2.1 Differential operators

For a real-analytic function f on $G(\mathbb {R})$ , the Lie algebra ${\mathfrak {sl}}_2(\mathbb {C})$ acts via

(2.1) $$ \begin{align} A(f)(g) := \partial_t f(g e^{tA}) \mid_{t = 0},~ A \in {\mathfrak{sl}}_2(\mathbb{C}). \end{align} $$

We define the raising and lowering operator

(2.2) $$ \begin{align} R := \frac{1}{2} \begin{pmatrix} 1 & i\\ i & -1 \end{pmatrix},~ L := \frac{1}{2} \begin{pmatrix} 1 & -i\\ -i & -1 \end{pmatrix}. \end{align} $$

If f is right $K_\infty $ -equivariant of weight k, then we have

$$ \begin{align*} \sqrt{v}^{-(k+2)} R(f)(g_\tau) = R_{\tau, k} (\sqrt{v}^{-k} f(g_\tau)),~\sqrt{v}^{-(k-2)} L(f)(g_\tau) = L_{\tau, k} (\sqrt{v}^{-k} f(g_\tau)), \end{align*} $$

where $R_{\tau , k}$ and $L_{\tau , k}$ are the usual raising and lowering operators given by

(2.3) $$ \begin{align} R_{\tau, k} := 2i \partial_\tau + \frac{k}{v},~L_{\tau, k} := -2iv^2 \partial_{\overline{\tau}}. \end{align} $$

We say that f is holomorphic, resp. anti-holomorphic, if $L(f) = 0$ , resp. $R(f) = 0$ .

For $ r\in \mathbb {N}_0$ and $k_1, k_2 \in {\tfrac {1}{2}}\mathbb {Z}$ , define

(2.4) $$ \begin{align} \begin{aligned} Q_{r, (k_1, k_2)}(X, Y) &:= \sum_{s = 0}^r \binom{r + k_1 - 1}{s} \binom{r + k_2 - 1}{r - s} X^{r-s}(-Y)^s,\\ \tilde{Q}_r(X, Y) &:= \frac{Q_{r, (1, 1)}(X, Y) (X + Y)}{X + (-1)^r Y} \end{aligned} \end{align} $$

in $\mathbb {Q}[X, Y]$ . We omit $(k_1, k_2)$ from the notation when it is $(1, 1)$ , in which case

$$ \begin{align*}Q_r(X, Y) = (X + Y)^r P_r\left ( \frac{ X - Y}{X+Y} \right ), \end{align*} $$

with $P_r(x)$ the r-th Legendre polynomial given explicitly by

(2.5) $$ \begin{align} P_r(x) = 2^{-r} \sum_{s = 0}^r \binom{r}{s}^2(x-1)^{r-s} (x+1)^s = (-1)^{r_0} \sum_{s = 0}^{r_0} \binom{r_0 - r - 1/2}{r_0 - s} \binom{r - r_0 - 1/2}{s} x^{r-2s}, \end{align} $$

where $r_0 := \lfloor r/2 \rfloor $ . The second identity comes from (3.133) on page 38 of [Reference GouldGou72] and direct calculation. We thank Zhiwei Sun for pointing us to this reference.

From the differential equation satisfied by $P_r$ , we have

(2.6) $$ \begin{align} ( \partial_X \partial_Y)(Q_r(X, Y)(X+Y)) = (r+1)(\partial_X + \partial_Y)Q_r(X, Y). \end{align} $$

For $A \in {\mathfrak {sl}}_2(\mathbb {C})$ , denote

(2.7) $$ \begin{align} A_1 = (A, 0),~ A_2 = (0, A) \end{align} $$

in ${\mathfrak {sl}}_2(\mathbb {C})^2$ . Then we have two operators

(2.8) $$ \begin{align} \mathrm{RC}_{r, (k_1, k_2)} := (-4\pi)^{-r} Q_{r, k_1, k_2}(R_1, R_2),~ \widetilde{\mathrm{RC}}_r := (-4\pi)^{-r} \tilde{Q}_r(R_1, R_2) \end{align} $$

on real-analytic functions on $G(\mathbb {R})^2$ . If $f: G(\mathbb {R})^2 \to \mathbb {C}$ is holomorphic and right $K_\infty ^2$ -equivariant of weight $(k_1, k_2)$ , then $\mathrm {RC}_{r, (k_1, k_2)}(f)^\Delta : G(\mathbb {R}) \to \mathbb {C}$ is holomorphic and right $K_\infty $ -equivariant of weight $k_1 + k_2 + 2r$ , and the operator $\mathrm {RC}$ is usually called the Rankin-Cohen operator. Here, $f^\Delta (g) := f(g^\Delta ) = f(g, g)$ is the restriction of f to the diagonal $G(\mathbb {R}) \subset G(\mathbb {R})^2$ . In fact, we have (see [Reference Bruinier, van der Geer, Harder and ZagierBvdGHZ08, Proposition 19])

$$ \begin{align*}\mathrm{RC}_{r, (k_1, k_2)}(f)^\Delta(g_z) = (2\pi i)^{-r} \left ( Q_r(\partial_{z_1}, \partial_{z_2}) f(g_{z_1}, g_{z_2})\right ) \mid_{z_1 = z_2 = z}. \end{align*} $$

For example, if $f(g_{z_1}, g_{z_2}) = {\mathbf {e}}(m_1 z_1 + m_2 z_2)$ , then

(2.9) $$ \begin{align} \mathrm{RC}_{r, (k_1, k_2)}(f)^\Delta(g_{z}) = Q_{r, (k_1, k_2)}(m_1, m_2) {\mathbf{e}}((m_1 + m_2)z). \end{align} $$

From Lemma 2.2 in [Reference LiLi23], we know that there are unique constants $c^{(r; k_1, k_2)}_{\ell } \in \mathbb {Q}$ such that

(2.10) $$ \begin{align} (4\pi)^{-r} (R_{1}^r f)^\Delta = \sum_{\ell = 0}^r c^{(r; k_1, k_2)}_\ell (4\pi)^{-r+\ell} R^{r-\ell}\mathrm{RC}_{\ell, (k_1, k_2)}(f)^\Delta \end{align} $$

whenever $k_1 + k_2 + 2r < 2$ .

