 0 & -1_{2m}\\ 0 & 0 & -1_r & 0\\ 0 & 1_r & 0 & 0\\ 1_{2m} & 0 & 0 & 0 \end{array}\right],\Psi=\left[\begin{array}{@{}cccc@{}} 0 & 0 & 0 & J_m'I_m'\\ 0 & -\alpha^{-1} & 0 & 0\\ 0 & 0 & 1_r & 0\\ -J_m'I_m' & 0 & 0 & 0 \end{array}\right]. \end{align*}$$

This induces the following isomorphism on $\mathbb {R}$ -groups:

$$\begin{align*}\mathfrak{i}:G(\mathbb{R})\stackrel{\sim}\longrightarrow G_{\infty}:=\{g\in\mathrm{GL}_{2n}(\mathbb{C}):g^{\ast}\phi_{\infty}g=\phi_{\infty},{^t\!g}\psi_{\infty}g=\psi_{\infty}\}, \end{align*}$$

$$\begin{align*}\phi_{\infty}=\left[\begin{array}{@{}cccc@{}} 0 & 0 & 0 & -1_{2m}\\ 0 & 0 & -1_r & 0\\ 0 & 1_r & 0 & 0\\ 1_{2m} & 0 & 0 & 0 \end{array}\right],\psi_{\infty}=\left[\begin{array}{@{}cccc@{}} 0 & 0 & 0 & J_m'\\ 0 & 1_r & 0 & 0\\ 0 & 0 & 1_r & 0\\ -J_m' & 0 & 0 & 0 \end{array}\right]. \end{align*}$$

Let $K_{\infty }$ be a maximal compact subgroup of $G(\mathbb {R})$ . As in the last subsection,

$$\begin{align*}\Omega=\{U\in\mathbb{C}_n^{2n}:-iU^{\ast}\phi_{\infty}U>0,{^t\!U}\psi_{\infty}U=0\}, \end{align*}$$

and define the symmetric space by

$$\begin{align*}\mathfrak{Z}=\mathfrak{Z}_n=\mathfrak{Z}_{m,r}=\{z\in\mathbb{C}_n^n:U(z)\in\Omega\},U(z)=\left[\begin{array}{@{}c@{}} z\\ u_0 \end{array}\right],u_0=\left[\begin{array}{@{}cc@{}} 0 & 1_r\\ 1_{2m} & 0 \end{array}\right]. \end{align*}$$

Explicitly,

$$\begin{align*}\mathfrak{Z}=\left\{z=(u,v,w):=\left[\begin{array}{@{}cc@{}} u & v\\ w{^t\!v}J_m' & w \end{array}\right]:\begin{array}{@{}c@{}} u\in\mathbb{C}_{2m}^{2m},v\in\mathbb{C}_{r}^{2m},w\in\mathbb{C}_{r^r},i(z^{\ast}-z)>0, \\ {^t\!w}w+1=0,uJ_{m}'+v{^t\!v}-{J^{\prime}_{m}}{^t\!U}=0. \end{array}\right\}. \end{align*}$$

The action of $G_{\infty }$ on $\mathfrak {Z}$ is given by

$$\begin{align*}g z=(az+bu_0)(cz+du_0)^{-1}u_0,\lambda(g,z)=u_0^{-1}(cz+du_0),g=\left[\begin{array}{@{}cc@{}} a & b\\ c & d \end{array}\right]\in G_{\infty}. \end{align*}$$

For $z_1,z_2\in \mathfrak {Z}$ , we set $\eta (z_1,z_2)=i(z_1^{\ast }-z_2),\delta (z_1,z_2)=\det (\eta (z_1,z_2))$ and $\eta (z)=\eta (z,z),\delta (z)=\delta (z,z)$ . We will take $z_0=i\cdot 1_n$ to be the origin of $\mathfrak {Z}$ and $K_{\infty }$ the subgroup of $G_{\infty }$ fixing $z_0$ . Then $g\mapsto \lambda (g,z_0)$ gives an isomorphism $K_{\infty }\cong U(n)=\{g\in \mathrm {GL}_n(\mathbb {C}):g^{\ast }g=1_n\}$ and our symmetric space $\mathfrak {Z}\cong G_{\infty }/K_{\infty }$ . We note that we are using the same notation $K_{\infty }$ for maximal subgroup of $G_{\infty }$ and its preimage in $G(\mathbb {R})$ .

2.2.3 The symmetric spaces $\mathfrak {H}$ and $\mathfrak {B}$

We now give another two useful realizations. They are much simpler than the symmetric space $\mathfrak {Z}$ and are useful in studying CM-points in Section 5.1. However, as the isomorphism between $G_{\infty }$ above and $G_{\infty }'$ below is rather complicated, the action of the $\mathbb {Q}$ -group G on the symmetric space is difficult to compute.

Note that G is isomorphic to

$$\begin{align*}G'=\{g\in\mathrm{GL}_n(\mathbb{B}):g^{\ast}\phi'g=\phi'\},\phi'=\left[\begin{array}{@{}ccc@{}} \zeta\cdot 1_m & 0 & 0\\ 0 & \zeta\cdot 1_r & 0\\ 0 & 0 & -\zeta\cdot 1_m \end{array}\right]. \end{align*}$$

By changing rows and columns, the map $\mathfrak {i}$ induces isomorphism

$$\begin{align*}\mathfrak{i}:G'\stackrel{\sim}\longrightarrow\mathfrak{G}'=\{g\in\mathrm{GL}_{2n}(\mathbb{K}):g^{\ast}\Phi' g=\Phi',{^t\!g}\Psi' g=\Psi'\}, \end{align*}$$

$$\begin{align*}\Phi'=J_n,\Psi'=\mathrm{diag}[1,1,1,-\alpha,-\alpha,-\alpha], \end{align*}$$

and

$$\begin{align*}\mathfrak{i}:G'(\mathbb{R})\cong G_{\infty}':=\{g\in\mathrm{GL}_{2n}(\mathbb{C}):g^{\ast}J_ng=J_n,{^t\!g}g=1_{2n}\}. \end{align*}$$

Take $u_0=1$ , and the symmetric space associated with this group is

$$\begin{align*}\mathfrak{H}=\mathfrak{H}_n=\{z\in\mathbb{C}_n^n:{^t\!z}z+1=0,i(z^{\ast}-z)>0\}. \end{align*}$$

This is an unbounded realization of type-D domain in [Reference Lan17]. The action of $G_{\infty }'$ on $\mathfrak {H}$ and the automorphy factor is given by

$$\begin{align*}gz=(az+b)(cz+d)^{-1},\lambda(g,z)=cz+d,g=\left[\begin{array}{@{}cc@{}} a & b\\ c & d \end{array}\right]. \end{align*}$$

For $z_1,z_2\in \mathfrak {H}$ , we set $\eta (z_1,z_2)=i(z_1^{\ast }-z_2)$ . We take $z_0=i_n:=i\cdot 1_n$ to be the origin of $\mathfrak {H}$ and $K_{\infty }'$ the subgroup of $G_{\infty }'$ fixing $z_0$ . Since $\eta (gz_0)=\eta (z_0)=2$ for $g\in K_{\infty }$ , $g\mapsto \lambda (g,z_0)$ gives an isomorphism $K_{\infty }'\cong U(n)$ and thus $\mathfrak {H}\cong G_{\infty }'/K_{\infty }'$ .

Let $T'=\frac {1}{\sqrt {2}}\left [\begin {array}{@{}cc@{}} i & -i\\ 1 & 1 \end {array}\right ]$ , and sending $g\mapsto T^{\prime -1}gT'$ , we have isomorphism

$$\begin{align*}\mathfrak{i}_{\infty}":G_{\infty}'\stackrel{\sim}\longrightarrow G_{\infty}"=\{g\in\mathrm{GL}_{2n}(\mathbb{C}):g^{\ast}\phi_{\infty}"g=\phi_{\infty}",{^t\!g}\psi_{\infty}"g=\psi_{\infty}"\}, \end{align*}$$

with

$$\begin{align*}\phi_{\infty}"=\left[\begin{array}{@{}cc@{}} i_n & 0\\ 0 & -i_n \end{array}\right],\psi_{\infty}"=\left[\begin{array}{@{}cc@{}} 0 & -i_n\\ -i_n & 0 \end{array}\right]. \end{align*}$$

Take $u_0=1$ , and the symmetric space associated with this group is defined as

$$\begin{align*}\mathfrak{B}=\mathfrak{B}_n=\{z\in\mathbb{C}_n^n:{^t\!z}=-z,zz^{\ast}<1_n\}. \end{align*}$$

This is a bounded domain of type $\mathfrak {R}_{\mathrm {III}}$ in [Reference Hua14]. The action of $G_{\infty }"$ on $\mathfrak {B}$ and the automorphy factor is given by

$$\begin{align*}gz=(az+b)(cz+d)^{-1},\lambda(g,z)=cz+d,g=\left[\begin{array}{@{}cc@{}} a & b\\ c & d \end{array}\right]. \end{align*}$$

For $z_1,z_2\in \mathfrak {H}$ , we set $\eta (z_1,z_2)=i(z_1^{\ast }z_2-1)$ . We take $z_0=0$ to be the origin of $\mathfrak {B}$ and $K_{\infty }"$ the subgroup of $G_{\infty }"$ fixing $z_0$ . Since $\eta (gz_0)=\eta (z_0)=-i$ for $g\in K_{\infty }$ , $g\mapsto \lambda (g,z_0)$ gives an isomorphism $K_{\infty }"\cong U(n)$ and thus $\mathfrak {H}\cong G_{\infty }"/K_{\infty }"$ . The relation between $\mathfrak {H}$ and $\mathfrak {B}$ can be given explicitly by the Cayley transform

$$\begin{align*}\mathfrak{H}\stackrel{\sim}\longrightarrow\mathfrak{B}:z\mapsto(z-i)(z+i)^{-1}. \end{align*}$$

Let $z_1,z_2\in \mathfrak {B}_n,\alpha \in G(\mathbb {R})$ as above, and $dz=(dz_{hk})$ be a matrix of the same shape as $z\in \mathbb {C}_n^n$ whose entries are $1$ -forms $dz_{hk}$ . Comparing

$$\begin{align*}\left[\begin{array}{@{}cc@{}} z_1 & 1\\ 1 & -\overline{z}_1 \end{array}\right]^{\ast}\left[\begin{array}{@{}cc@{}} 1_n & 0\\ 0 & -1_n \end{array}\right]\left[\begin{array}{@{}cc@{}} z_2 & 1\\ 1 & -\overline{z}_2 \end{array}\right]=\left[\begin{array}{@{}cc@{}} z_1^{\ast}z_2-1 & z_1^{\ast}+\overline{z}_2\\ z_2+{}^t\!z_1 & 1-{}^t\!z_1\overline{z}_2 \end{array}\right]=\left[\begin{array}{@{}cc@{}}z_1^{\ast}z_2-1 & z_1^{\ast}-z_2^{\ast}\\ z_2-z_1 & 1-{}^t\!z_1\overline{z}_2\end{array}\right], \end{align*}$$

$$\begin{align*}\left[\begin{array}{@{}cc@{}} \alpha z_1 & 1\\ 1 & -\overline{\alpha z}_1 \end{array}\right]^{\ast}\left[\begin{array}{@{}cc@{}} 1_n & 0\\ 0 & -1_n \end{array}\right]\left[\begin{array}{@{}cc@{}} \alpha z_2 & 1\\ 1 & -\overline{\alpha z}_2 \end{array}\right]=\left[\begin{array}{@{}cc@{}}(\alpha z_1)^{\ast}(\alpha z_2)-1 & (\alpha z_1)^{\ast}-(\alpha z_2)^{\ast}\\ \alpha z_2-\alpha z_1 & 1-{}^t\!(\alpha z_1)(\overline{\alpha z}_2)\end{array}\right], \end{align*}$$

and using the fact (which can be obtained from the property of $U(z)$ )

$$\begin{align*}\alpha\left[\begin{array}{@{}cc@{}} z & 1\\ 1 & -\overline{z} \end{array}\right]=\left[\begin{array}{@{}cc@{}} \alpha z & 1\\ 1 & -\overline{\alpha z} \end{array}\right]\left[\begin{array}{@{}cc@{}} \lambda(\alpha,z) & 0\\ 0 &\overline{\lambda(\alpha,z)} \end{array}\right], \end{align*}$$

we have

$$\begin{align*}\alpha z_2-\alpha z_1={^t\!\lambda}(\alpha,z_1)^{-1}(z_2-z_1)\lambda(\alpha,z_2)^{-1}. \end{align*}$$

Therefore,

$$\begin{align*}d(\alpha z)={^t\!\lambda(\alpha,z)}^{-1}\cdot dz\cdot \lambda(\alpha,z)^{-1}. \end{align*}$$

Since the jacobian of the map $z\mapsto \alpha z$ is $j(\alpha ,z)^{-n+1}$ , the differential form

$$\begin{align*}\mathbf{d}z=\delta(z)^{-n+1}\prod_{h\leq k}[(i/2)dz_{hk}\wedge d\overline{z}_{hk}] \end{align*}$$

is an invariant measure. If we have another realization $\mathcal {H}$ (e.g., $\mathfrak {Z,H}$ ) with identification $\rho :\mathcal {H}\to \mathcal {B}$ , we then define $\mathbf {d}z:=\mathbf {d}(\rho (z))$ with $z\in \mathcal {H}$ to be the differential form on $\mathcal {H}$ . Clearly, this is also an invariant measure.