2.2 Quadratic space associated to real quadratic field

Let $F = \mathbb {Q}(\sqrt {D})$ be a real quadratic field, which becomes a $\mathbb {Q}$ -quadratic space of signature $(1, 1)$ with respect to the quadratic form

$$ \begin{align*}Q_a (\lambda):= a {\mathrm{Nm}}(\lambda) ,~ \lambda \in F \end{align*} $$

for any $a \in \mathbb {Q}^\times $ . We denote this quadratic space by $V_a $ and identify

(2.11) $$ \begin{align} \begin{aligned} \iota_a: V_a (\mathbb{R}) = F \otimes_{\mathbb{Q}} \mathbb{R} &\cong \mathbb{R}^2 \\ (\lambda_1, \lambda_2) &\mapsto (\lambda_1, {\mathrm{sgn}}(a )\lambda_2)\sqrt{|a |}. \end{aligned} \end{align} $$

This is an isometry, where the quadratic form on $\mathbb {R}^2$ is given by $Q((x_1, x_2)) = x_1x_2$ . The special orthogonal group $H_a := \mathrm {SO}({V_a })$ satisfies

$$ \begin{align*}H_a (\mathbb{Q}) \cong F^1 ( \cong F^\times/\mathbb{Q}^\times), \end{align*} $$

where $ F^1 := \{\mu \in F: {\mathrm {Nm}}(\mu ) = 1\}$ acts on V via multiplication. Furthermore, we identify

$$ \begin{align*}H_a (\mathbb{R}) \cong \mathbb{R}^\times, \end{align*} $$

where $t \in \mathbb {R}^\times $ acts on $V_a (\mathbb {R}) = F \otimes _{\mathbb {Q}} \mathbb {R} \cong \mathbb {R}^2$ via $(x_1, x_2) \mapsto (t x_1, t^{-1}x_2)$ . Note that the invariant measure on $H(\mathbb {R}) \cong \mathbb {R}^\times $ is $\frac {dt}{|t|}$ . We have a character ${\mathrm {sgn}} : H_a (\mathbb {R}) \cong \mathbb {R}^\times \to \{\pm 1\}$ . Denote

$$ \begin{align*}H_a (\mathbb{R})^+ := \ker({\mathrm{sgn}}) \cong \mathbb{R}_{> 0} \end{align*} $$

its kernel, which is the connected component of $H_a(\mathbb {R})$ containing the identity. We also denote

(2.12) $$ \begin{align} H_a (\mathbb{Q})^+ := H_a (\mathbb{R})^+ \cap H_a (\mathbb{Q}) \end{align} $$

which is an index 2 subgroup of $H_a (\mathbb {Q})$ .

Remark 2.1. Let $\chi = \chi _{E/F}$ be a Hecke character associated to a quadratic extension $E/F$ . Suppose $E/\mathbb {Q}$ is biquadratic. Then $\chi \mid _{\mathbb {A}^\times }$ is trivial and $\chi $ factors through the map ${\mathrm {Nm}}^-: \mathbb {A}_F^\times \to H_a (\mathbb {A})$ ; that is, there exists $\varrho = \varrho _{E/F}: H_a (\mathbb {A}) \to \{\pm 1\}$ such that

(2.13) $$ \begin{align} \varrho \circ {\mathrm{Nm}}^- = \chi. \end{align} $$

Note that $\varrho $ is odd if and only if E is totally imaginary. We also denote the compact subgroup

(2.14) $$ \begin{align} K_\varrho := H_a(\hat{\mathbb{Z}}) \cap \mathrm{ker}(\varrho). \end{align} $$

Note that $H_a (\mathbb {Q})\backslash H_a (\hat {\mathbb {Q}})/K_\varrho $ is a finite set.

2.3 The Weil representation and theta functions

Let $(V, Q)$ be a rational quadratic space of signature $(p, q)$ , and $H_V := {\mathrm {SO}}(V)$ . For a subfield $E \subset \mathbb {C}$ , we denote ${\mathcal {S}}(\hat V; E)$ , resp. ${\mathcal {S}}(V_p; E)$ , to denote the space of Schwartz functions on $\hat V := V(\hat {\mathbb {Q}})$ , resp. $V_p := V(\mathbb {Q}_p)$ , with values in E, which is an E-vector space. We omit E from the notation if it is $\mathbb {Q}$ . However, we write ${\mathcal {S}}(V(\mathbb {R}))$ and ${\mathcal {S}}(V(\mathbb {A}))$ for the space of Schwartz function on $V(\mathbb {R})$ and $V(\mathbb {A})$ valued in $\mathbb {C}$ , respectively.

For a latticeFootnote ³ $L \subset V$ , we denote $L^\vee \subset V$ its dual lattice, $\hat L := L \otimes \hat {\mathbb {Z}}$ , $\hat L^\vee := L^\vee \otimes \hat {\mathbb {Z}}$ and

(2.15) $$ \begin{align} L_{m, \mu} := \{\lambda \in L+ \mu: Q(\lambda) = m\} \end{align} $$

for $m \in \mathbb {Q}, \mu \in L^\vee /L$ . The finite dimensional E-subspace

(2.16) $$ \begin{align} {\mathcal{S}}(L; E) := \{\phi \in {\mathcal{S}}(\hat V; E): \phi \text{ is } \hat{L}\text{-invariant with support on } \hat L^\vee\} \subset {\mathcal{S}}(\hat V; E), \end{align} $$

is spanned by $\{\phi _{L + \mu } : \mu \in \hat L^\vee /\hat L \cong L^\vee /L\}$ with

(2.17) $$ \begin{align} \phi_{L + \mu} := \operatorname{Char}(\hat L + \mu) \in {\mathcal{S}}(\hat V). \end{align} $$

For full sublattices $M \subset L \subset V$ , it is clear that

(2.18) $$ \begin{align} {\mathcal{S}}(L; E) \subset {\mathcal{S}}(M; E) \subset {\mathcal{S}}(\hat V; E). \end{align} $$

As above, we also denote ${\mathcal {S}}(L) := {\mathcal {S}}(L; \mathbb {Q})$ . Furthermore, since

$$ \begin{align*}{\mathcal{S}}(\hat V) = \bigcup_{L \subset V \text{ lattice}} {\mathcal{S}}(L), \end{align*} $$

we have ${\mathcal {S}}(\hat V; E) = {\mathcal {S}}(\hat V) \otimes _{\mathbb {Q}} E$ for any subfield $E \subset \mathbb {C}$ .