2.3 Doubling embedding

We keep the notation as before and consider two groups

$$\begin{align*}G_{n_1}=\left\{g\in\mathrm{GL}_{n_1}(\mathbb{B}):g^{\ast}\phi_1 g=\phi_1\right\},\phi_1=\left[\begin{array}{@{}ccc@{}} 0 & 0 & -1_{m_1}\\ 0 & \zeta\cdot 1_r & 0\\ 1_{m_1} & 0 & 0 \end{array}\right], \end{align*}$$

$$\begin{align*}G_{n_2}=\left\{g\in\mathrm{GL}_{n_2}(\mathbb{B}):g^{\ast}\phi_2 g=\phi_2\right\},\phi_2=\left[\begin{array}{@{}ccc@{}} 0 & 0 & -1_{m_2}\\ 0 & \zeta\cdot 1_r & 0\\ 1_{m_2} & 0 & 0 \end{array}\right], \end{align*}$$

with $n_1=2m_1+r,n_2=2m_2+r$ . We always assume that $m_1\geq m_2>0$ . We set $N=n_1+n_2 = 2m_1+2m_2+2r$ , and consider the map

$$\begin{align*}G_{n_1}\times G_{n_2}\to G^{\omega}=\{g\in\mathrm{GL}_{N}(\mathbb{B}):g^{\ast}\omega g=\omega\},\omega=\left[\begin{array}{@{}cc@{}} \phi_1 & 0\\ 0 & -\phi_2 \end{array}\right], \end{align*}$$

by sending $g_1\times g_2\mapsto \mathrm {diag}[g_1,g_2]$ . Note that $R^{\ast }\omega R=J_{N/2} := \left [\begin {array}{@{}cc@{}} 0 & -1_{N/2}\\ 1_{N/2} & 0 \end {array}\right ]$ ,

with

$$\begin{align*}R=\left[\begin{array}{@{}cccccc@{}} 1_{m_1} & 0 & 0 & 0 & 0 & 0\\ 0 & 1/2 & 0 & 0 &-\zeta^{-1} & 0\\ 0 & 0 & 0 & 1_{m_2} & 0 & 0\\ 0 & 0 & -1_{m_1} & 0 & 0 & 0\\ 0 & -1/2 & 0 & 0 & -\zeta^{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 1_{m_2} \end{array}\right]. \end{align*}$$

Composing the above map with $g\mapsto R^{-1}gR$ , we obtain an embedding

$$\begin{align*}\rho:G_{n_1}\times G_{n_2}\to G^{\omega}\to G_N=\{g\in\mathrm{GL}_{N}(\mathbb{B}):g^{\ast}J_{N/2} g=J_{N/2}\}. \end{align*}$$

We can thus view $G_{n_1}\times G_{n_2}$ as a subgroup of $G_N$ . To ease notation, we write $G_{n_1}\times G_{n_2}$ as its image in $G_N$ under $\rho $ and $\beta \times \gamma $ for $\beta \in G_{n_1},\gamma \in G_{n_2}$ as an element in $G_N$ under the embedding $\rho $ .

Now we consider a special case of this embedding, namely the case where ${m_1=m_2=m,n_1=n_2=n}$ . To ease the notation, we always omit the subscript “n” and keep the subscript $N=2n$ . The embedding $\rho $ then induces

$$\begin{align*}\rho:G_{\infty}\times G_{\infty}\to G_{N\infty},g_1\times g_2\mapsto R^{-1}\mathrm{diag}[g_1,g_2]R, \end{align*}$$

with

$$\begin{align*}R=R_{\infty}=\left[\begin{array}{@{}cccccccc@{}} 1_{2m} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1/2 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 1/2 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1_{2m} & 0 & 0 & 0\\ 0 & 0 & 0 & -1_{2m} & 0 & 0 & 0 & 0\\ 0 & -1/2 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & -1/2 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1_{2m} \end{array}\right], \end{align*}$$

where the entries with $\pm 1,\pm 1/2$ should be understood as $\pm I_r,\pm \frac {1}{2}I_r$ . Let $\mathfrak {Z},\mathfrak {Z}_N$ be symmetric spaces associated with $G_{\infty },G_{N\infty }$ . We are now going to define an associated embedding of symmetric space $\iota :\mathfrak {Z}\times \mathfrak {Z}\to \mathfrak {Z}_N$ . We first define the embedding

$$\begin{align*}\Omega_1\times\Omega_2\to\Omega_N,U_1\times U_2\mapsto R^{-1}\left[\begin{array}{@{}cc@{}} U_1 & 0\\ 0 & \mathfrak{J}\overline{U_2} \end{array}\right],\mathfrak{J}=\left[\begin{array}{@{}cccc@{}} J_{m}' & 0 & 0 & 0\\ 0 & 0 & -1_r & 0\\ 0 & 1_r & 0 & 0\\ 0 & 0 & 0 & J_{m}' \end{array}\right]. \end{align*}$$

Let $z_1= (u_1,v_1,w_1),z_2=(u_2,v_2,w_2) \in \mathfrak {Z}$ . The image of $U(z_1)\times U(z_2) \in \Omega _1 \times \Omega _2 $ under this map is

$$\begin{align*}\left[\begin{array}{@{}cccc@{}} u_1 & v_1 & 0 & 0\\ w_{1}{}^t\!v_1J_{m}' & w_1 & 0 & 1\\ 0 & 1 & -\overline{w}_2v_2^{\ast}J_{m}' & -\overline{w}_2\\ 0 & 0 & -J_{m}'\overline{u}_2 & -J_{m}'\overline{v}_2\\ 1 & 0 & 0 & 0\\ 0 & \frac{1}{2} & \frac{\overline{w}_2v_2^{\ast}J_{m}'}{2} & \frac{\overline{w}_2}{2}\\ \frac{-w_1{}^t\!v_1J_{m}'}{2} & \frac{-w_1}{2} & 0 & \frac{1}{2}\\ 0 & 0 & J_{m}' & 0 \end{array}\right]=:\left[\begin{array}{@{}c@{}} A(z_1,z_2)\\ B(z_1,z_2) \end{array}\right]. \end{align*}$$

We then define the embedding of symmetric space as

$$\begin{align*}\iota:\mathfrak{Z}\times\mathfrak{Z}\to\mathfrak{Z}_N,\,\,\,\,z_1\times z_2\mapsto A(z_1,z_2)B(z_1,z_2)^{-1}S, \end{align*}$$

where $S:=\mathrm {diag}[1,1/2,1/2,1]$ .

Explicitly, if we write $w^{-1}_0:=1+w_1\overline {w}_2,w_0':=1-w_1\overline {w}_2$ , then

$$\begin{align*}\iota(z_1,z_2)= \end{align*}$$

$$\begin{align*}\left[\begin{array}{@{}cccc@{}} u_1-v_1\overline{w}_2w_0w_1{}^t\!v_1J_{m}' & v_1w_1^{-1}w_0w_1 & -v_1\overline{w}_2w_0 & -v_1w_1^{-1}w_0w_1\overline{w}_2v_2^{\ast}\\[2pt]2w_0w_1{}^t\!v_1J_{m}' & 2w_0w_1 & w_0'w_0 & -2w_0w_1\overline{w}_2v_2^{\ast}\\[2pt]-2\overline{w}_2w_0w_1{}^t\!v_1J_{m}' & w_1^{-1}w_0'w_0w_1 & -2\overline{w}_2w_0 & -2w_1^{-1}w_0w_1\overline{w}_2v_2^{\ast}\\[2pt] -J_{m}'\overline{v}_2w_0w_1{}^t\!v_1J_{m}' & -J_{m}'\overline{v}_2w_0w_1 & -J_{m}'\overline{v}_2w_0 & -J_{m}'\overline{u}_2J_{m}^{\prime-1}+J_{m}'\overline{v_2}w_0w_1\overline{w}_2v_2^{\ast} \end{array}\right]. \end{align*}$$

Here, we note that we have “normalized” our embedding by S so that $\iota $ maps the origin of $\mathfrak {Z}\times \mathfrak {Z}$ to the “origin” of $\mathfrak {Z}_N$ . That is, $\iota (z_0\times z_0)=i\cdot 1_N=:Z_0$ , where $z_0$ and $Z_0$ are the origins of $\mathfrak {Z}$ and $\mathfrak {Z}_N$ , respectively.

For example, in the case where $r=0$ , then the embedding is quite simple, namely $\iota (z_1,z_2) = \mathrm {diag}[u_1,-u^*_2]$ , where in the case of $r=1$ , it is given by (note that in this case $w_1=w_2 =i$ )

$$\begin{align*}\iota(z_1,z_2) = \left[\begin{array}{@{}cccc@{}} u_1-\frac{1}{2}v_1{}^t\!v_1J_m' & \frac{1}{2}v_1 & -\frac{i}{2}v_1 & \frac{i}{2}v_1v_2^{\ast}\\[2pt] i{}^t\!v_1J_m' & i & 0 & -v_2^{\ast}\\[2pt] -{}^t\!v_1J_{m}' & 0 & i & iv_2^{\ast}\\[2pt] -\frac{i}{2}J_{m}'\overline{v}_2{}^t\!v_1J_{m}' & -\frac{i}{2}J_m'\overline{v}_2 & -\frac{1}{2}J_{m}'\overline{v}_2 & -u_2^{\ast}-\frac{1}{2}J_m'\overline{v}_2v_2^{\ast} \end{array}\right]. \end{align*}$$

We now show that the embedding of the symmetric spaces is compatible with the embedding of the groups.

Proof By our definition of embedding and the action,

$$\begin{align*}\left[\begin{array}{@{}c@{}} \iota(g_1z_2,g_2z_2)\\ 1 \end{array}\right]=R^{-1}\left[\begin{array}{@{}cc@{}} U(g_1z_1) & 0\\ 0 & \mathfrak{J}\overline{U(g_2z_2)} \end{array}\right]B(g_1z_1,g_2z_2)^{-1}S \end{align*}$$

$$\begin{align*}=R^{-1}\left[\begin{array}{@{}cc@{}} g_1 & 0\\ 0 & g_2 \end{array}\right]RR^{-1}\left[\begin{array}{@{}cc@{}} U(z_1) & 0\\ 0 & \mathfrak{J}\overline{U(z_2)} \end{array}\right]\left[\begin{array}{@{}cc@{}} \lambda(g_1,z_1) & 0\\ 0 & \overline{\lambda(g_2,z_2)} \end{array}\right]^{-1}B(g_1z_1,g_2z_2)^{-1}S \end{align*}$$

$$\begin{align*}=\left[\begin{array}{@{}c@{}} \rho(g_1,g_2)\iota(z_1,z_2)\\ 1 \end{array}\right]\lambda(\rho(g_1,g_2),\iota(z_1,z_2))\times \end{align*}$$

$$\begin{align*}S^{-1}B(z_1,z_2)\left[\begin{array}{@{}cc@{}} \lambda(g_1,z_1) & 0\\ 0 & \overline{\lambda(g_2,z_2)} \end{array}\right]^{-1}B(g_1z_1,g_2z_2)^{-1}S. \end{align*}$$

We must have

$$\begin{align*}\lambda(\rho(g_1,g_2),\iota(z_1,z_2))S^{-1}B(z_1,z_2)\left[\begin{array}{@{}cc@{}} \lambda(g_1,z_1) & 0\\ 0 & \overline{\lambda(g_2,z_2)} \end{array}\right]^{-1}B(g_1z_1,g_2z_2)^{-1}S=1, \end{align*}$$

and the desired result follows. Taking the determinant, we also obtain (2).

Suppose $z_1=g_1z_0,z_2=g_2z_0$ for $g_1\in G_{1\infty },g_2\in G_{2\infty }$ with $z_0$ the origin of $\mathfrak {Z}_1$ or $\mathfrak {Z}_2$ . Then

$$\begin{align*}\delta(\iota(z_1,z_2))&=\delta(\rho(g_1,g_2)\iota(z_0,z_0))=|j(\rho(g_1,g_2),\iota(z_0,z_0))|^{-2}\delta(\iota(z_0,z_0))\\[6pt]&=|j(g_1,z_0)j(g_2,z_0)\det(B(z_0,z_0)^{-1}B(g_1z_0,g_2z_0))|^{-2}\delta(z_0)\delta(z_0)\\[6pt]&=|\det(B(z_1,z_2))|^{-2}\delta(z_1)\delta(z_2).\\[-34pt] \end{align*}$$

3.1 Modular forms and Fourier–Jacobi expansion

Fix an integral ideal $\mathfrak {n}$ as in the previous section.

Definition 3.1 A holomorphic function $f:\mathfrak {Z}\to \mathbb {C}$ is called a quaternionic modular form for a congruence subgroup $\Gamma $ and weight $k \in \mathbb {N}$ if for all $\gamma \in \Gamma $ ,

$$\begin{align*}f(\gamma z)=j(\gamma,z)^kf(z). \end{align*}$$

We note here that since we are assuming $m \geq 2$ , we do not need any condition at the cusps due to Koecher’s principle (see [Reference Krieg16, Lemma 1.5]).

Denote the space of such functions by $M_k(\Gamma )$ . Here, we are using the realization $(G_{\infty },\mathfrak {Z})$ for our symmetric space. In fact, the definition is independent of the choice of realizations in the following sense. If we choose another realization $\mathcal {H}$ (e.g., $\mathfrak {H},\mathfrak {B}$ ) with identification $\rho :\mathcal {H}\to \mathfrak {Z}$ , then with notation as in equation (2.2), to a function ${f:\mathfrak {Z}\to \mathbb {C}}$ , we associate a function g on $\mathcal {H}$ by setting $g(z)=\det (\mu (z))^{-k}f(\rho (z))$ . Then $f:\mathfrak {Z}\to \mathbb {C}$ is a modular form if and only if $g:\mathcal {H}\to \mathbb {C}$ is a modular form.

We write $S:=S(\mathbb {Q}):=\{X\in M_m(\mathbb {B}):X^{\ast }=X\}$ for the (additive) algebraic group of hermitian matrices. We use $S^+$ (resp. $S_+$ ) denote the subgroup of S consisting of positive-definite (resp. positive) elements. For a fractional ideal $\mathfrak {a}\subset \mathbb {B}$ , we set ${S(\mathfrak {a})=S\cap M_m(\mathfrak {a})}$ . Denote $e_{\infty }(z):=\mathrm {exp}(2\pi iz)$ for $z\in \mathbb {C}$ and $\lambda =\frac {1}{2}\mathrm {tr}$ . For $f\in M_k(\Gamma )$ and $\gamma \in G$ , there is a Fourier–Jacobi expansion of the form

$$\begin{align*}(f|_k\gamma)(z)=\sum_{\tau\in S_+}c(\tau,\gamma,f;v,w)e(\lambda(\mathfrak{i}(\tau)u)),z=(u,v,w)\in\mathfrak{Z}. \end{align*}$$

In particular, for $\gamma =1$ , we simply write

$$\begin{align*}f(z)=\sum_{\tau\in S_+}c(\tau;v,w)e(\lambda(\mathfrak{i}(\tau)u)). \end{align*}$$

We call f a cusp form if $c(\tau ,\gamma ,f;v,w)=0$ for every $\gamma \in G$ and every $\tau $ such that $\det (h)=0$ . The space of cusp form is denoted by $S_k(\Gamma )$ .

Given a function $\mathbf {f}:G(\mathbb {A})\to \mathbb {C}$ , we can, by abusing the notation, also view it as a function $\mathbf {f}:G(\mathbb {A}_{\mathbf {h}})\times \mathfrak {Z}\to \mathbb {C}$ by setting $\mathbf {f}(g_{\mathbf {h}},z):=j(g_z,z_0)^k\mathbf {f}(g_{\mathbf {h}}g_z)$ with $z=g_z\cdot z_0$ .

Definition 3.2 A function $\mathbf {f}:G(\mathbb {A})\to \mathbb {C}$ is called a quaternionic modular form of weight k, level $\mathfrak {n}$ if:

(1) viewed as a function $\mathbf {f}:G(\mathbb {A}_{\mathbf {h}})\times \mathfrak {Z}\to \mathbb {C}$ , $\mathbf {f}(g_{\mathbf {h}},z)$ is holomorphic in z,

(2) for $\alpha \in G(\mathbb {Q}),k_{\infty }\in K_{\infty }$ , and $k\in K_1(\mathfrak {n})$ ,

$$\begin{align*}\mathbf{f}(\alpha gk_{\infty}k)=j(k_{\infty},z_0)^{-k}\mathbf{f}(g), \end{align*}$$

or equivalently,

(2’) viewed as a function $\mathbf {f}:G(\mathbb {A}_{\mathbf {h}})\times \mathfrak {Z}\to \mathbb {C}$ , for $\alpha \in G(\mathbb {Q}),k\in K_1(\mathfrak {n})$ , we have

$$\begin{align*}\mathbf{f}(\alpha g_{\mathbf{h}}k,\alpha z)=j(\alpha,z)^k\mathbf{f}(g_{\mathbf{h}},z). \end{align*}$$

We will denote the space of such functions by $\mathcal {M}_k(K_1(\mathfrak {n}))$ .

We call $\mathbf {f}\in \mathcal {M}_k(K_1(\mathfrak {n}))$ a cusp form if

$$\begin{align*}\int_{U(\mathbb{Q})\backslash U(\mathbb{A})}\mathbf{f}(ug)\mathbf{d}u=0, \end{align*}$$

for all unipotent radicals U of all proper parabolic subgroups of G. The space of cusp forms will be denoted by $\mathcal {S}_k(K_1(\mathfrak {n}))$ .

It is well known that the above two definitions are related by

$$\begin{align*}\mathcal{M}_k(K_1(\mathfrak{n}))\cong\bigoplus_{j}M_k(\Gamma^j_1(N)),\qquad\mathcal{S}_k(K_1(\mathfrak{n}))\cong\bigoplus_{j}S_k(\Gamma^j_1(N)). \end{align*}$$

Write $\mathbf {f}\leftrightarrow (f_0,f_1,{\ldots },f_h)$ for the correspondence under above maps. Here, $f_j(z)=\mathbf {f}(t_j,z)=j(g_z,i)^k\mathbf {f}(t_jz)$ with $z=g_z\cdot i$ .

When $n=2m,r=0$ , then the Fourier–Jacobi expansion becomes the usual Fourier expansion. For $x\in \mathbb {Q}_v,v\in \mathbf {h}$ , define $e_v(x)=e_{\infty }(-y)$ with $y\in \bigcup _{n=1}^{\infty }p^{-n}\mathbb {Z}$ such that $x-y\in \mathbb {Z}_v$ . Set $e_{\mathbb {A}}(x)=e_{\infty }(x_{\infty })\prod _{v\in \mathbf {h}}e_v(x_v)$ . Let $\mathbf {f}\in \mathcal {M}_k(K_1(\mathfrak {n}))$ . For $g\in G(\mathbb {A})$ , we write it as $g=\gamma t_jkp_{\infty }k_{\infty }$ with $\gamma \in G(\mathbb {Q})$ , $k\in K_1(\mathfrak {n}),k_{\infty }\in K_{\infty }$ . Take $t_j$ of the form $\left [\begin {array}{@{}cc@{}} q_j & \sigma _j\hat {q}_j\\ 0 & \hat {q}_j \end {array}\right ]$ with $q_j\in \mathrm {GL}_m(\mathbb {B}_{\mathbf {h}}),\sigma _j\in S(\mathbb {A}_{\mathbf {h}})$ and $p_{\infty }=\left [\begin {array}{@{}cc@{}} q_{\infty } & \sigma _{\infty }\hat {q}_{\infty }\\ 0 & \hat {q}_{\infty } \end {array}\right ]$ with $q_{\infty }\in \mathrm {GL}_m(\mathbb {B}_{\infty }),\sigma _{\infty }\in S(\mathbb {R})$ . Set $q=q_jq_{\infty },\sigma =\sigma _j\sigma _{\infty }$ , then $\mathbf {f}$ has a Fourier expansion of the form

$$\begin{align*}\mathbf{f}(g)=j(k_{\infty},z_0)^{-k}\sum_{\tau\in S}\det(q_{\infty})^{-k}c(\tau,q;\mathbf{f})e_{\infty}(\lambda(q^{\ast}\tau q)z_0)e_{\mathbb{A}}(\lambda(\tau\sigma)). \end{align*}$$

We call $c(\tau ,q;\mathbf {f})$ the Fourier coefficients of $\mathbf {f}$ .

For two modular forms $f,h\in M_k(\Gamma )$ , we define the Petersson inner product by

$$\begin{align*}\langle f,h\rangle=\int_{\Gamma\backslash\mathfrak{Z}}f(z)\overline{h(z)}\delta(z)^k\mathbf{d}z, \end{align*}$$

whenever the integral converges. For example, this is well defined when one of $f,g$ is a cusp form. Adelically, for $\mathbf {f,h}\in \mathcal {M}_k(K_1(\mathfrak {n}))$ , we define

$$\begin{align*}\langle\mathbf{f,h}\rangle=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\mathbf{f}(g)\overline{\mathbf{h}(g)}\mathbf{d}g. \end{align*}$$

Here, $\mathbf {d}g$ , an invariant differential of $G(\mathbb {A})$ , is given as follows: $\mathbf {d}g=\mathbf {d}g_{\mathbf {h}}\mathbf {d}g_{\infty }$ , where $\mathbf {d}g_{h}$ is the canonical measure on $G(\mathbb {A}_{\mathbf {h}})$ normalized such that the volume of $K_1(\mathfrak {n})$ is $1$ and $\mathbf {d}g_{\infty }=\mathbf {d}(g_{\infty }z_0)$ with $\mathbf {d}z$ an invariant differential of $\mathfrak {Z}$ .

Viewing $\mathbf {f,h}$ as functions $\mathbf {f,g}:G(\mathbb {A}_{\mathbf {h}})\times \mathfrak {Z}\to \mathbb {C}$ , we have

$$\begin{align*}\langle\mathbf{f,h}\rangle=\int_{G(\mathbb{Q})\backslash (G(\mathbb{A}_{\mathbf{h}})/K_1(\mathfrak{n})\times\mathfrak{Z})}\mathbf{f}(g,z)\overline{\mathbf{h}(g,z)}\delta(z)^k\mathbf{d}g_{\mathbf{h}}\mathbf{d}z. \end{align*}$$

Again, these integrals are well defined if one of $\mathbf {f,h}$ is a cusp form. If $\mathbf {f}\leftrightarrow (f_j), \mathbf {h}\leftrightarrow (h_j)$ , then

$$\begin{align*}\langle\mathbf{f,h}\rangle=\sum_{j}\langle f_j,h_j\rangle. \end{align*}$$

3.2 Hecke operators and L-functions

In the rest of the paper, we make the assumption that all finite places v with $v\nmid \mathfrak {n}$ are split in $\mathbb {B}$ . We define the groups

$$\begin{align*}E=\prod_{v\in\mathbf{h}}\mathrm{GL}_m(\mathcal{O}_v),\,\,\,M=\{x\in\mathrm{GL}_m(\mathbb{B})_{\mathbf{h}}:x_v\in M_m(\mathcal{O}_v)\}. \end{align*}$$

Let $\mathfrak {X}=\prod _v\mathfrak {X}_v$ be a subgroup of $G(\mathbb {A}_{\mathbf {h}})$ with $\mathfrak {X}_v=K_v$ if $v|\mathfrak {n}$ and $\mathfrak {X}_v=G_v$ if otherwise. Define the Hecke algebra $\mathbb {T}=\mathbb {T}(K_1(\mathfrak {n}),\mathfrak {X})$ be the $\mathbb {Q}$ -algebra generated by double coset $[K_1(\mathfrak {n})\xi K_1(\mathfrak {n})]$ with $\xi \in \mathfrak {X}$ . Given $\mathbf {f}\in \mathcal {M}_k(K_1(\mathfrak {n}))$ , the Hecke operator $[K_1(\mathfrak {n})\xi K_1(\mathfrak {n})]$ acts on $\mathbf {f}$ by

$$\begin{align*}(\mathbf{f}|[K_1(\mathfrak{n})\xi K_1(\mathfrak{n})])(g)=\sum_{y\in Y}\mathbf{f}(gy^{-1}), \end{align*}$$

where Y is a finite subset of $G_{\mathbf {h}}$ such that

$$\begin{align*}K_1(\mathfrak{n})\xi K_1(\mathfrak{n})=\bigcup_{y\in Y}K_1(\mathfrak{n})y. \end{align*}$$

We say that $\mathbf {f}\in \mathcal {S}_k(K_1(\mathfrak {n}))$ is an eigenform if there exist some numbers $\lambda _{\mathbf {f}}(\xi )\subset \mathbb {C}$ called eigenvalues such that

$$\begin{align*}\mathbf{f}|[K_1(\mathfrak{n})\xi K_1(\mathfrak{n})]=\lambda_{\mathbf{f}}(\xi)\mathbf{f}\text{ for all }\xi\in\mathfrak{X}. \end{align*}$$

We use the notation $l(\xi ):=\det (r)$ if $\xi \in G(\mathcal {O}_{\mathbf {h}})\mathrm {diag}[\hat {r},1,r]G(\mathcal {O}_{\mathbf {h}})$ with $r\in M$ . For a Hecke character $\chi $ , we define the series

$$\begin{align*}D(s,\mathbf{f},\chi)=\sum_{\xi\in K_1(\mathfrak{n})\backslash\mathfrak{X}/K_1(\mathfrak{n})}\lambda_{\mathbf{f}}(\xi)\chi^{\ast}(l(\xi))l(\xi)^{-s},\text{ }\mathrm{Re}(s)\gg0. \end{align*}$$

Here, $\chi ^{\ast }$ is the associated Dirichlet character of $\chi $ . We further define the L-function by

(3.1) $$ \begin{align} L(s,\mathbf{f},\chi)=\Lambda_{\mathfrak{n}}(s,\chi)D(s,\mathbf{f},\chi),\qquad\Lambda_{\mathfrak{n}}(s,\chi)=\prod_{i=0}^{n-1}L_{\mathfrak{n}}(2s-2i,\chi^2). \end{align} $$

Here, the subscript $\mathfrak {n}$ means the Euler factors at $v|\mathfrak {n}$ are removed.

Define $\mathbb {T}_v=\mathbb {T}(K_v,\mathfrak {X}_v)$ be the local counterpart of $\mathbb {T}$ so $\mathbb {T}=\otimes ^{\prime }_v\mathbb {T}_v$ . Obviously, $\mathbb {T}_v$ is trivial if $v|\mathfrak {n}$ . For $v\nmid \mathfrak {n}$ , by our assumptions, we can identify the local group $G_v$ with local orthogonal group, as in equation (2.1). Such local Hecke algebra is discussed in [Reference Shimura33] where a Satake map is constructed

$$\begin{align*}\omega:\mathbb{T}_v\to\mathbb{Q}[t_1,...,t_{m_v},t_1^{-1},{\ldots},t_{m_v}^{-1}], \end{align*}$$

where $m_v$ is the local Witt index of $G(\mathbb {Q}_v)$ and $2n=2m_v+r_v$ as in equation (2.1). Given an eigenform $\mathbf {f}$ , the map $\xi \mapsto \lambda _{\mathbf {f}}(\xi )$ induces homomorphism $\mathbb {T}_v\to \mathbb {C}$ which are parameterized by Satake parameters

$$\begin{align*}\alpha_{1,v}^{\pm 1},{\ldots},\alpha_{m_v,v}^{\pm 1}. \end{align*}$$

The L-function then has an Euler product expression

$$\begin{align*}L(s,\mathbf{f},\chi)=\prod_{p\nmid\mathfrak{n}}L_p(s,\mathbf{f},\chi), \end{align*}$$

with $L_p(s,\mathbf {f},\chi )$ given by

$$\begin{align*}\prod_{i=1}^{\frac{r_v}{2}}(1-\chi(p)p^{2i+2m_v-2-2s})^{-1}\prod_{i=1}^{m_v}\left((1-\alpha_{i,p}\chi(p)p^{m_v+r_v-2-s})(1-\alpha_{i,p}^{-1}\chi(p)p^{m_v-s})\right)^{-1}. \end{align*}$$

In particular, if $r=0$ or $\alpha \in \mathbb {Q}_v^{\times 2}$ , then $G(\mathbb {Q}_v)$ is totally isotropic (i.e., $m_v=n,r_v=0$ ) so that

$$\begin{align*}L_p(s,\mathbf{f},\chi)=\prod_{i=1}^n\left((1-\alpha_{i,p}\chi(p)p^{n-2-s})(1-\alpha_{i,p}^{-1}\chi(p)p^{n-s})\right)^{-1}. \end{align*}$$

Here, we write p for the prime corresponding to some place v in the notation above. Finally, it is known that $L(s,\mathbf {f},\chi )$ is absolutely convergent for $Re(s)> 2n-1$ (see [Reference Shimura33, Proposition 17.4]).