Suppose $V = V_1 \oplus V_2$ is a decomposition of rational quadratic spaces. For any lattice $L_i \subset V_i$ , we have ${\mathcal {S}}(L_1 \oplus L_2; E) = {\mathcal {S}}(L_1; E) \otimes {\mathcal {S}}(L_2; E) \subset {\mathcal {S}}(\hat V_1; E) \otimes {\mathcal {S}}(\hat V_2; E)$ via the natural restriction map. This also gives us

(2.19) $$ \begin{align} {\mathcal{S}}(\hat V; E) = {\mathcal{S}}(\hat V_1; E) \otimes {\mathcal{S}}(\hat V_2; E) = \bigoplus_{L_1 \subset V_1,~ L_2 \subset V_2 \text{ lattices}} {\mathcal{S}}(L_1; E) \otimes {\mathcal{S}}(L_2; E). \end{align} $$

Combining with equation (2.18), we see that for any given $\phi \in {\mathcal {S}}(\hat V; E)$ , we can find a lattice $L = L_1 \oplus L_2 \subset V$ such that $L_i \subset V_i$ and

(2.20) $$ \begin{align} \phi \in {\mathcal{S}}(L; E) = {\mathcal{S}}(L_1; E) \otimes {\mathcal{S}}(L_2; E). \end{align} $$

Let $\tilde G(\mathbb {A}) := \mathrm {Mp}_2(\mathbb {A})$ be the metaplectic cover of $G(\mathbb {A})$ . The group $\tilde G(\mathbb {A}) \times H_V(\mathbb {A})$ acts on ${\mathcal {S}}(V(\mathbb {A}))$ via the Weil representation $\omega = \omega _{V, \psi }$ (see [Reference KudlaKud94, section 5] for explicit formula). For each prime $p \le \infty $ , we also have the local Weil representation $\omega _p$ of $G(\mathbb {Q}_p)$ acting on ${\mathcal {S}}(V(\mathbb {Q}_p); \mathbb {C})$ . Then $\omega _f := \otimes _{p < \infty } \omega _p$ gives a representation of $G(\hat {\mathbb {Q}})$ on ${\mathcal {S}}(\hat V; \mathbb {C})$ .

For any lattice $L \subset V$ , the subspace ${\mathcal {S}}(L; \mathbb {C})$ defined in (2.16) is a $K_f$ -invariant subspace with $K_f := G(\hat {\mathbb {Z}})$ . It has a unitary pairing $\langle , \rangle $ with the vector space $\mathbb {C}[L^\vee /L] := \oplus _{\mu \in L^\vee /L} \mathbb {C} \mathfrak {e}_\mu $ given by

(2.21) $$ \begin{align} \langle \mathfrak{e}_{\mu} , \phi_{\mu'} \rangle := \langle \mathfrak{e}_{\mu} , \phi_{\mu'} \rangle_L := \begin{cases} 1,& \mu = \mu',\\ 0,&\text{ otherwise.} \end{cases} \end{align} $$

More generally, if $L = L_1 \oplus L_2$ , we have $L^\vee /L = L_1^\vee /L_1 \oplus L_2^\vee / L_2$ and $\mathbb {C}[L^\vee /L] \cong \mathbb {C}[L_1^\vee /L_1] \otimes \mathbb {C}[L_2^\vee /L_2]$ . Therefore, we can extend the pairing above to

(2.22) $$ \begin{align} \langle \cdot, \cdot\rangle_{L_2} : \mathbb{C}[L^\vee/L] \times {\mathcal{S}}(L_2; \mathbb{C}) \to \mathbb{C}[L_1^\vee/L_1],~ ( \mathfrak{v}_1 \otimes \mathfrak{v}_2 ,\phi) \mapsto \langle\mathfrak{v}_2 ,\phi \rangle_{}\mathfrak{v}_1 \end{align} $$

with $\mathfrak {v}_i \in \mathbb {C}[L_i^\vee /L_i]$ and $\phi \in {\mathcal {S}}(L_2; \mathbb {C})$ .

With respect to the perfect pairing in (2.21), the unitary dual of $\omega _f$ is the representation $\rho _L$ on $\mathbb {C}[L^\vee /L]$ given by

(2.23) $$ \begin{align} \begin{aligned} \rho_L(n(1))(\mathfrak{e}_\mu) &:= {\mathbf{e}}(Q(\mu)) \mathfrak{e}_\mu, \\ \rho_L(w)(\mathfrak{e}_\mu) &:= \frac{{\mathbf{e}}(-(p-q)/8)}{\sqrt{|L^\vee/L|}} \sum_{\mu \in L^\vee/L} {\mathbf{e}}(-(\mu, \mu')) \mathfrak{e}_{\mu'}, \end{aligned} \end{align} $$

where $(p, q)$ is the signature of $V(\mathbb {R})$ . This is the Weil representation on finite quadratic modules used by Borcherds in [Reference BorcherdsBor98]. If we identify ${\mathcal {S}}(L; \mathbb {C})$ and $\mathbb {C}[L^\vee /L]$ with $\mathbb {C}^{|L^\vee /L|}$ via the bases $\{\phi _\mu : \mu \in \hat L^\vee /\hat L\}$ and $\{\mathfrak {e}_\mu : \mu \in L^\vee / L\}$ , respectively, then $\omega _f = \overline {\rho _L} = \rho _L^{-1}$ . For full sublattices $M \subset L$ , the following trace map

(2.24) $$ \begin{align} \operatorname{Tr}^L_M: \mathbb{C}[L^\vee/L] \to \mathbb{C}[M^\vee/M],~ \mathfrak{e}_\mu \mapsto \frac{1}{[L:M]}\sum_{h \in M^\vee/M,~ h \equiv \mu \bmod{L}} \mathfrak{e}_h \end{align} $$

intertwines the Weil representation and is compatible with the inclusion in (2.18) in the sense that

(2.25) $$ \begin{align} \langle \operatorname{Tr}^L_M(\mathfrak{v}), \phi \rangle_M = \langle \mathfrak{e}, \phi \rangle_L \end{align} $$

for any $\mathfrak {v} \in \mathbb {C}[L^\vee /L], \phi \in {\mathcal {S}}(L; \mathbb {C})$ .

The following result will be very useful for us later.

Lemma 2.2. For any prime p, the local Weil representation $\omega _p$ descends to ${\mathcal {S}}(V(\mathbb {Q}_p); \mathbb {Q}(\zeta _{p^\infty }))$ with

(2.26) $$ \begin{align} \mathbb{Q}(\zeta_{p^\infty}) := \bigcup_{n \ge 1} \mathbb{Q}(\zeta_{p^n}) \subset \mathbb{Q}^{\mathrm{ab}} \end{align} $$

the maximal abelian extension of $\mathbb {Q}$ ramified only at p.

As usual, we let $H_{k, L}(\Gamma )$ denote the space of harmonic Maass forms valued in $\mathbb {C}[L^\vee /L]$ of weight $k \in \frac {1}{2}\mathbb {Z}$ and representation $\rho _L$ on a congruence subgroup $\Gamma \subset \operatorname {SL}_2(\mathbb {Z})$ (see [Reference Bruinier and FunkeBF04, section 3]). It contains the subspace $M^!_{k, L}(\Gamma )$ of vector-valued weakly holomorphic modular forms. Post-composing with $\operatorname {Tr}^L_M$ in (2.24) induces a map $H_{k, L}(\Gamma ) \to H_{k, M}(\Gamma )$ , which preserves holomorphicity and rationality of holomorphic part Fourier coefficients. If $L^\vee /L$ is trivial (resp. $\Gamma = \operatorname {SL}_2(\mathbb {Z})$ ), then we drop L (resp. $\Gamma $ ) from the above notations. Furthermore, we let

(2.27) $$ \begin{align} M^{!, \infty}_{k}(\Gamma):= \{f \in M^!_k(\Gamma) : f \text{ is holomorphic away from the cusp } \infty \} \end{align} $$

and denote for $f(\tau ) = \sum _{m \in \mathbb {Q},~ \mu \in L^\vee /L} c(m, \mu ) q^m \mathfrak {e}_\mu \in M^!_{k, L}$

(2.28) $$ \begin{align} \mathrm{prin}(f) := \sum_{m \in \mathbb{Q}_{< 0},~ \mu \in L^\vee/L} c(m, \mu) q^m \mathfrak{e}_\mu \end{align} $$

the principal part of f.