4.1 Siegel Eisenstein series and its Fourier expansion

We fix an integer $0\leq t\leq m$ , and for $x\in G$ , we write

$$\begin{align*}x=\left[\begin{array}{@{}ccccc@{}} a_1 & a_2 & b_1 & c_1 & c_2\\ a_3 & a_4 & b_2 & c_3 & c_4\\ g_1 & g_2 & e & f_1 & f_2\\ h_1 & h_2 & l_1 & d_1 & d_2\\ h_3 & h_4 & l_2 & d_3 & d_4 \end{array}\right] \end{align*}$$

with block size $(t,m-t,r,t,m-t)\times (t,m-t,r,t,m-t)$ . The t-Klingen parabolic subgroup of $G_{n}$ is defined as

$$\begin{align*}P_{n}^t=\{x\in G:a_2=g_2=h_2=h_3=h_4=l_2=d_3=0\}. \end{align*}$$

Clearly, we have $P_n^m=G_{n}$ . We define a projection map $\pi _t:P_n^t\to G_{2t+r}$ by

$$\begin{align*}\left[\begin{array}{@{}ccccc@{}} a_1 & 0 & b_1 & c_1 & c_2\\ a_3 & a_4 & b_2 & c_3 & c_4\\ g_1 & 0 & e & f_1 & f_2\\ h_1 & 0 & l_1 & d_1 & d_2\\ 0 & 0 & 0 & 0 & d_4 \end{array}\right]\mapsto\left[\begin{array}{@{}ccc@{}} a_1 & b_1 & c_1\\ g_1 & e & f_1\\ h_1 & l_1 & d_1 \end{array}\right]. \end{align*}$$

In particular, if $r=0$ , then $n=2m$ . In this case, the parabolic subgroup for $t=0$ ,

$$\begin{align*}P_n:=P^0_n=\left\{\left[\begin{array}{@{}cc@{}} a & b\\ c & d \end{array}\right]\in G_{n}:c=0\right\}, \end{align*}$$

is called Siegel parabolic subgroup. We now fix weight $l \in \mathbb {N}$ , and let $\chi $ be a Hecke character whose conductor divides $\mathfrak {n}$ . The Siegel-type Eisenstein series is defined as

$$\begin{align*}\mathbf{E}_l(x,s)=\mathbf{E}^m_l(x,s;\chi)=\sum_{\gamma\in P_n\backslash G_{n}}\varphi(\gamma x,s), \end{align*}$$

with

$$\begin{align*}\varphi(x,s)=\chi_{\mathbf{h}}(\det(d_p))^{-1}j(x,z_0)^{-l}|\det(d_p)|^{-s}_{\mathbf{h}}|j(x,z_0)|^{l-s}, \end{align*}$$

if $x=pkk_{\infty }\in P_n(\mathbb {A})K_1(\mathfrak {n})K_{\infty }$ and $\varphi (x,s)=0$ if otherwise. Let $J_{\mathbf {h}}\in G_{n}(\mathbb {A})$ be an element defined by $J_v=J_m$ for $v\in \mathbf {h}$ and $J_{\infty }=1$ . We set $\mathbf {E}_l^{\ast }(x,s)=\mathbf {E}_l(xJ_{\mathbf {h}}^{-1},s)$ . Then we have the following proposition (see, for example, [Reference Bouganis3]).

Proposition 4.1 $\mathbf {E}_l^{\ast }(x,s)$ has a Fourier expansion of form

$$\begin{align*}\mathbf{E}_l^{\ast}\left(\left[\begin{array}{@{}cc@{}} q & \sigma\hat{q}\\ 0 & \hat{q} \end{array}\right],s\right)=\sum_{h\in S}c(h,q,s)e_{\mathbb{A}}(\lambda(h\sigma)), \end{align*}$$

where $q\in \mathrm {GL}_m(\mathbb {B}_{\mathbb {A}})$ and $\sigma \in S(\mathbb {A})$ . The Fourier coefficient $c(h,q,s)\neq 0$ only if $(q^{\ast }hq)_v\in T(\mathbb {Q}_v)\cap M_m(\mathfrak {n}_v^{-1})$ for all $v\in \mathbf {h}$ . In this case, we have

$$\begin{align*}c(h,q,s)\kern1.2pt{=}\kern1.2pt A(n)\chi(\det(q_{\mathbf{h}}))^{-1}\det(q_{\infty})^s|\det(q)|_{\mathbf{h}}^{2m-1-s}\alpha_{\mathfrak{n}}(q_{\mathbf{h}}^{\ast}hq_{\mathbf{h}},s,\chi)\xi(q_{\infty}q_{\infty}^{\ast},h,s+l,s-l). \end{align*}$$

Here:

(1) $A(n)\in \overline {\mathbb {Q}}^{\times }$ is a constant depending on n.

(2) If h has rank r, then

$$\begin{align*}\alpha_{\mathfrak{n}}(q^{\ast}hq,s,\chi)=\frac{\prod_{i=1}^{m-r}L_{\mathfrak{n}}(2s-4m+2r+2i+1,\chi^2)}{\prod_{i=0}^{m-1}L_{\mathfrak{n}}(2s-2i,\chi^2)}\prod_{p}P_{h,q,p}(\chi^{\ast}(p)p^{-s}), \end{align*}$$

where $P_{h,q,p}(X)\in \mathcal {O}[X]$ and $P_{h,q,p}=1$ if $\det (h)\in 2^{m+1}\mathbb {Z}_p^{\times }$ . Here, p is the prime corresponding to v.

(3) Let p be the number of positive eigenvalues of h, q the number of negative eigenvalues of h, and $t=m-p-q$ , then for $y\in S_{\infty }^+,h\in S_{\infty }$ ,

$$\begin{align*}\xi(y,h,s+l,s-l)=\frac{\Gamma_t(2s-2m+1)}{\Gamma_{m-q}(s+l)\Gamma_{m-p}(s-l)}\omega(y,h,s+l,s-l), \end{align*}$$

where

$$\begin{align*}\Gamma_m(s)=\pi^{m(m-1)}\prod_{i=0}^{m-1}\Gamma(s-2i),m\in\mathbb{Z}, \end{align*}$$

and $\omega $ is holomorphic with respect to $s+l,s-l$ . In particular, when $p=m$ ,

$$\begin{align*}\xi(y,h,2l,0)=2^{2-2m}(2\pi i)^{2ml}\Gamma_m^{-1}(2l)\det(h)^{l-\frac{2m-1}{2}}e(i\lambda(hy)). \end{align*}$$

For $g\in G(\mathbb {A})$ , write it as $g=\gamma t_jkp_{\infty }k_{\infty }$ with $\gamma \in G(\mathbb {Q}),k\in \eta K_0(\mathfrak {n})\eta ^{-1},k_{\infty }\in K_{\infty }$ , and $t_j=\left [\begin {array}{@{}cc@{}} q_j & \sigma _j\hat {q}_j\\ 0 & \hat {q}_j \end {array}\right ],p_{\infty }=\left [\begin {array}{@{}cc@{}} q_{\infty } & \sigma _{\infty }\hat {q}_{\infty }\\ 0 & \hat {q}_{\infty } \end {array}\right ]$ . Set $q=q_jq_{\infty },\sigma =\sigma _j\sigma _{\infty }$ , then by modularity property, $\mathbf {E}_l^{\ast }(g,s)$ can be written as

$$\begin{align*}\mathbf{E}_l^{\ast}(g,s)=j(k_{\infty},z_0)^{-l}\sum_{h}c(h,q,s)e_{\mathbb{A}}(\lambda(h\sigma)), \end{align*}$$

with $c(h,q,s)$ in the above propositions. We are interested in the special values $\mathbf {E}_l^{\ast }(g,s)$ for $s=l$ . From the Fourier expansion, we have the following proposition by counting poles and degree of $\pi $ in those Gamma functions.

Proposition 4.2 Assume that $l>n-1$ . Then the Fourier coefficients $c(h,q,l)\neq 0$ unless $h>0$ and in this case

$$\begin{align*}\det(q_{\infty})^{-l}c(h,q,l)=C\cdot e_{\infty}(i\lambda(q^{\ast}hq)). \end{align*}$$

Here, up to a constant in $\overline {\mathbb {Q}}$ , $C=\chi (\det (q_{\mathbf {h}}))^{-1}|\det (q)|_{\mathbf {h}}^{2m-1-l}$ . In particular, $\mathbf {E}_l^{\ast }(g,l)$ is holomorphic in the sense that when viewed as a function on $G(\mathbb {A}_{\mathbf {h}})\times \mathfrak {Z}$ , $\mathbf {E}_l^{\ast }(g_{\mathbf {h}},z,l)$ is holomorphic in z.

We note, in particular, that the proposition implies that also $\mathbf {E}_l(g,l)$ is holomorphic in the above sense since $\mathbf {E}_l^{\ast }(x,s)=\mathbf {E}_l(xJ_{\mathbf {h}}^{-1},s)$ .

4.2 Coset decompositions

Let

$$\begin{align*}\rho:G_n\times G_n\to G_{N},g_1\times g_2\mapsto R^{-1}\mathrm{diag}[g_1,g_2]R, \end{align*}$$

be the doubling embedding defined before. To ease the notation, we may omit the subscript n. Denote $P_N$ for the Siegel parabolic subgroup of $G_N$ and $P^t=P_{n}^t$ the t-parabolic subgroup of G.

Proposition 4.3 For $0\leq t\leq m$ , let $\tau _t$ be the element of $G_N$ given by

$$\begin{align*}\tau_t=\left[\begin{array}{@{}cccccc@{}} 1_{m} & 0 & 0 & 0 & 0 & 0\\ 0 & 1_r & 0 & 0 & 0_r & 0\\ 0 & 0 & 1_{m} & 0 & 0 & 0\\ 0 & 0 & e_t & 1_{m} & 0 & 0\\ 0 & 0_r & 0 & 0 & 1_r & 0\\ e_t^{\ast} & 0 & 0 & 0 & 0 & 1_{m} \end{array}\right],e_t=\left[\begin{array}{@{}cc@{}} 1_t & 0\\ 0 & 0 \end{array}\right]\in\mathbb{B}^{m}_{m}. \end{align*}$$

Then $\tau _t$ form a complete set of representatives of $P_N\backslash G_N/G\times G$ .

Proof This can be proved similarly to the proofs of Lemmata 4.1 and 4.2 in [Reference Shimura30]. Let $W=\{w\in \mathbb {B}^N_{2N}:wJ_nw^{\ast }=0,\mathrm {rank}(w)=N\}$ , then $P_N\backslash G_N\cong \mathrm {GL}_N\backslash W$ . Therefore, it suffices to find representatives of $\mathrm {GL}_N\backslash W/G\times G$ . Let $w=\left [\begin {array}{@{}cccccc@{}} a_1 & b_1 & c_1 & a_2 & b_2 & c_2 \end {array}\right ]\in W$ of column size $(m,r,m,m,r,m)$ . The condition $wJ_nw^{\ast }=0$ is equivalent to $wR^{\ast }\omega Rw^{\ast }=0$ . Explicitly, $wR^{\ast }=\left [\begin {array}{@{}cc@{}}x& y\end {array}\right ]$ with

$$\begin{align*}x=\left[\begin{array}{@{}ccc@{}} a_1 & \frac{b_1}{2}+b_2\zeta^{-1} & a_2 \end{array}\right],y=\left[\begin{array}{@{}ccc@{}} -c_1 & -\frac{b_1}{2}+b_2\zeta^{-1} & c_2 \end{array}\right],x\phi_1 x^{\ast}=y\phi_2 y^{\ast}. \end{align*}$$

Multiplying by some element in $\mathrm {GL}_N$ , we can assume that

$$\begin{align*}x\phi_1 x^{\ast}=y\phi_2 y^{\ast}=\left[\begin{array}{@{}ccc@{}} 0 & 0 & e_t\\ 0 & \zeta I_r & 0\\ e_t^{\ast} &0 & 0 \end{array}\right]\text{ or }\left[\begin{array}{@{}ccc@{}} 0 & 0 & e_t\\ 0 & 0_r & 0\\ e_t^{\ast} &0 & 0 \end{array}\right],0\leq t\leq m. \end{align*}$$

Let $V=\mathbb {B}^{n}$ with standard basis $\{\epsilon _i\}$ and denote $x_i$ be the ith row of x. Let $\tilde {U}$ be the subspace of V spanned by basis $\epsilon _i$ with $i\leq t$ or $m+r\leq i\leq m+r+t$ and $\tilde {U}^{\perp }$ the subspace spanned by other basis, so $V=\tilde {U}\oplus \tilde {U}^{\perp }$ . Let $\theta $ be the restriction of $\phi _1$ on $\tilde {U}$ and $\eta $ the restriction on $\tilde {U}^{\perp }$ , then we can write $(V,\phi _1)=(\tilde {U},\theta )\oplus (\tilde {U}^{\perp },\eta )$ . Assume $x\phi x^{\ast }$ is given by the first matrix as above, let U be the subspace of V spanned by vector $x_i$ with $i\leq m$ or $m+r\leq i\leq m+r+t$ , and let $U'$ be the subspace spanned by other $x_i$ . We also denote $U^{\perp }=\{v\in V:u\phi v^{\ast }=0\text { for any }u\in U\}$ , then $V=U\oplus U^{\perp }$ . Then there exists an automorphism $\gamma $ of $(V,\phi )$ such that $U\gamma =\tilde {U},U^{\perp }\gamma =\tilde {U}^{\perp }$ . Now $U'\gamma \subset \tilde {U}^{\perp }$ is a totally $\eta $ -isotropic subspace; thus; there exists an automorphism $\gamma '$ of $(\tilde {U}^{\perp },\eta )$ such that $U'\gamma \gamma '\subset \sum _{m+t+r+1\leq i\leq n}\mathbb {B}\epsilon _i$ . Viewing $\gamma '$ as an automorphism of $(V,\phi )$ and putting $g_1=\gamma \gamma '$ , we have (similarly for y)

$$\begin{align*}xg_1=\left[\begin{array}{@{}ccccc@{}} 1_t & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & u\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1_t & 0\\ 0 & 0 & 0 & 0 & v \end{array}\right],yg_2=\left[\begin{array}{@{}ccccc@{}} 1_t & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & u'\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1_t & 0\\ 0 & 0 & 0 & 0 & v' \end{array}\right]. \end{align*}$$

We further modify

$$\begin{align*}g_1\mapsto g_1\left[\begin{array}{@{}ccccc@{}} 0 & 0 & 0 & 1_t & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ -1_t & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 \end{array}\right]\text{ so }xg_1=\left[\begin{array}{@{}ccccc@{}} 0 & 0 & 0 & -1_t & 0\\ 0 & 0 & 0 & 0 & u\\ 0 & 0 & 1 & 0 & 0\\ 1_t & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & v \end{array}\right]. \end{align*}$$

Therefore,

$$\begin{align*}w(g_1\times g_2)=\left[\begin{array}{cccccccccc} 0 & 0 & 0 & -1_t & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & u & 0 & 0 & u'\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\zeta & 0 & 0\\ 1_t & 0& 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & v & 0 & 0 & v' \end{array}\right]. \end{align*}$$

By our assumption $\mathrm {rank}(w)=N$ , we must have $\left [\begin {array}{@{}cc@{}}u & u'\\ v & v'\end {array}\right ]$ is of full rank. If $x\phi _1x^{\ast }$ equals the second matrix as above, then by the same argument, we can obtain a similar result but without the term $1$ in the middle of $xg_1$ and $yg_2$ , which contradicts the assumption $\mathrm {rank}(w)=N$ . Suppose $\left [\begin {array}{@{}cc@{}} a & b\\ c & d \end {array}\right ]=\left [\begin {array}{@{}cc@{}}u & u'\\ v & v'\end {array}\right ]^{-1}$ , and multiplying

$$\begin{align*}\left[\begin{array}{@{}ccccc@{}} -1_t & 0 & 0 & 0 & 0\\ 0 & a & 0 & 0 & b\\ 0 & 0 & -\zeta^{-1} & 0 & 0\\ 0 & 0 & 0 & 1_t & 0\\ 0 & c & 0 & 0 & d \end{array}\right]\in\mathrm{GL}_N, \end{align*}$$

on the left, we then get the desired form in the proposition.

Put $V_t=\tau _t(G_{n}\times G_{n})\tau _t^{-1}\cap P_N$ . Then, by straightforward computation,

$$\begin{align*}V_t=\{\beta\times\gamma\in P_{n}^t\times P_{n}^t:\kappa_t\pi_t(\beta)=\pi_t(\gamma)\kappa_t\},\kappa_t=\left[\begin{array}{@{}ccc@{}} 0 & 0 & -1_t\\ 0 & 1_r & 0\\ 1_t & 0 & 0 \end{array}\right]. \end{align*}$$

To simplify the computation, we may also use the modified representatives $\tilde {\tau }_t=\tau _t(1_{n}\times (\kappa _t\times 1_{2m-2t}))$ for $0\leq t\leq m$ . Put $\tilde {V}_t=\tilde {\tau }_t(G_{n}\times G_{n})\tilde {\tau }_t^{-1}\cap P_N$ . Then

$$\begin{align*}\tilde{V}_t=\{\beta\times\gamma\in P_{n}^t\times P_{n}^t:\pi_t(\beta)=\pi_t(\gamma)\}. \end{align*}$$

One easily shows that (compare with Lemma 4.3 in [Reference Shimura30])

$$\begin{align*}P_{n}^t\times P_{n}^t=\bigcup_{\xi\in G_{2t+r}}\tilde{V}_t((\xi\times 1_{2m-2t})\times 1_{2n})=\bigcup_{\xi\in G_{2t+r}}\tilde{V}_t(1_{2n}\times(\xi\times 1_{2m-2t})), \end{align*}$$

$$\begin{align*}P^N\tilde{\tau}_t(G_{n}\times G_{n})=\bigcup_{\xi,\beta,\gamma}P^N\tilde{\tau}_t((\xi\times 1_{2m-2t})\beta\times\gamma)=\bigcup_{\xi,\beta,\gamma}P^N\tilde{\tau}_t(\beta\times(\xi\times 1_{2m-2t})\gamma), \end{align*}$$

where $\xi $ runs over $G_{2t+r}$ and $\beta ,\gamma $ run over $P_{n}^t\backslash G_{n}$ . In particular,

$$\begin{align*}P^N\tilde{\tau}_{m}(G_{n}\times G_{n})=\bigcup_{\xi\in G_{n}}P^N\tilde{\tau}_{m}(\xi\times 1_{n})=\bigcup_{\xi\in G_{n}}P^N\tilde{\tau}_{m}(1_n\times\xi). \end{align*}$$

We denote $K_1(\mathfrak {n})=\prod _vK_v$ be the open compact subgroup of $G_n(\mathbb {A}_{\mathbf {h}})$ defined before. We also consider the open compact subgroup of $G_N$ , $K_{1}^N(\mathfrak {n})=\prod _{v \in \mathbf {h}}K_{v}^N$ with

$$\begin{align*}K_{v}^N(\mathfrak{n})=\left\{\gamma=\left[\begin{array}{@{}cc@{}} a & b\\ c & d \end{array}\right]\in G_N(\mathbb{A}_{v})\cap M_N(\mathcal{O}_v):\gamma\equiv 1_N\text{ mod }\mathfrak{n}_v\right\}. \end{align*}$$

Proof Let $\xi =\left [\begin {array}{@{}ccc@{}} a & b & c\\ g & e & f\\ h & l & d \end {array}\right ]\in G_n$ where blocks has size $(m,r,m)\times (m,r,m)$ . We calculate that

$$\begin{align*}\tilde{\tau}_m(\xi\times 1)\tilde{\tau}_m^{-1}=\left[\begin{array}{@{}cccccc@{}} \ast &\ast &\ast &\ast &\ast &\ast \\ \ast &\ast &\ast &\ast &\ast &\ast \\ \ast &\ast &\ast &\ast &\ast &\ast \\ -h & -l & -d+1 & d & \frac{l\zeta}{2} & 0\\ -\zeta^{-1}g & -\zeta^{-1}(e-1) & -\zeta^{-1}f & \zeta^{-1}f & \frac{e+1}{2} & 0\\ a-1 & b & c & -c & \frac{-b\zeta}{2} & 1 \end{array}\right]. \end{align*}$$

Suppose $\tilde {\tau }_m(\xi \times 1)\tilde {\tau }_m^{-1}\in P_N(\mathbb {Q}_v)K^N_v$ , then there exist $p\in P_N(\mathbb {Q}_v)$ , say

$$\begin{align*}\left[\begin{array}{@{}cccccc@{}} \ast &\ast &\ast &\ast &\ast &\ast \\ \ast &\ast &\ast &\ast &\ast &\ast \\ \ast &\ast &\ast &\ast &\ast &\ast \\ 0 & 0 & 0 & p_{11} & p_{12} & p_{13}\\ 0 & 0 & 0 & p_{21} & p_{22} & p_{23}\\ 0 & 0 & 0 & p_{31} & p_{31} & p_{33} \end{array}\right], \end{align*}$$

such that $p\tilde {\tau }_m(\xi \times 1)\tilde {\tau }_m^{-1}\in K^N_v$ . It is obvious that $p_{13}\equiv p_{23}\equiv 0\text { mod }\mathfrak {c}$ and $p_{33}\equiv 1\text { mod }\mathfrak {c}$ . Comparing the third and fourth columns, we have

$$\begin{align*}p_{11}(-d+1)+p_{12}(-\zeta^{-1}f)+p_{13}c\equiv 0\text{ mod }\mathfrak{n},p_{11}d+p_{12}(\zeta^{-1}f)+p_{13}(-c)\equiv 1\text{ mod }\mathfrak{n}; \end{align*}$$

$$\begin{align*}p_{21}(-d+1)+p_{22}(-\zeta^{-1}f)+p_{23}c\equiv 0\text{ mod }\mathfrak{n},p_{21}d+p_{22}(\zeta^{-1}f)+p_{23}(-c)\equiv 0\text{ mod }\mathfrak{n}; \end{align*}$$

$$\begin{align*}p_{31}(-d+1)+p_{32}(-\zeta^{-1}f)+p_{33}c\equiv 0\text{ mod }\mathfrak{n},p_{31}d+p_{32}(\zeta^{-1}f)+p_{33}(-c)\equiv 0\text{ mod }\mathfrak{n}. \end{align*}$$

This forces $p_{11}\equiv 1\text { mod }\mathfrak {n}$ and $p_{21}\equiv p_{31}\equiv 0\text { mod }\mathfrak {n}$ . Comparing the second and fifth columns and using our assumption on $\mathfrak {n}$ , we have

$$\begin{align*}p_{11}(-l)+p_{12}(-\zeta^{-1}(e-1))+p_{13}b\equiv0\text{ mod }\mathfrak{n},p_{11}(l\zeta)+p_{12}(e+1)+p_{13}(-b\zeta)\equiv0\text{ mod }\mathfrak{n}; \end{align*}$$

$$\begin{align*}p_{21}(-l)+p_{22}(-\zeta^{-1}(e-1))+p_{23}b\equiv0\text{ mod }\mathfrak{n},p_{21}(l\zeta)+p_{22}(e+1)+p_{23}(-b\zeta)\equiv2\text{ mod }\mathfrak{n}; \end{align*}$$

$$\begin{align*}p_{31}(-l)+p_{32}(-\zeta^{-1}(e-1))+p_{33}b\equiv0\text{ mod }\mathfrak{n},p_{31}(l\zeta)+p_{32}(e+1)+p_{33}(-b\zeta)\equiv0\text{ mod }\mathfrak{n}. \end{align*}$$

The second line shows that $p_{22}\equiv e\equiv 1\text { mod }\mathfrak {n}$ and then $p_{12}\equiv p_{32}\equiv 0\text { mod }\mathfrak {n}$ by other formulas. Therefore, $p\in P_N(\mathbb {Q}_v)\cap K^N_v$ . From the above identities, we already have

$$\begin{align*}d-1\equiv f\equiv c\equiv l\equiv e-1\equiv b\equiv 0\text{ mod }\mathfrak{n}. \end{align*}$$

The claim $\xi \in K_v$ then follows from the above identities together with

$$\begin{align*}p_{i1}(-h)+p_{i2}(-\zeta^{-1}g)+p_{i3}(a-1)\equiv 0\text{ mod }\mathfrak{n},i=1,2,3.\\[-37pt] \end{align*}$$

4.3 Integral representation

Let $\chi $ be a Hecke character whose conductor divides the fixed ideal $\mathfrak {n}$ . Recall that we write $N=2n$ . We define $\sigma \in G_N(\mathbb {A}_{\mathbf {h}})$ as $\sigma _v := 1$ , the identity matrix, if $v\nmid \mathfrak {n}$ , and $\sigma _v = \tilde {\tau }_m$ if $v|\mathfrak {n}$ .

We then consider the weight k Siegel-type Eisenstein series (twisted by $\tilde {\tau }_m$ ) on $G_N$ defined as

$$\begin{align*}\mathbf{E}(\mathfrak{g},s):=E^n_k(\mathfrak{g}\,\sigma^{-1},s;\chi),\,\,\,\,\mathfrak{g} \in G_N(\mathbb{A}). \end{align*}$$

By decomposition of $G_N$ . we can write

$$\begin{align*}\mathbf{E}(\mathfrak{g},s)=\sum_{t=0}^m\sum_{\gamma\in P_N\backslash P_N\tilde{\tau}_t(G\times G)}\phi(\gamma\mathfrak{g},s)=:\sum_{t=0}^m\mathbf{E}_t(\mathfrak{g},s). \end{align*}$$

Assume $\mathfrak {g}=h\times g$ with $h,g\in G(\mathbb {A})$ , and let $\mathbf {f}\in S_k(K_1(\mathfrak {n}))$ be a cusp form.

We are going to study the integral

$$\begin{align*}\int_{G(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\mathbf{E}(g\times h,s)\mathbf{f}(h)\mathbf{d}h. \end{align*}$$

Proposition 4.5 Assume that $t<m$ , then

$$\begin{align*}\int_{G(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\mathbf{E}_t(g\times h,s)\mathbf{f}(h)\mathbf{d}h=0. \end{align*}$$

Proof Let $U_t$ be the unipotent radical of $P^t_n$ . Then the integral equals

$$\begin{align*}\int_{G(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\sum_{\xi\in G_{2t+r}}\sum_{\beta,\gamma\in P^t_n\backslash G}\phi(\tilde{\tau}_t((\xi\times 1_{2n-2t})\beta g\times\gamma h),s)\mathbf{f}(h)\mathbf{d}h \end{align*}$$

$$\begin{align*}=\int_{P^t_n\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\sum_{\xi\in G_{2t+r}}\sum_{\gamma\in P^t_n\backslash G_n}\phi(\tilde{\tau}_t((\xi\times 1_{2n-2t}) g\times\gamma h),s)\mathbf{f}(h)\mathbf{d}h \end{align*}$$

$$\begin{align*}=\int_{U_t(\mathbb{A})P^t_n(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\int_{U_t(\mathbb{Q})\backslash U_t(\mathbb{A})}\sum_{\xi,\gamma}\phi(\tilde{\tau}_t(\xi ng\times\gamma h),s)\mathbf{f}(nh)\mathbf{d}n\mathbf{d}h. \end{align*}$$

Since $\xi $ normalizes $U_t$ and $\tilde {\tau }_t(n\times 1)\in P_n$ , this equals

$$\begin{align*}=\int_{U_t(\mathbb{A})P^t_n(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\int_{U_t(\mathbb{Q})\backslash U_t(\mathbb{A})}\sum_{\xi,\gamma}\phi(\tilde{\tau}_t(\xi g\times\gamma h),s)\mathbf{f}(nh)\mathbf{d}n\mathbf{d}h, \end{align*}$$

which vanishes by the cuspidality of $\mathbf {f}$ .

Therefore,

$$\begin{align*}\int_{G(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\mathbf{E}(g\times h,s)\mathbf{f}(h)\mathbf{d}h=\int_{G(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\mathbf{E}_m(g\times h,s)\mathbf{f}(h)\mathbf{d}h \end{align*}$$

$$\begin{align*}=\int_{G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\phi(\tilde{\tau}_m(g\times h),s)\mathbf{f}(h)\mathbf{d}h. \end{align*}$$

The infinite part is calculated in following lemma.

Lemma 4.6 For $k + Re(s)> 2n+1$ , we have

$$\begin{align*}\int_{G_{\infty}/K_{\infty}}\phi_{\infty}(\tilde{\tau}_m(g_{\infty}\times h_{\infty}),s)\mathbf{f}(h_{\mathbf{h}}\cdot h_{\infty})\mathbf{d}h_{\infty}=c_k(s)\mathbf{f}(h_{\mathbf{h}}\cdot g_{\infty}), \end{align*}$$

with

$$\begin{align*}c_k(s) &= \alpha(s) \pi^{\frac{n(n-1)}{2}} \frac{\Gamma(s+k-2n+3) \Gamma(s+k-2n+5) \ldots \Gamma(s+k-1)}{\Gamma(s+k-n+2) \Gamma(s+k-n+3) \ldots \Gamma(s+k)}\\[6pt]&=\alpha(s) \pi^{\frac{n(n-1)}{2}} \frac{\prod_{j=0}^{q-1} \Gamma(s+k-n+1-t-2j)}{\prod_{j=0}^{q-1} \Gamma(s+k-2j)}, \end{align*}$$

where $n=2q+t$ with $q \in \mathbb {N}$ and $t \in \{0,1\}$ and $\alpha (s)$ is a holomorphic function on $s \in \mathbb {C}$ such that $\alpha (\lambda ) \in \overline {\mathbb {Q}}$ for all $\lambda \in \mathbb {Q}$ .