In [Reference McGrawMcG03, Theorem 4.3], McGraw extended the representation $\rho _L$ to the metaplectic cover of ${\mathrm {GL}}_2$ . To simplify the notation, we recall it here for lattices with even rank, in which case this extension factors through ${\mathrm {GL}}_2$ . Using the short exact sequence

$$ \begin{align*}1 \to \operatorname{SL}_2 \to {\mathrm{GL}}_2 \stackrel{\det}{\to} \mathbb{G}_m \to 1,\end{align*} $$

we can identify ${\mathrm {GL}}_2 \cong \operatorname {SL}_2 \rtimes T$ . Then $\omega _f$ extends to a $\mathbb {Q}$ -linear action of ${\mathrm {GL}}_2(\hat {\mathbb {Q}}) = {\mathrm {GL}}_2(\mathbb {Q}){\mathrm {GL}}_2(\hat {\mathbb {Z}})$ on ${\mathcal {S}}(\hat V; \mathbb {Q}^{\mathrm {ab}})$ via

(2.29) $$ \begin{align} (\omega_f(g, t(a))\phi)(x) := (\omega_f(g)\sigma_a(\phi))(x) = \sigma_a( (\omega_f(t(a)^{-1}gt(a))(\phi))(x)) \end{align} $$

for $ g \in \operatorname {SL}_2(\hat {\mathbb {Q}}), a \in \hat {\mathbb {Q}}^\times , \phi \in {\mathcal {S}}(\hat V; \mathbb {Q}^{\mathrm {ab}})$ , where $\sigma _a \in \operatorname {Gal}(\mathbb {Q}^{\mathrm {ab}}/\mathbb {Q})$ satisfies $\sigma _a(\psi _f(a')) = \psi _f(aa')$ for all $a , a' \in \hat {\mathbb {Q}}^\times $ and acts on ${\mathcal {S}}(\hat V; \mathbb {Q}^{\mathrm {ab}})$ via its action on $\mathbb {Q}^{\mathrm {ab}}$ . This gives us

(2.30) $$ \begin{align} {\mathcal{S}}(\hat V) = {\mathcal{S}}(\hat V; \mathbb{Q}^{\mathrm{ab}})^{T(\hat{\mathbb{Z}})}. \end{align} $$

For each $p < \infty $ , this gives an extension of $\omega _p$ to a $\mathbb {Q}$ -linear action of ${\mathrm {GL}}_2(\mathbb {Q}_p)$ on ${\mathcal {S}}(V_p; \mathbb {Q}(\zeta _{p^\infty }))$ , which satisfies

(2.31) $$ \begin{align} {\mathcal{S}}(V_p) = {\mathcal{S}}(V_p; \mathbb{Q}(\zeta_{p^\infty}))^{T(\mathbb{Z}_p)}. \end{align} $$

For $\varphi \in {\mathcal {S}}(V(\mathbb {A}))$ , we have the theta function

(2.32) $$ \begin{align} \theta_V(g, h, \varphi) := \sum_{x \in V(\mathbb{Q})} (\omega(g) \varphi)(h^{-1} x) \end{align} $$

for $(g, h) \in (G \times H_V)(\mathbb {A})$ as usual. For a lattice $L \subset V$ , we also denote

(2.33) $$ \begin{align} \Theta_L(\tau, h) := v^{-(p-q)/4}\sum_{\mu \in L^\vee/L} \theta_V(g_\tau, h, \phi_\mu\phi_\infty) \phi_\mu \end{align} $$

the vector-valued theta function with $\phi _\infty $ the Gaussian.

2.4 CM points and higher Green functions

We follow [Reference Bruinier, Kudla and YangBKY12] and [Reference Bruinier, Ehlen and YangBEY21] to recall CM points and higher Green functions. Let $(\mathrm {V}, Q)$ be a rational quadratic space of signature $(n, 2)$ , and ${\tilde {H}} = {\tilde {H}}_{\mathrm {V}} := \mathrm {GSpin}(\mathrm {V})$ . For an open compact subgroup $K \subset {\tilde {H}}(\hat {\mathbb {Q}})$ , we have the associated Shimura variety $X_K$ , whose $\mathbb {C}$ -points are given by

$$ \begin{align*}X_K(\mathbb{C}) = {\tilde{H}}(\mathbb{Q}) \backslash( \mathbb{D}\times {\tilde{H}}(\hat{\mathbb{Q}}) / K). \end{align*} $$

Here, $\mathbb {D} = \mathbb {D}^+ \sqcup \mathbb {D}^-$ is the symmetric space associated to $\mathrm {V}(\mathbb {R})$ . For $m \in \mathbb {Q}$ and $\varphi \in {\mathcal {S}}(\hat {\mathrm {V}}; \mathbb {C})$ , one can define the special divisor $Z(m, \varphi )$ on $X_K$ (see, for example, [Reference Bruinier, Ehlen and YangBEY21, section 2]).

The CM points on $X_K$ can be described as follows. For a totally real field F of degree d with real embeddings $\sigma _j, 1 \le j \le d$ , denote $\alpha _j := \sigma _j(\alpha )$ for $\alpha \in F$ . Suppose $\alpha _{j_0} < 0$ for some $j_0$ and $\alpha _j> 0$ when $j \neq j_0$ . Then a CM quadratic extension $E/F$ becomes an F-quadratic space $W = W_\alpha = E$ with respect to the quadratic form $Q_\alpha := \alpha {\mathrm {Nm}}_{E/F}$ . Suppose there is an isometric embedding as in (1.5). Then the subspace $W \otimes _{F, \sigma _{j_0}} \mathbb {R} \subset \mathrm {V}(\mathbb {R})$ is a negative 2-plane and determines a point $z_0^\pm \in \mathbb {D}^\pm $ with a choice of orientation. For convenience, we denote

(2.34) $$ \begin{align} z_0 := z_0^+. \end{align} $$

The group $\mathrm {Res}_{F/\mathbb {Q}}({\mathrm {SO}}(W))$ is contained in ${\mathrm {SO}}(\mathrm {V})$ , whose preimage in ${\tilde {H}}_{\mathrm {V}}$ is a torus denoted by $T_W$ . Note that $T_W(\mathbb {Q}) = E^\times /F^1$ . We denote the CM cycle on $X_K$ associated to $T_W$ by