Proof Note that

$$\begin{align*}j(\tilde{\tau}_m(g_{\infty}\times h_{\infty}),z_0\times z_0)=j(\tilde{\tau}_m,g_{\infty}z_0\times h_{\infty}z_0 )j(g_{\infty},z_0)\overline{j(h_{\infty},z_0)}. \end{align*}$$

Put $w=h_{\infty }z_0,z=g_{\infty }z_0$ , then since

$$\begin{align*}f(w)=j(h_{\infty},z_0)^{k}\mathbf{f}(h_{\infty}),j(\tilde{\tau}_m,z\times w)=\delta(w,z), \end{align*}$$

the integral becomes

$$\begin{align*}j(g_{\infty},z_0)^{-k}\delta(z)^{\frac{s-k}{2}}\int_{\mathfrak{Z}}\delta(w,z)^{-k}|\delta(w,z)|^{k-s}\delta(w)^{\frac{k+s}{2}}f(w)dw. \end{align*}$$

This kind of integral is calculated in [Reference Shimura31, Appendix A2] and [Reference Hua14]. In particular, it is shown there that for $k+ Re(s)> n+\frac {1}{2}$ ,

$$\begin{align*}\int_{\mathfrak{Z}}\delta(w,z)^{-k}|\delta(w,z)|^{-2s}\delta(w)^{s+k}f(w) dw = \widetilde{c}_k(s) f(z) \delta(z)^{-s}, \end{align*}$$

where $\widetilde {c}_k(s)$ is a function on s which does not depend on f. Indeed, as it is explained in [Reference Shimura31], the quantity $\widetilde {c}_k(s)$ is independent of f and it is equal to

$$\begin{align*}\widetilde{c}_k(s) = \alpha(s) \int_{\mathfrak{B}} \det(I + z \bar{z})^{s+k} \textbf{d}z, \end{align*}$$

where $\textbf {d}z$ is the invariant measure on the bounded domain and is given as $\textbf {d}z = \det (I + z \bar {z})^{-n+1} dz$ , and $\alpha (s)$ is a holomorphic function on s such that $\alpha (\lambda ) \in \overline {\mathbb {Q}}$ for all $\lambda \in \mathbb {Q}$ (actually it can be made precise, but we do not need it here). But this last integral has been computed in [Reference Hua14, p. 46] from which we obtain that

$$\begin{align*}\widetilde{c}_k(s) = \alpha(s) \pi^{\frac{n(n-1)}{2}} \frac{\Gamma(2\lambda +1) \Gamma(2 \lambda +3) \ldots \Gamma(2 \lambda +2n-3)}{\Gamma(2\lambda +n) \Gamma(2\lambda +n+1) \ldots \Gamma(2\lambda + 2n -2)}, \end{align*}$$

where $\lambda = s+k -n+1$ .

Setting now $s \mapsto \frac {s-k}{2}$ , we obtain that

$$\begin{align*}j(g_{\infty},z_0)^{-k}\delta(z)^{\frac{s-k}{2}}\int_{\mathfrak{Z}}\delta(w,z)^{-k}|\delta(w,z)|^{k-s}\delta(w)^{\frac{k+s}{2}}f(w)dw = \end{align*}$$

$$\begin{align*}\widetilde{c}_k((s-k)/2)j(g_{\infty},z_0)^{-k}f(z)=c_k(s)\mathbf{f}(h_{\mathbf{h}}\cdot g_{\infty}), \end{align*}$$

where we have set $c_k(s) := \widetilde {c}_k((s-k)/2)$ .

By changing variables, it remains to calculate

$$\begin{align*}\int_{K_1(\mathfrak{n})\backslash G(\mathbb{A}_{\mathbf{h}})}\phi_{\mathbf{h}}(\tilde{\tau}_m(h\times 1),s)\mathbf{f}(gh^{-1})\mathbf{d}h. \end{align*}$$

Note that for $v|\mathfrak {n}$ , $\phi _{v}$ is nonzero unless $h_{v}\in K_v$ . Hence, it remains to consider the integral over $\prod _{v\nmid \mathfrak {n}}K_v\backslash G(\mathbb {Q}_v)$ . These unramified integrals are well known by [Reference Lapid and Rallis19, Reference Shimura33, Reference Yamana37]. Indeed, by the Cartan decomposition, we can write

$$\begin{align*}\begin{aligned} G(\mathbb{Q}_v)&=\coprod_{\substack{e_1,{\ldots},e_{m_v}\in\mathbb{Z}\\0\leq e_1\leq {\ldots}\leq e_{m_v}}}K_{e_{1},{\ldots},e_{m_v}},\\ K_{e_1,{\ldots},e_{m_v}}&=K_v\mathrm{diag}[p_v^{e_1},{\ldots},p_v^{e_{m_v}},1_{r_v},p_v^{-e_1},{\ldots},p_v^{-e_{m_v}}]K_v, \end{aligned} \end{align*}$$

where $m_v$ is the local Witt index of $G(\mathbb {Q}_v)$ and $p_v$ the prime corresponds to v. Note that by definition of $\phi $ ,

$$\begin{align*}\phi_v(\tilde{\tau}_m(h_v\times 1))=(\chi_v(p_v)p_v^{-s})^{e_1+\cdots+e_{m_v}}. \end{align*}$$

Assume that $\mathbf {f}$ is an eigenform such that $\mathbf {f}|[K_{e_1,{\ldots },e_{m_v}}]=\lambda _{\mathbf {f}}(K_{e_1,{\ldots },e_{m_v}})\mathbf {f}$ . Then the integral over $K_v\backslash G(\mathbb {Q}_v)$ contributes a Dirichlet series

$$\begin{align*}\sum_{\substack{e_1,{\ldots},e_{m_v}\in\mathbb{Z}\\0\leq e_1\leq \cdots\le e_{m_v}}}\lambda_{\mathbf{f}}(K_{e_1,{\ldots},e_{m_v}})(\chi_v(p_v)p_v^{-s})^{e_1+\cdots+e_{m_v}}. \end{align*}$$

In conclusion, we obtain

$$\begin{align*}\prod_{v\nmid\mathfrak{n}}\sum_{\substack{e_1,{\ldots},e_{m_v}\in\mathbb{Z}\\0\leq e_1\leq {\ldots}\le e_{m_v}}}\lambda_{\mathbf{f}}(K_{e_1,{\ldots},e_{m_v}})(\chi_v(p_v)p_v^{-s})^{e_1+\cdots+e_{m_v}}\cdot\mathbf{f}(g)=D(s,\mathbf{f},\chi)\mathbf{f}(g). \end{align*}$$

We summarize the discussion in the following theorem.

Theorem 4.7 Let $\mathbf {f}\in \mathcal {S}_k^n(K_1(\mathfrak {n}))$ be an eigenform, and assume that $\mathfrak {n}$ is coprime to $(2\zeta )$ . Then

$$\begin{align*}\int_{G(\mathbb{Q})\backslash G(\mathbb{A})/K_1(\mathfrak{n})K_{\infty}}\mathbf{E}(g\times h,s)\mathbf{f}(h)\mathbf{d}h=c_k(s)D(s,\mathbf{f},\chi)\mathbf{f}(g). \end{align*}$$

5.1 CM-points

We introduce the following setting, with some small repetition of what we have discussed so far.

We let $\mathbb {B}$ be a definite quaternion algebra over $\mathbb {Q}$ , $T^{\ast }=-T\in M_n(\mathbb {B})$ a skew-hermitian matrix, and define the algebraic group

$$\begin{align*}G:=G(T):=\{g\in\mathrm{SL}_n(\mathbb{B}):g^{\ast}Tg=T\}. \end{align*}$$

Let $K_i$ , $i=1,\ldots ,n$ , be imaginary quadratic fields and consider the CM algebra $Y=K_1\times \cdots \times K_n$ and $Y^1=\{y\in Y:yy^{\rho }=1\}$ with $\rho $ induced by the nontrivial involutions (i.e., complex conjugation) on each $K_i$ . We are interested in embeddings $h:Y^1\to G(T)$ . Clearly, $h(Y^1)\subset G(T)(\mathbb {R})$ and $(Y^1 \otimes _{\mathbb {Q}} \mathbb {R})^{\times }$ is a compact subgroup of $G(T)(\mathbb {R})$ . Let us show that there always exists such an embedding.

Without loss of generality, we may write $T=\mathrm {diag}[a_1,{\ldots },a_n]$ in diagonal form. We then select as imaginary quadratic fields $K_i:=\mathbb {Q}(a_i)$ , for $i=1,\ldots , n$ , and define the embedding

$$\begin{align*}h:Y^1\to G(T),(y_1,{\ldots},y_n)\mapsto\mathrm{diag}[y_1,{\ldots},y_n]. \end{align*}$$

Back to our general considerations, we select an imaginary quadratic field K which is different from $K_i$ ’s above, and splits $\mathbb {B}$ . It is easy to see that that there exists always such a field K. We now fix an embedding $M_n(\mathbb {B})\to M_{2n}(K)$ . Denote the image of T in $M_{2n}(K)$ by $\mathcal {T}$ and the unitary group

$$\begin{align*}U(\mathcal{T}):=\{g\in\mathrm{GL}_{2n}(K):g^{\ast}\mathcal{T}g=\mathcal{T}\}. \end{align*}$$

We note that the group of $\mathbb {R}$ -points of $U(\mathcal {T})$ is isomorphic to

$$\begin{align*}U(n,n)=\left\{g\in\mathrm{GL}_{2n}(\mathbb{C}):g^{\ast}\left[\begin{array}{@{}cc@{}} i\cdot 1_n & 0\\ 0 & -i\cdot 1_n \end{array}\right]g=\left[\begin{array}{@{}cc@{}} i\cdot 1_n & 0\\ 0 & -i\cdot 1_n \end{array}\right]\right\}. \end{align*}$$

Its action on the bounded domain (see, for example, [Reference Shimura32]),

$$\begin{align*}\mathcal{B}=\{z\in M_n(\mathbb{C}):1-z^{\ast}z>0\}, \end{align*}$$

is defined by $gz=(az+b)(cz+d)^{-1}$ for $g=\left [\begin {array}{@{}cc@{}} a & b\\ c & d \end {array}\right ]$ , with the obvious block matrices. The two factors of automorphy are given by $\lambda (g,z)=\overline {c}{^t\!z}+\overline {d}$ , and $\mu (g,z)=cz+d.$ The embedding $M_n(\mathbb {B})\to M_{2n}(K)$ induces an embedding $\mathfrak {i}:G(T)\to U(\mathcal {T})$ which is compatible with natural inclusion $\iota :\mathfrak {B}\to \mathcal {B}$ . We will view $G(T)$ (resp. $\mathfrak {B}$ ) as a subgroup (resp. subspace) of $U(\mathcal {T})$ (resp. $\mathcal {B}$ ) under this embedding.

We call a point fixed by some $h(Y^1)$ as above a CM-point, and we note that this definition does not depend on the choice of the field K. For example, take ${T=\mathrm {diag}[\zeta \cdot 1_m,\zeta \cdot 1_r,-\zeta \cdot 1_m]}$ . This is the group $G'$ in ection 2, and we have described its embedding into unitary group and the action on $\mathfrak {B}$ explicitly there. Let h and Y be as above, and one easily checks that $0$ is the fixed point of $h(Y^1)$ and thus a CM-point.

We now want to attach some CM periods to our CM-points. We will do this by relating our definition with the notion of CM-points of unitary groups. Indeed, our selection of the field K allows us to view our group as a subgroup of a unitary group, and hence an embedding $\mathfrak {B} \hookrightarrow \mathcal {B}$ . Our next aim is to relate the just-defined CM-points in $\mathfrak {B}$ with the well-studied, as in [Reference Shimura32], CM-points of $\mathcal {H}$ . It is here that we employ the idea of Shimura, which was used in [Reference Shimura25] (see also [Reference Shimura26, Section 7]) to study CM-points in general type-C domains.

Let $w \in \mathfrak {B}\subset \mathcal {B}$ be a CM-point fixed by $h(Y^{1}) \subset G\subset U(n,n)$ . Then, for such a point, we have that

$$\begin{align*}\Lambda(\alpha,w) p(x,w) = p(x \alpha ,w),\,\,\,\,\,\, \alpha \in h(Y^1),\,\,\,x \in \mathbb{C}^{2n}, \end{align*}$$

where $\Lambda (\alpha ,w) \in GL_{2n}(\mathbb {C})$ and $p(x,z) : \mathbb {C}^{2n} \times \mathcal {B} \rightarrow \mathbb {C}^{2n}$ are the maps defined in [Reference Shimura32, §4.7]. In this way, we can obtain an embedding $Y \rightarrow End_{\mathbb {C}}(\mathbb {C}^{2n})$ by sending ${\alpha \mapsto \Lambda (\alpha , w)}$ where we have used the fact that Y is spanned by $Y^1$ over $\mathbb {Q}$ . We now extend this to an injection h of $K \otimes _{\mathbb {Q}} Y \cong \mathcal {S}:=\mathcal {S}_1 \times \cdots \times \mathcal {S}_n$ into $End(\mathbb {C}^{2n})$ , where $\mathcal {S}_i = K K_i$ . Indeed, we set

$$\begin{align*}h(\beta \otimes \alpha) p(x,w) = p(\beta x \alpha, w) = p(x \beta \alpha, w) = p(x \alpha \beta, w). \end{align*}$$

That is, the point w can be seen as a fixed point of $\mathcal {S}^{1} \otimes _{\mathbb {Q}} \mathbb {R}$ , where $\mathcal {S}^{1} =\{ s \in \mathcal {S} \,\,\,|\,\,\, s s^{\rho } = 1 \} $ with $\rho $ the involution on $\mathcal {S}$ induced by the complex conjugation on $KK_i$ . Hence, w is a CM-point in $\mathcal {B}$ defined in [Reference Shimura32, §4.11] for unitary groups. In particular, w has entries in $\overline {\mathbb {Q}}$ by [Reference Shimura32, Lemma 4.13].