(2.35) $$ \begin{align} Z(W) := T_W(\mathbb{Q}) \backslash (\{z_0^\pm\} \times T_W(\hat{\mathbb{Q}})/K_W) \subset X_K \end{align} $$

with $K_W := K \cap T_W(\hat {\mathbb {Q}})$ . It is defined over F, and its Galois conjugates are the CM cycles $Z(W(j))$ with $1 \le j \le d$ , where $W(j)$ is the neighborhood F-quadratic spaces at $\sigma _j$ of admissible incoherent $\mathbb {A}_F$ -quadratic space $\mathbb {W}$ associated to W (see [Reference Bruinier and YangBY11, Reference Bruinier, Kudla and YangBKY12] for details). Note that $W(j) = W_{\alpha (j)}$ for some $\alpha (j) \in F^\times $ and $\alpha (j_0) = \alpha $ . For totally positive $t \in F$ , we define the ‘Diff’ set

(2.36) $$ \begin{align} \mathrm{Diff}(W, t) := \{\mathfrak{p}: W_{\mathfrak{p}} \text{ does not represent }t\} \end{align} $$

following [Reference KudlaKud97]. Note that this set is finite and odd (see [Reference Yang and YinYY19, Proposition 2.7]).

When F is real quadratic (i.e. $d = 2$ ), for $\alpha \in F^\times $ with ${\mathrm {Nm}}(\alpha ) < 0$ , we set

(2.37) $$ \begin{align} {\alpha^\vee} := \alpha(2) \in F^\times. \end{align} $$

Then $({\alpha ^\vee })^\vee = \alpha $ . Note that ${\alpha ^\vee }$ is not necessarily the Galois conjugate of $\alpha $ !

Denote

(2.38) $$ \begin{align} \sigma_0 := \frac{n}{4} - \frac{1}{2}. \end{align} $$

Let $L \subset \mathrm {V}$ be an even integral lattice such that K stabilizes $\hat L$ . For $\mu \in L^\vee /L$ and $m \in \mathbb {Z} + Q(\mu )$ , we write $Z(m, \mu ) := Z(m, \phi _\mu )$ . The automorphic Green function on $X_K \backslash Z(m, \mu )$ is defined by

(2.39) $$ \begin{align} \Phi_{m, \mu}(z, h, s) := 2 \frac{\Gamma(s + \sigma_0 )}{\Gamma(2s)} \sum_{ \lambda \in h(L_{m, \mu})} \left ( \frac{m}{Q(\lambda_{z^\perp})} \right )^{s + \sigma_0} F\left ( s + \sigma_0 , s - \sigma_0, 2s; \frac{m}{Q(\lambda_{z^\perp})} \right ), \end{align} $$

for $\mathrm {Re}(s)> \sigma _0 + 1$ , where $F(a, b, c; z)$ is the Gauss hypergeometric function [Reference Abramowitz and StegunAS64, Chapter 15]. At $Z(m, \mu )$ , the function $\Phi _{m, \mu }$ has logarithmic singularity.

At $s = \sigma _0 + 1 + r$ with $r \in \mathbb {N}$ , the function $\Phi _{m, \mu }(z, h, s)$ is called a higher Green function. For a harmonic Maass form $f = \sum _{m, \mu } c(m, \mu ) q^{-m} \phi _\mu + O(1) \in H_{k - 2r, L}$ with $k:= -2\sigma _0$ , define

(2.40) $$ \begin{align} \Phi_f^r(z, h) := r! \sum_{m> 0,~ \mu \in L'/L} c(m, \mu) m^r \Phi_{m, \mu} (z, h, \sigma_0 + 1 + r) \end{align} $$

to be the associated higher Green function. Following from the work of Borcherds [Reference BorcherdsBor98] and generalization by Bruinier [Reference BruinierBru02] (also see [Reference Bruinier, Ehlen and YangBEY21, Proposition 4.7]), the function $\Phi _f^r$ has the following integral representation:

(2.41) $$ \begin{align} \begin{aligned} \Phi_f^r(z, h) &= (4\pi)^{-r} \lim_{T \to \infty} \int_{{\mathcal{F}}_T}^{} \langle R^r_\tau f(\tau), \overline{\Theta_L(\tau, z, h)} \rangle d \mu(\tau)\\ &= (-4\pi)^{-r} \lim_{T \to \infty} \int_{{\mathcal{F}}_T}^{} \langle f(\tau), \overline{R^r_\tau \Theta_L(\tau, z, h)} \rangle d \mu(\tau), \end{aligned} \end{align} $$

where ${\mathcal {F}}_T$ is the truncated fundamental domain of $\operatorname {SL}_2(\mathbb {Z}) \backslash \mathbb {H}$ at height $T> 1$ and $d\mu $ is the invariant measure given in (3.20). It has logarithmic singularity along the special divisor

(2.42) $$ \begin{align} Z_f := \sum_{m> 0,~ \mu \in L^\vee/L} c(-m, \mu) Z(m, \mu) \end{align} $$

on $X_K$ . Note that $[z_0, h] \in Z(W) \cap Z_f$ if and only if

(2.43) $$ \begin{align} h(L_{m, \mu}) \cap z_0^\perp \neq \emptyset \end{align} $$

for some $m, \mu $ with $c(-m, \mu ) \neq 0$ .

2.5 Product of modular curves as a Shimura variety

We follow and slightly modify [Reference Yang and YinYY19, section 3] to express $X_0(N) \times X_0(N)$ as $\mathrm {O}(2, 2)$ orthogonal Shimura variety. Consider $(\mathrm {V}, Q) = (M_2(\mathbb {Q}), \det )$ , and the lattice

$$ \begin{align*}L := \left\{ \begin{pmatrix} a & b\\ Nc & d \end{pmatrix}: a, b, c, d \in \mathbb{Z} \right\} \subset \mathrm{V} \end{align*} $$

for any $N \in \mathbb {N}$ . Then the dual lattice $L^\vee $ is given by

$$ \begin{align*}L^\vee := \left\{ \begin{pmatrix} a & b/N\\ c & d \end{pmatrix}: a, b, c, d \in \mathbb{Z} \right\} \subset \mathrm{V}, \end{align*} $$

and $L^\vee /L \cong (\mathbb {Z}/N\mathbb {Z})^2$ is isomorphic to that of a scaled hyperbolic plane.