Remark 5.2 Following [Reference Shimura32, Section 4], let $\Omega =\{K,\Psi ,L,\mathcal {T},\{u_i\}_{i=1}^s\}$ be a PEL type and $\mathcal {F}(\Omega )$ family of polarized abelian varieties of PEL type. The abelian varieties in $\mathcal {F}(\Omega )$ are parameterized by $\mathcal {B}$ . More precisely, there is a bijection

$$\begin{align*}\Gamma\backslash\mathcal{B}\stackrel{\sim}\longrightarrow\mathcal{F}(\Omega), \Gamma=\{\gamma\in U(\mathcal{T}):L\gamma=L,u_i\gamma-u_i\in L\}. \end{align*}$$

As in [Reference Shimura24], we can define $\Omega '=\{\mathbb {B},\Psi ',L,T,\{u_i\}_{i=1}^s\}$ for quaternions, and $\mathcal {F}(\Omega ')$ are parameterized by $\mathfrak {B}$ . The natural inclusion $\mathcal {F}(\Omega ')\to \mathcal {F}(\Omega )$ is compatible with ${\mathfrak {B}\to \mathcal {B}}$ . Moreover, similar to [Reference Garrett6, Reference Shimura25], we actually have an embedding of canonical models between $\Gamma \backslash \mathfrak {B}$ and $\Gamma '\backslash \mathcal {B}$ for certain congruence subgroups $\Gamma ,\Gamma '$ .

As we have remarked, CM-points for unitary groups have been extensively studied in [Reference Shimura32, Chapter II]. We recall some of their properties. For $\alpha \in \mathcal {S}^1$ , we put $\psi (\alpha ) := \lambda (h(\alpha ),w) \in GL_n(\mathbb {C})$ , $\phi (\alpha ):= \mu (h(\alpha ),w) \in GL_n(\mathbb {C})$ , and $\Phi (\alpha ) = \mathrm {diag}[ \psi (\alpha ), \phi (\alpha )] \in GL_{2n}(\mathbb {C})$ . We can then find $B,C \in GL_{n}(\overline {\mathbb {Q}})$ (see [Reference Shimura32, p. 78]) such that for all $\alpha \in \mathcal {S}$ ,

$$\begin{align*}B \psi(\alpha) B^{-1} = \mathrm{diag}[\psi_1(\alpha), \ldots, \psi_n(\alpha)],\,\,\,\,\,C \phi(\alpha) C^{-1} = \mathrm{diag}[\phi_1(\alpha), \ldots, \phi_n(\alpha)], \end{align*}$$

for some ring homomorphism $\phi _i, \psi _i : \mathcal {S} \rightarrow \mathbb {C}$ , where we have $\mathbb {Q}$ -linearly extended $\psi $ and $\phi $ , from $\mathcal {S}^1$ to $\mathcal {S}$ . We set

$$\begin{align*}\mathfrak{p}_{\infty} (w) := C^{-1} \mathrm{diag}[p_{\mathcal{S}}(\phi_1, \Phi) , \ldots, p_{\mathcal{S}}(\phi_n,\Phi)] C \in GL_n(\mathbb{C}), \end{align*}$$

$$\begin{align*}\mathfrak{p}_{\infty \rho} (w) := B^{-1} \mathrm{diag}[p_{\mathcal{S}}(\psi_1, \Phi) , \ldots, p_{\mathcal{S}}(\psi_n,\Phi)] B \in GL_n(\mathbb{C}), \end{align*}$$

where the CM periods $p_{\mathcal {S}}(\psi _i,\Phi ) \in \mathbb {C}^{\times }$ and $p_{\mathcal {S}}(\phi _i,\Phi ) \in \mathbb {C}^{\times }$ are defined as in [Reference Shimura32, p. 78]. Actually, we should remark here that the periods $p_{\mathcal {S}}(\psi _i,\Phi )$ , $p_{\mathcal {S}}(\phi _i,\Phi )$ are uniquely determined up to elements in $\overline {\mathbb {Q}}^{\times }$ , but this is sufficient for our applications.

We now use the fact that $w\in \mathfrak {B}\subset \mathcal {B}$ is a CM-point for both $(Y,h)$ and also for $(\mathcal {S},h)$ . Note that $\psi (\alpha ) = \phi (\alpha )$ for $\alpha \in Y^1 \subset \mathcal {S}^1$ . Indeed, for $\alpha \in G(\mathbb {R})$ , we have that (see [Reference Shimura25, equation (2.18.9)]) $\left [ \begin {matrix} a & b\\ c& d \end {matrix} \right ] = \left [ \begin {matrix} \overline {d} & -\overline {c}\\ -\overline {b}& \overline {a} \end {matrix} \right ]$ , and hence, in particular, we have that $\lambda (\alpha ,z) = \mu (\alpha ,z)$ since ${^t\!z} = - z$ . In particular, the values $\psi (\alpha ) = \phi (\alpha ) = \lambda (\alpha ,w) = \mu (\alpha ,w)$ for $\alpha \in Y^1$ , that is, the restrictions of $\phi $ and $\psi $ to $Y^1$ , are independent of the choice of the field K. Furthermore, we note that $\psi (\alpha ) = \phi (\alpha )$ for all $\alpha \in K$ with $\alpha \overline {\alpha } =1$ seen as elements of $U(n,n)$ , i.e., $\alpha 1_{2n} \in U(n,n)$ .

In the following lemma, we use the notation $I_Y, J_Y, J_{\mathcal {S}_j}$ as defined in [Reference Shimura32, p. 77].

Lemma 5.3 With notation as above, for all $1 \leq i \leq n$ , we have that

$$\begin{align*}p_{\mathcal{S}}(\psi_i,\Phi) = p_Y(\text{Res}_{\mathcal{S}/Y}(\psi_i), \Phi') =p_Y(\text{Res}_{\mathcal{S}/Y}(\phi_i), \Phi')= p_{\mathcal{S}}(\phi_i,\Phi) , \end{align*}$$

where $\Phi ' = \text {Res}_{\mathcal {S}/Y} \phi = \text {Res}_{\mathcal {S}/Y} \psi \in I_Y$ .

Proof Let us write $\Phi = \sum _{j=1}^{n} \Phi _j$ with $\Phi _j \in I_{\mathcal {S}_j}$ and $\Phi ' = \sum _{j=1}^n \Phi _j'$ , with $\Phi _j' \in I_{K_j}$ . Then we have that $\Phi _j = \text {Inf}_{\mathcal {S}_j/K_j} (\Phi ^{\prime }_j)$ . Indeed, first, we observe that $\Psi = \sum _{j=1}^{n} \text {Res}_{\mathcal {S}_j/K} \Phi _j \in I_K$ (see [Reference Shimura32, p. 85]), where $\Psi $ as in Remark 5.2. Moreover, we know that $\Phi = \phi + \psi $ with $\phi ,\psi \in I_{\mathcal {S}}$ as above, and we have seen that $\psi = \overline {\phi }$ when restricted to K via $K \hookrightarrow Y \otimes _{\mathbb {Q}} K = \mathcal {S}$ . But, on the other hand, we have seen that $\psi = \phi $ when restricted to Y, from which we obtain that $\Phi _j = \Phi _j' \otimes \tau + \Phi _j' \otimes \overline {\tau }$ , where $\tau $ is a fixed embedding of $K \hookrightarrow \mathbb {C}$ (i.e., a CM type for K). Since $\mathcal {S}_j = K_j \otimes _{\mathbb {Q}} K$ , the claim that $\Phi _j = \text {Inf}_{\mathcal {S}_j/K_j} (\Phi ^{\prime }_j)$ now follows.

The statement of the lemma is now obtained from the inflation-restriction properties of the periods (see [Reference Shimura32, p. 84]):

$$\begin{align*}p_{\mathcal{S}}(\psi_i,\Phi) = \prod_{j=1}^n p_{\mathcal{S}_j}(\psi_{ij},\Phi_j) = \prod_{j=1}^n p_{K_j}(\text{Res}_{\mathcal{S}_j/K_j}(\psi_{ij}), \Phi^{\prime}_j)=p_Y(\text{Res}_{\mathcal{S}/Y}(\psi_i), \Phi'), \end{align*}$$

where $\psi _{ij} \in J_{\mathcal {S}_j}$ induced by $\psi _i \in J_{\mathcal {S}} = \bigcup _{j=1}^n J_{\mathcal {S}_j}$ . Similarly follows also the other equality.

The above lemma shows that we have $\mathfrak {p}_{\infty } (w) = \mathfrak {p}_{\infty \rho } (w)$ for $w \in \mathfrak {B}$ and they are independent of the choice of the imaginary quadratic field K we chose above (and hence of the embedding to the unitary group). We then simply define $\mathfrak {p}(w)=\mathfrak {p}_{\infty }(w)=\mathfrak {p}_{\infty \rho }(w)$ for the period attached to CM-point $w\in \mathfrak {B}$ . By [Reference Shimura32, Proposition 11.5] and the definition of periods, we immediately have:

1. (1) The coset $\mathfrak {p}(w) GL_n(\overline {\mathbb {Q}})$ is determined by the point $w \in \mathfrak {B}$ independently of the embedding $(Y,h)$ chosen above.

2. (2) $\mathfrak {p} (\gamma w) GL_n(\overline {\mathbb {Q}}) = \lambda (\gamma , w) \mathfrak {p}(w) GL_n(\overline {\mathbb {Q}})$ for all $\gamma \in G(\mathbb {Q})$ .

5.2 Algebraic modular forms

We keep the notation from before. In particular, we write G for $G(T)$ and we have an embedding $\mathfrak {i}:G \to U(\mathcal {T})$ as above. For the following considerations, we need to augment our definition of modular forms from scalar-valued to vector-valued.

We start with a $\overline {\mathbb {Q}}$ -rational representation $\omega :\mathrm {GL}_n(\mathbb {C})\to \mathrm {GL}(V)$ . Given a function $f:\mathfrak {B}\to V$ and $g\in G$ , define $(f|_{\omega }g)(z)=\omega (\lambda (g,z))^{-1}f(gz)$ . For a congruence subgroup $\Gamma $ , the space of modular forms $M_{\omega }(\Gamma )$ consists of holomorphic function with the property $f|_{\omega }\gamma =f$ for all $\gamma \in \Gamma $ . Put $M_{\omega }=\bigcup M_{\omega }(\Gamma )$ where the union is over all congruence subgroups, and

$$\begin{align*}\mathfrak{A}_{\omega}=\bigcup_e\{g^{-1}f:f\in M_{\tau_e},0\neq g\in M_e\}, \end{align*}$$

$$\begin{align*}\mathfrak{A}_{\omega}(\Gamma)=\{h\in\mathfrak{A}_{\omega}:h|_{\omega}\gamma=h\text{ for }\gamma\in\Gamma\}, \end{align*}$$

where e runs over $\mathbb {Z}$ and $\tau _e$ denotes the representation defined by $\tau _e(x)=\det (x)^e\omega (x)$ .

We can compare the definition for our group with the unitary group. Let $\omega :\mathrm {GL}_n(\mathbb {C})\times \mathrm {GL}_n(\mathbb {C})\to \mathrm {GL}(V)$ be $\overline {\mathbb {Q}}$ -rational representation. Denote $\mathcal {A}_{\omega },\mathcal {A}_{\omega }(\Gamma )$ for modular function spaces for unitary group as in [Reference Shimura32, §5.3]. The composition of $\omega $ with the diagonal embedding $\mathrm {GL}_n\to \mathrm {GL}_n\times \mathrm {GL}_n$ gives a representation $\omega :\mathrm {GL}_n(\mathbb {C})\to \mathrm {GL}(V)$ . Clearly, if $f\in \mathcal {A}_{\omega }$ , then its pullback $f\circ \iota \in \mathfrak {A}_{\omega }$ is a quaternionic modular form. Moreover, $f\in \mathcal {A}_{\omega }(\overline {\mathbb {Q}})$ and $f\circ \iota $ is finite, then the pullback $f\circ \iota \in \mathfrak {A}_{\omega }(\overline {\mathbb {Q}})$ .

Even though we have provided a definition of algebraicity for modular forms on the bounded domain $\mathfrak {B}$ , we can transfer it also to the other realization of the symmetric spaces discussed in Section 2. Indeed, with the notation of Section 2.2, suppose we are given two of these realizations $(\mathfrak {i}_1,\Phi _1,H_1,K_1,\mathcal {G}_1)$ and $(\mathfrak {i}_2,\Phi _2,H_2,K_2,\mathcal {G}_2)$ , with $\mathfrak {i}_1=\mathfrak {i}_2$ both induced from an algebraic embedding $M_n(\mathbb {B})\to M_{2n}(K)$ , where K is an imaginary quadratic field which splits $\mathbb {B}$ . In particular, the matrix R in equation (2.2) has algebraic entries, and hence we obtain that the bijective map $\rho : \mathcal {H}_1\to \mathcal {H}_2$ as defined there is algebraic, in the sense that maps algebraic points of $\mathcal {H}_1$ to algebraic points of $\mathcal {H}_2$ . We also conclude from this that $\mu (z)$ , as defined in the same equation, is algebraic if z is. In particular, given any realization $\mathcal {H}$ , there is a bijection $\rho : \mathcal {H} \rightarrow \mathfrak {B} $ . We define the CM-points on $\mathcal {H}$ to be the inverse image with respect to $\rho $ of the CM-points of $\mathfrak {B}$ .

As we have discussed, every vector-valued modular form $g : \mathcal {H} \rightarrow V$ corresponds uniquely to a modular form $f:\mathfrak {B} \to V$ by the rule $g(z)=\omega (\mu (z))^{-1}f(\rho (z))$ . So it is enough to now observe that if w is a CM-point of $\mathcal {H}$ , which by definition means $\rho (w)$ is a CM-point of $\mathfrak {B}$ , and we have established above $\mu (w) \in GL_n(\overline {\mathbb {Q}})$ . Hence, we can use the same periods $\mathfrak {P}_{\omega }(w)$ for both f and g.