For $g_j \in \operatorname {SL}_2(\mathbb {Q})$ and $\Lambda \in \mathrm {V}(\mathbb {Q})$ , the map

(2.44) $$ \begin{align} \Lambda \mapsto g_1 \Lambda g_2^{-1} \end{align} $$

gives $\operatorname {SL}_2 \times \operatorname {SL}_2 \cong \operatorname {Spin}(\mathrm {V})$ and identifies $H_{\mathrm {V}}$ as a subgroup of ${\mathrm {GL}}_2 \times {\mathrm {GL}}_2$ [Reference Yang and YinYY19, section 3.1]. Let $K(N) := K(\Gamma _0(N)) \subset {\mathrm {GL}}_2(\hat {\mathbb {Z}})$ be the open compact subgroup in [Reference Yang and YinYY19, section 3.1] and $K := (K(N) \times K(N)) \cap H_{\mathrm {V}}(\hat {\mathbb {Q}})$ . Then the map

(2.45) $$ \begin{align} w: \mathbb{H}^2 \to \mathbb{D}^+,~ (z_1, z_2) \mapsto \mathbb{R} \Re\left(\begin{smallmatrix} z_1 & -z_1z_2\\ 1 & -z_2 \end{smallmatrix}\right) + \mathbb{R} \Im\left(\begin{smallmatrix} z_1 & -z_1z_2\\ 1 & -z_2 \end{smallmatrix}\right) \end{align} $$

induces an isomorphism

(2.46) $$ \begin{align} X_0(N) \times X_0(N) \cong X_K,~ (z_1, z_2) \mapsto [w(z_1, z_2), 1] \end{align} $$

with $X_K$ the Shimura variety for $H_{\mathrm {V}}$ .

Under the map (2.44), the inverse images of the discriminant kernel $\Gamma _L \subset {\mathrm {SO}}(L)$ are

$$ \begin{align*}\Gamma^\Delta_0(N) := \left\{ (g_1, g_2) \in \Gamma_0(N)^2: g_1 g_2 \in \Gamma_1(N) \right\} \subset \Gamma_0(N)^2, \end{align*} $$

which contains $ \Gamma _1(N) \times \Gamma _1(N)$ and is a normal subgroup of $\Gamma _0(N)^2$ satisfying

$$ \begin{align*}\Gamma_0(N)^2 / \Gamma^\Delta_0(N) \cong (\mathbb{Z}/N\mathbb{Z})^\times,~ (\left(\begin{smallmatrix} a_1 & *\\ * & * \end{smallmatrix}\right), \left(\begin{smallmatrix} a_2 & *\\ * & * \end{smallmatrix}\right)) \mapsto a_1a_2^{} \bmod{N}. \end{align*} $$

The group ${\mathrm {SO}}(L^\vee /L) := {\mathrm {SO}}(L)/\Gamma _L \cong \Gamma _0(N)^2/\Gamma ^\Delta _0(N) \cong (\mathbb {Z}/N\mathbb {Z})^\times $ acts on $L^\vee /L \cong (\mathbb {Z}/N\mathbb {Z})^2$ via

$$ \begin{align*}\alpha \cdot (b, c) := (\alpha b, \alpha^{-1} c), \end{align*} $$

and the induced linear map on $\mathbb {C}[L^\vee /L]$ intertwines the Weil representation $\rho _L$ .

Now given $f \in M^!_{k}(\Gamma _0(N))$ for $k \in 2\mathbb {Z}$ , we can lift it to a vector-valued modular form in $M^!_{k, \rho _L}$ via the following map:

(2.47) $$ \begin{align} \mathrm{vv}: M^!_{k}(\Gamma_0(N)) \to M^!_{k, \rho_L},~ f \mapsto \sum_{\Gamma_0(N) \backslash \mathrm{SL}_2(\mathbb{Z})} (f \mid_{k} \gamma ) \rho_L(\gamma)^{-1} \cdot \mathfrak{e}_{0}. \end{align} $$

This map and its generalizations are well-studied (see, for example, [Reference ScheithauerSch09]), whose properties are summarized in the following result.

Lemma 2.3. When $k < 0$ , we have

(2.48) $$ \begin{align} \mathrm{prin}(\mathrm{vv}(f)) = \mathrm{prin}(f) \mathfrak{e}_{0}. \end{align} $$

for all $f \in M^{!, \infty }_{k}(\Gamma _0(N))$ , on which space the map $\mathrm {vv}$ is an isomorphism with the inverse given by

(2.49) $$ \begin{align} g = \sum_{\mu \in L^\vee/L} g_\mu \mathfrak{e}_\mu \mapsto g_{0}. \end{align} $$

Furthermore, it preserves the rationality of the Fourier expansion at the cusp infinity.

As a consequence, we can relate the higher Green function $G^{\Gamma _0(N)}_{r+1, f}$ from the introduction to one on the Shimura variety $X_K$ .

Corollary 2.4. Under the isomorphism (2.46), we have

(2.50) $$ \begin{align} G^{\Gamma_0(N)}_{r+1, f}(z_1, z_2) = - \Phi_{\mathrm{vv}(f)}^r([w(z_1, z_2), 1]) \end{align} $$

with $G^{\Gamma _0(N)}_{r+1, f}$ the higher Green function defined in (1.4) for $f \in M^{!, \infty }_{-2r}(\Gamma _0(N))$ with $r> 0$ .

Proof. Under (2.46), the divisor

(2.51) $$ \begin{align} Z(m, 0) = \Gamma_0(N)^2 \backslash \{(z_1, z_2) \in \mathbb{H}^2: (\left(\begin{smallmatrix} z_1 & -z_1z_2\\ 1 & -z_2 \end{smallmatrix}\right) , x) = 0 \text{ for some } x \in L_{m, 0}\} \end{align} $$

on $X_K$ is simply the m-th Hecke correspondence $T_m$ on $X_0(N) \times X_0(N)$ . Therefore, the two sides of (2.50) have logarithmic singularity along the same divisor. Using Corollary 4.2 and Theorem 4.4 of [Reference Bruinier, Ehlen and YangBEY21], we see that their difference is a smooth function in $L^2(X_0(N)^2)$ and an eigenfunction of the Laplacians in $z_1$ and $z_2$ with eigenvalue $r(1-r) < 0$ . By fixing $z_2$ , this difference is an eigenfunction of the Laplacian on $X_0(N)$ with negative eigenvalue, which vanishes identically. This holds for any $z_2 \in X_0(N)$ , and we obtain (2.50).

Remark 2.5. Following Section V.4 of [Reference Gross and ZagierGZ86], we call a set of integers $\{\lambda _m: m \in \mathbb {N}\}$ a relation for $S_{2-k}(\Gamma _0(N))$ if only finitely many $\lambda _m$ are nonzero and

$$ \begin{align*}\sum_{m \ge 1} \lambda_m a_m = 0 \end{align*} $$

for all $\sum _{m \ge 1} a_m q^m \in S_{2-k}(\Gamma _0(N))$ . Since $g_0 \in S_{2-k}(\Gamma _0(N))$ for all $g \in S_{2-k, L^-}$ , we have

$$ \begin{align*}\{f_P, g \} := \mathrm{CT}\left (\sum_{\mu \in L^\vee/L} f_{P, \mu} g_\mu\right ) = 0 \end{align*} $$

for $f_P = \sum _{m \ge 1} \lambda _m q^{-m} \mathfrak {e}_0$ . By Serre duality [Reference BorcherdsBor99], there exists $f \in M^!_{k, L}$ with $\mathrm {prin}(f) = f_P$ .

Suppose $E_1, E_2$ are imaginary quadratic fields such that $E = E_1E_2$ is biquadratic containing a real quadratic field F (i.e., $E_1 \neq E_2$ ). Then for any CM points $z_j \in E_j$ , the point $(z_1, z_2) \in \mathbb {H}^2$ is sent to $Z(W_\alpha ) \cup Z(W_{\alpha ^\vee }) \subset X_K$ under the isomorphism in (2.46) (see Section 3.2 in [Reference Yang and YinYY19] for details).

2.6 Eisenstein series

We recall coherent and incoherent Eisenstein series for the group $G = \operatorname {SL}_2$ following [Reference Bruinier, Kudla and YangBKY12]. Let F be a totally real field of degree d and discriminant $D_F$ , $E/F$ be a quadratic CM extension with absolute discriminant $D_{E}$ and $\chi = \chi _{E/F} = \otimes _{v \le \infty } \chi _v$ the associated Hecke character. For a standard section $\Phi \in I(s, \chi )$ with

(2.52) $$ \begin{align} I(s, \chi) := \operatorname{Ind}^{G(\mathbb{A}_F)}_{B(\mathbb{A}_F)}(|\cdot|^s \chi) = \bigotimes_{v \le \infty} I_v(s, \chi_v),~ I_v(s, \chi_v) := \operatorname{Ind}^{G(F_v)}_{B(F_v)}(|\cdot|_v^s \chi_v), \end{align} $$

we can form the Eisenstein series

$$ \begin{align*}E^*(g, s, \Phi) := \Lambda(s + 1, \chi) E(g, s, \Phi),~ E(g, s, \Phi) := \sum_{\gamma \in B\backslash \operatorname{SL}_2(F)} \Phi(\gamma g, s), \end{align*} $$

where $\Lambda (s, \chi )$ is the completed L-function for $\chi $ (see equation (4.6) in [Reference Bruinier, Kudla and YangBKY12]). When $\Phi = \otimes _v \Phi _v$ , the Eisenstein series $E(g, s, \Phi )$ has the Fourier expansion

$$ \begin{align*}E^*(g,s , \Phi) = E^*_0(g,s , \Phi) + \sum_{{t} \in F^\times} E^*_{t}(g,s , \Phi),~ \end{align*} $$

and for $t \in F^\times $ ,

$$ \begin{align*}E^*_{t}(g, s, \Phi) = \prod_v W^*_{{t}, v}(g, s, \Phi_v), \end{align*} $$

where $W^*_{t, v}$ is the normalized local Whittaker function defined by

(2.53) $$ \begin{align} \begin{aligned} W^*_{{t}, v}(g_v, s, \Phi_v) &:= |D_E/D_F|_v^{-(s+1)/2} L(s + 1, \chi_v) \int_{F_v} \Phi_v(w n(b) g_v, s) \psi_v(-{t} b) db. \end{aligned} \end{align} $$

For simplicity, we denote

(2.54) $$ \begin{align} W^*_{t, v}(\Phi_v) := W^*_{t, v}(1, 0, \Phi_v),~ W^{*, \prime}_{t, v}(\Phi_v) := \partial_s W^*_{t, v}(1, s, \Phi_v) \mid_{s = 0}. \end{align} $$

We will be interested in the case when $\Phi $ is a Siegel-Weil section.

Given $\alpha \in F^\times $ , we view $W_\alpha = E$ as an F-quadratic space with quadratic form $Q_\alpha (z) := \alpha z\bar z$ . Denote $\omega _\alpha $ the associated Weil representation. We have an $\operatorname {SL}_2$ -equivariant map $\lambda _\alpha : {\mathcal {S}}(W_\alpha (\mathbb {A}_F)) \to I(0, \chi )$ given by

(2.55) $$ \begin{align} \lambda_\alpha(\phi) (g) = (\omega_\alpha(g) \phi)(0). \end{align} $$

At each place v, there are local versions of $\omega _\alpha $ and $\lambda _\alpha $ as well, denoted by $\omega _{\alpha , v}$ and $\lambda _{\alpha , v}$ . When $\Phi = \lambda _\alpha (\phi )$ , resp. $\Phi _v = \lambda _{\alpha , v}(\phi _v)$ , we replace $\Phi $ , resp. $\Phi _v$ , from the notations above by $\phi $ , resp. $\phi _v$ .

Let $Z(W_\alpha )$ be the CM points on $X_K$ as in Section 2.4 and $W_{\alpha , \mathbb {Q}} \subset \mathrm {V}$ the rational quadratic space as in (1.5). A special case of the Siegel-Weil formula (see [Reference Bruinier, Kudla and YangBKY12, Theorem 4.5]) gives us

(2.56) $$ \begin{align} \theta(g, Z(W_\alpha), \phi) = C \cdot E(g^\Delta, 0, \phi) \end{align} $$

for any $\phi \in {\mathcal {S}}(W_{\alpha , \mathbb {Q}}(\mathbb {A})) = {\mathcal {S}}(W_\alpha (\mathbb {A}_F))$ . Here, $C = \deg (Z(W_\alpha ))/2$ .

Suppose only the j-th real embedding of $\alpha $ is negative. Denote and with $-1$ at the j-th slot. The sections and are coherent and incoherent, respectively. For all $\phi \in {\mathcal {S}}(W_\alpha (\hat F); \mathbb {C})$ , the Eisenstein series $E^*(g, s, \Phi (j))$ is holomorphic of weight at $s = 0$ and

(2.57)

is called a coherent Eisenstein series. However, the Eisenstein series $E^*(g, s, \Phi )$ vanishes at $s = 0$ , and its derivative

(2.58) $$ \begin{align} E^{*, \prime}(\tau, \phi) := \partial_s {\mathrm{Nm}}(v)^{-1/2} E^{*}(g_\tau, s, \Phi) \mid_{s = 0} \end{align} $$

is called an incoherent Eisenstein series, which is related to the coherent Eisenstein series via the differential equation [Reference Bruinier, Kudla and YangBKY12, Lemma 4.3]

(2.59)

for all $1 \le j \le d$ . Furthermore, it has the Fourier expansion

(2.60) $$ \begin{align} E^{*, \prime}(\tau, \phi) = \mathcal{E}(\tau, \phi) + \phi(0) \Lambda(0, \chi) \log {\mathrm{Nm}}(v) + \mathcal{E}^*(\tau, \phi), \end{align} $$

where $\mathcal {E}^*(\tau , \phi )$ has exponential decay near the cusp infinity and

$$ \begin{align*}\mathcal{E}(\tau, \phi) = a_0(\phi) + \sum_{t \in F,~ t \gg 0} a_t(\phi) {\mathbf{e}}(\operatorname{Tr}(t \tau)). \end{align*} $$