In particular, the algebraicity as defined for bounded domains can be transferred to the unbounded domain $\mathfrak {Z}$ . Let now $f:\mathfrak {Z}\to \mathbb {C}$ be a weight k modular form defined in Section 3, we can take its Fourier–Jacobi expansion. We denote the Fourier–Jacobi coefficients by $c(\tau ,f;v,w)$ . When $r=0,n=2m$ , we simply denote it by $c(\tau ,f)$ .

Proof This can be proved similarly as [Reference Shimura32, Propositions 11.11, 11.15, and 26.8]. See also [Reference Milne20, Proposition 7.2] for (1) and [Reference Garrett6] for (2) and (3).

We briefly explain the proof for (3). Let $r=0,n=2m$ , and $f\in M_k(\Gamma )$ for a congruence subgroup $\Gamma $ . Let V be the model of $\Gamma \backslash \mathfrak {Z}$ defined over $\overline {\mathbb {Q}}$ , then $\mathfrak {A}_0(\Gamma )$ can be identified with the function field of V. By the same method in [Reference Shimura32, Sections 6, 7], one can show that $g\in \mathfrak {A}_0(\Gamma )$ if and only if g has algebraic Fourier coefficients. We will reduce our problem for $f\in M_k$ to $\mathfrak {A}_0$ similarly to what is done in the proof of [Reference Shimura32, Proposition 11.11].

Let $\mathcal {W}$ be a dense subset of CM-points in $\mathfrak {Z}$ . We first assume that $\det (\mathfrak {p}(w))^{-k}f(w) \in \overline {\mathbb {Q}}$ for all $w\in \mathcal {W}$ where f is finite. Note that there exists a function $U\in \mathfrak {A}_k(\overline {\mathbb {Q}})$ on $\mathfrak {Z}$ holomorphic in w with $\det (U)(w)\neq 0$ . Indeed, denote $\mathcal {H}$ for the unbounded realization of $\mathcal {B}$ via the Cayley transform, and we can simply put $U(z)=R(z)$ for $z\in \mathfrak {Z}\hookrightarrow \mathcal {H}$ with R the function in [Reference Shimura32, Proposition 9.11]. We set $g:=\det (U)^{-k}f$ , and note that

$$\begin{align*}g(w) = \det\left(U(w)^{-1} \mathfrak{p}(w)\right)^k \det(\mathfrak{p}(w))^{-k} f(w). \end{align*}$$

But now we have that $U(w)^{-1}\mathfrak {p}(w)$ is $\overline {\mathbb {Q}}$ -rational since this holds for the function R in unitary case. That is, $g(w)$ is $\overline {\mathbb {Q}}$ -rational for every CM-point w where g is finite; thus, $g\in \mathfrak {A}_0(\overline {\mathbb {Q}})$ . Since $ f = \det (U)^{-k}g$ , we obtain that f also has algebraic Fourier expansion.

For the other direction, we keep the same notation. If $f \in M_k$ has algebraic Fourier expansion, then $g \in \mathfrak {A}_{0}(\overline {\mathbb {Q}})$ . For every CM-point w, we may choose the function U above such that U is finite at w and $U(w)$ is invertible. If f is finite at w, then so is g and $g(w)$ is $\overline {\mathbb {Q}}$ rational. The equality of $g(w)$ as above then shows that $\det (\mathfrak {p}(w))^{-k} f(w)$ is $\overline {\mathbb {Q}}$ -rational, and hence $f\in M_k(\overline {\mathbb {Q}})$ .

We end this subsection by giving a definition for adelic modular forms. Let $\mathbf {f}\in \mathcal {M}_k$ be an (adelic) modular form of scalar weight k, i.e., we are taking $\omega =\det ^k$ . We say that $\mathbf {f}$ is algebraic, denoted by $\mathbf {f}\in \mathcal {M}_k(\overline {\mathbb {Q}})$ if for a dense subset $\mathcal {W}$ of CM-points in $\mathfrak {Z}$ , $\mathfrak {P}_k(w)^{-1}\mathbf {f}(g_{\mathbf {h}}g)\in \overline {\mathbb {Q}}$ for all $w\in \mathcal {W}$ . Since $j(g,w)\in \overline {\mathbb {Q}}$ for CM-point w, this is the same as all component $f_j$ under correspondence $\mathbf {f}\leftrightarrow (f_0,{\ldots },f_h)$ are algebraic.

Let $r=0,n=2m$ , and keep the notation for Eisenstein series in previous sections. Let $\mathbf {E}_l(g,s)$ be a Siegel Eisenstein series for group $G_n$ . By explicit computation of Fourier expansion, we have $\mathbf {E}_l^{\ast }(g,l)\in \mathcal {M}_l(\overline {\mathbb {Q}})$ . Clearly, we also have $\mathbf {E}_l(g,l)\in \mathcal {M}_l(\overline {\mathbb {Q}})$ .

5.3 Differential operators and nearly holomorphic functions

In this subsection, we summarize some of the result of [Reference Shimura29] (see also [Reference Shimura32, Chapter 3]) on differential operators on type-D domains and then apply these operators to Siegel-type Eisenstein series. We will be working with the bounded realization of our symmetric space, but thanks to the remark above, we can transfer the definitions from one realization to the other. We set

$$\begin{align*}\mathfrak{B}=\{z\in\mathbb{C}_n^n:{^t\!z}=- z,z^{\ast}z<1_n\},\,\,\mathfrak{T}:=\{z\in\mathbb{C}_n^n:{^t\!z}=-z\},\,\,\eta(z):=1-z^{\ast}z. \end{align*}$$

Here, $\mathfrak {T}$ is the tangent space of $\mathfrak {B}$ at the origin $0$ .

Given a positive integer d and two finite-dimensional complex vector spaces W and V, we denote by $Ml_d(W,V)$ the vector space of all $\mathbb {C}$ -multilinear maps of $W\times \cdots \times W$ (d copies) into V and $S_d(W,V)$ the vector space of all homogeneous polynomial maps of W into V of degree d. We omit the symbol V if $V=\mathbb {C}$ . Given a representation $\omega :\mathrm {GL}_n(\mathbb {C})\to V$ , we define a representation $\{\omega \otimes \tau ^d,Ml_d(\mathfrak {T},V)\}$ by

$$\begin{align*}[(\omega\otimes\tau^d)(a)h](u_1,{\ldots},u_d)=\omega(a)h({^t\!a}u_1a,{\ldots},{^t\!a}u_da), \end{align*}$$

for $a\in \mathrm {GL}_n(\mathbb {C}),h\in Ml_d(\mathfrak {T},V),u_i\in \mathfrak {T}$ . In particular, taking $d=1$ and $\omega $ the trivial representation, we define the representation $\{\tau ,S_1(\mathfrak {T})\}$ of $\mathrm {GL}_n(\mathbb {C})$ by $[\tau (a)h](u)=h({^t\!a}ua)$ for $h\in S_1(\mathfrak {T}),u\in \mathfrak {T}$ .

Take an $\mathbb {R}$ -rational basis $\{\epsilon _{\nu }\}$ of $\mathfrak {T}$ over $\mathbb {C}$ and for $u\in \mathfrak {T}=\sum _{\nu }u_{\nu }\epsilon _{\nu }$ . For $z\in \mathfrak {B}$ , write $z=\sum _{\mu }z_{\nu }\epsilon _{\nu }$ . For $f\in C^{\infty }(\mathfrak {B},V)$ , we define $\mathfrak {D}f,\overline {\mathfrak {D}}f,\mathfrak {C}f\in C^{\infty }(\mathfrak {B},S_1(\mathfrak {T},V))$ by

$$\begin{align*}(\mathfrak{D}f)(u)=\sum_{\nu}u_{\nu}\frac{\partial f}{\partial z_{\nu}},\,\,(\overline{\mathfrak{D}}f)(u)=\sum_{\nu}u_{\nu}\frac{\partial f}{\partial \overline{z}_{\nu}},\,\,(\mathfrak{C}f)(u)=(\mathfrak{D}f)({^t\!\eta}(z)u\eta(z)). \end{align*}$$

We further define $\mathfrak {D}^df,\overline {\mathfrak {D}}^df,\mathfrak {C}^df$ by

$$\begin{align*}\mathfrak{D}^df=\mathfrak{DD}^{d-1}f,\,\,\overline{\mathfrak{D}}^df=\overline{\mathfrak{D}}\overline{\mathfrak{D}}^{d-1}f,\,\,\mathfrak{C}^df=\mathfrak{CC}^{d-1}f,\,\,\mathfrak{D}^0f=\overline{\mathfrak{D}}^0f=\mathfrak{C}f=f. \end{align*}$$

And, define $\mathfrak {D}_{\omega }^df\in C^{\infty }(\mathfrak {B},S_d(\mathfrak {T},V))$ by

$$\begin{align*}\mathfrak{D}_{\omega}^df=(\omega\otimes\tau^d)(\eta(z))^{-1}\mathfrak{C}^d[\omega(\eta(z))f]. \end{align*}$$

We now recall the important fact, due to Hua, Schmid, Johnson, and Shimura (see, for example, [Reference Shimura27]), that the representation $\{\tau ^d,S_d(\mathfrak {T})\}$ is the direct sum of irreducible representations and each irreducible constituent has multiplicity one. In particular, for each $\mathrm {GL}_n(\mathbb {C})$ -stable subspace $Z\subset S_d(\mathfrak {T})$ , we can define the projection map $\phi _Z$ of $S_d(\mathfrak {T})$ onto Z. Define $\mathfrak {D}_{\omega }^Zf\in C^{\infty }(\mathfrak {B},Z\otimes V)$ by $\mathfrak {D}_{\omega }^Zf=\phi _Z\mathfrak {D}_{\omega }^df$ .

We now set $r(z):=-\eta (z)^{-1}\overline {z}$ . Let d be a nonnegative integer and $\{\omega ,V\}$ the representation as before. A function $f\in C^{\infty }(\mathfrak {B},V)$ is called nearly holomorphic of degree d if it can be written as a polynomial in r, of degree less than d, with V-valued holomorphic functions on $\mathfrak {B}$ as coefficients. We denote the space of such functions by $\mathfrak {N}^d(\mathfrak {B},V)$ . Let $\mathfrak {N}^d_{\omega }$ be the space consisting of functions satisfying the modular properties as in $M_{\omega }$ but now replacing the holomorphic condition with nearly holomorphic. For a congruence subgroup $\Gamma $ , we can similarly define the space $\mathfrak {N}^d_{\omega }(\Gamma )$ . An exact same argument in the proof of [Reference Shimura32, Lemma 14.3] shows that this space is finite-dimensional over $\mathbb {C}$ .

Suppose V is $\overline {\mathbb {Q}}$ -rational. A function $f\in \mathfrak {N}_{\omega }^d$ is called algebraic, denoted by $f\in \mathfrak {N}_{\omega }^d(\overline {\mathbb {Q}})$ , if $\mathfrak {P}_{\omega }(w)^{-1}f(w)$ is $\overline {\mathbb {Q}}$ -rational for $w\in \mathcal {W}=\{g\cdot 0:g\in G(\overline {\mathbb {Q}})\}$ . Put $\mathfrak {N}_{\omega }^d(\Gamma ,\overline {\mathbb {Q}})=\mathfrak {N}_{\omega }^d(\overline {\mathbb {Q}})\cap \mathfrak {N}_{\omega }^d(\Gamma )$ . The proof of the following lemma is the same as the one in [Reference Shimura32, Theorem 14.9].

We now extend the above definitions to adelic modular forms. Let $\mathbf {f}\in \mathcal {M}_k$ and viewing it as a function on $G(\mathbb {A}_{\mathbf {h}})\times \mathfrak {B}$ by setting $\mathbf {f}(g_{\mathbf {h}},z)=j(g_z,z_0)^k\mathbf {f}(g_{\mathbf {h}}g_z)$ with $z=g_z\cdot 0\in \mathfrak {B}$ . Then $\mathfrak {D}_k\mathbf {f},\mathfrak {D}_k^Z\mathbf {f}$ is defined as applying differential operators on $z\in \mathfrak {B}$ . A function $\mathbf {f}:G(\mathbb {A}_{\mathbf {h}})\times \mathfrak {B}\to \mathbb {C}$ is called nearly holomorphic if it is nearly holomorphic in $z\in \mathfrak {B}$ . We can then define the space $\mathcal {N}_k^d$ and of nearly holomorphic modular forms as before. Similarly, we can define subspace $\mathcal {N}_k^d(\overline {\mathbb {Q}})$ . These definitions are equivalent to all components in the correspondence $\mathbf {f}\leftrightarrow (f_1,{\ldots },f_h)$ being nearly holomorphic or algebraic nearly holomorphic.

We now apply the differential operators to Siegel-type Eisenstein series and show that it is nearly holomorphic for certain values of s. We will keep the notation of Section 4, and so in particular $\mathbf {E}_l(g,s)$ is the Siegel-type Eisenstein series associated with group $G_n,n=2m$ of weight l and character $\chi $ .

Proof For this, we use [Reference Shimura27, Theorem 2D], which classifies the irreducible representations of $(\tau ^{mp}, S_{mp}(\mathfrak {T}))$ . In particular, for $p\in \mathbb {Z}$ and a weight q, we can define the operator $\Delta _q^p$ by $\Delta _q^p\mathbf {f}=(\mathfrak {D}_{\omega }^Z\mathbf {f})(\psi )$ with $\omega =\det ^q,Z=\mathbb {C}\psi \subset S_{mp}(\mathfrak {T})$ , and $\psi =\det ^{p/2}$ . Here, the square root of the determinant denotes the Pfaffian of the skew-symmetric matrix. Then

$$\begin{align*}\Delta_q^p\mathcal{N}_q^t(\overline{\mathbb{Q}})\subset\pi^{mp}\mathcal{N}_{q+p}^{t+mp}(\overline{\mathbb{Q}}). \end{align*}$$

We have shown that $\mathbf {E}_l(g,l)\in \mathcal {M}_l(\overline {\mathbb {Q}})$ , so $\Delta _l^p\mathbf {E}_l(g,l)\in \pi ^{mp}\mathcal {N}_{l+p}^{mp}(\overline {\mathbb {Q}})$ . Take $p=l-\mu $ , then by the explicit formula in [Reference Shimura27, Theorem 4.3], we have

$$\begin{align*}\Delta_{\mu}^p\mathbf{E}_{\mu}(g,\mu)=C\cdot\mathbf{E}_l(g,\mu),\,\,\,\text{with}\,\,C\in\overline{\mathbb{Q}}^{\times}. \end{align*}$$

This concludes the proof of the proposition.