Here, $a_0(\phi )$ is an explicit constant (see (2.24) in [Reference Yang and YinYY19]) and

(2.61) $$ \begin{align} \begin{aligned} a_t(\phi) &:= \begin{cases} (-i)^d \tilde{W}_t(\phi) \log {\mathrm{Nm}}(\mathfrak{p}) , & \text{ if } \mathrm{Diff}(W_\alpha, t) = \{\mathfrak{p}\},\\ 0,& \text{ otherwise.} \end{cases} \end{aligned} \end{align} $$

The coefficient $\tilde {W}_t(\phi )$ is given by (see [Reference Yang and YinYY19, Proposition 2.7])

(2.62) $$ \begin{align} \tilde{W}_t(\phi) = 2^d \frac{W^{*, \prime}_{t, \mathfrak{p}}(\phi_{\mathfrak{p}})}{\log {\mathrm{Nm}}(\mathfrak{p})} \prod_{v \nmid \mathfrak{p} \infty} W^*_{t, v}(\phi_v) \in \mathbb{Q}(\phi) \end{align} $$

when $\mathrm {Diff}(W_\alpha , t) = \{\mathfrak {p}\}$

2.7 Hecke’s cusp form

Denote $\omega _a = \omega _{V_a , \psi }$ the Weil representation and $\theta _a := \theta _{V_a}$ the theta function as in (2.32). For a bounded, integrable function $\rho : H_a (\mathbb {Q}) \backslash H_a (\mathbb {A})/K \to \mathbb {C}$ , consider the following theta lift:

(2.63) $$ \begin{align} \vartheta_a (g, \varphi, \rho) := \int_{[H_a ]} \theta_a (g, h, \varphi) \rho(h) dh. \end{align} $$

The measure $dh$ is the product measure of the local measures $dh_p$ , where $dh_p$ is normalized such that the maximal compact subgroup in $H_a(\mathbb {Q}_p)$ has volume 1. Such integral was first considered by Hecke in [Reference HeckeHec27] when $\rho = \varrho $ is an odd, continuous character – that is,

(2.64) $$ \begin{align} \varrho(h) = {\mathrm{sgn}}(h_\infty) \varrho_f(h_f) \end{align} $$

with $\varrho _f$ a continuous character on $H_a(\hat {\mathbb {Q}})$ and $\varphi = \varphi ^\pm _\infty \varphi _f$ with

(2.65) $$ \begin{align} \varphi^\pm_\infty(x_1, x_2) := (x_1 \pm x_2) e^{-\pi (x_1^2 + x_2^2)} \in {\mathcal{S}}(\mathbb{R}^2). \end{align} $$

Notice that $\varphi ^\pm _\infty $ satisfies $\varphi ^\pm _\infty (-x_1, -x_2) = - \varphi ^\pm _\infty (x_1, x_2)$ .

In this case, the m-th Fourier coefficient of $\vartheta _a $ is given by

$$ \begin{align*}W_m(\varphi, \varrho )(g) := \int_{[N]} \vartheta_a (n g, \varphi, \varrho) {\psi(-mn)}dn \end{align*} $$

for $m \in \mathbb {Q}$ . To evaluate it, we apply the usual unfolding trick

$$ \begin{align*} W_m(\varphi, \varrho )(g) &= \int_{[N]} \int_{[H_a ]}\sum_{\lambda \in V_a (\mathbb{Q})} (\omega_a (ng) \varphi)(h^{-1} \lambda) \varrho(h) dh {\psi(-mn)}dn\\ &= \int_{[H_a ]}\sum_{\lambda \in V_{a , m}(\mathbb{Q})} (\omega_a (g)\varphi)(h^{-1} \lambda) \varrho(h)dh\\ &= \sum_{\lambda \in H_{a }(\mathbb{Q}) \backslash V_{a , m}(\mathbb{Q})} \int_{H_{a , \lambda}(\mathbb{Q}) \backslash H_a (\mathbb{A})} (\omega_a (g)\varphi)( h^{-1} \lambda) \varrho(h)dh. \end{align*} $$

When $m = 0$ , we have $\lambda = 0$ since $V_a $ is anisotropic and $\varphi (0) = \varphi ^\pm _\infty (0)\varphi _f(0) = 0$ . When $Q_a (\lambda ) = m \neq 0$ , the group $H_{a , \lambda }$ is trivial. We can then write $g = g_\tau g_f$ with $g_\tau = n(u)m(\sqrt {v}) \in G(\mathbb {R})$ and $g_f \in G(\hat {\mathbb {Q}})$ and obtain

$$ \begin{align*}W_m(\varphi, \varrho )(g) = \sum_{\lambda \in H_{a }(\mathbb{Q}) \backslash V_{a , m}(\mathbb{Q})} \int_{\mathbb{R}^\times} (\omega_a (g_\tau)\varphi^\pm_\infty) (t^{-1}(\iota_a(\lambda))) \frac{dt}{t} \int_{ H_a (\hat{\mathbb{Q}})} \varrho_f(h) (\omega_a (g_f)\varphi_f)(h^{-1}\lambda) dh. \end{align*} $$

The group $H_a (\mathbb {Q})$ acts on $V_{a , m}(\mathbb {Q})$ transitively. The archimedean integral can be evaluated as

$$ \begin{align*} \int_{\mathbb{R}^\times} (\omega_a (g_\tau)\varphi^\pm_\infty)( t^{-1}(\iota_a(\lambda)))\frac{dt}{t} &= 2 {\mathbf{e}}(m u) {\frac{v}{\sqrt {|a|}}} \int_{0}^\infty (t^{-1} \lambda_1 \pm {\mathrm{sgn}}(a)t\lambda_2) e^{-\pi \frac{v}{|a|}(t^{-2}\lambda_1^2 + t^2\lambda_2^2)} \frac{dt}{t}\\& = 2 v {\mathbf{e}}(m u){\mathrm{sgn}}(\lambda_1) \sqrt{|m|} \int^\infty_0 (t^{-1} \pm {\mathrm{sgn}}(m) t)e^{-\pi v|m|(t^{-2} + t^2)} \frac{dt}{t}. \end{align*} $$

This is 0 if $\pm {\mathrm {sgn}}(m) < 0$ by the change of variable $t \mapsto 1/t$ and can be otherwise evaluated using the lemma below.

Lemma 2.6. For any $\beta> 0$ , we have

$$ \begin{align*}\int^\infty_0 (t^{-1} + t) e^{-\beta\pi(t^{-2} + t^2)} \frac{dt}{t} = \frac{e^{-2\pi \beta}}{\sqrt{\beta}}. \end{align*} $$

Therefore, we have

(2.66) $$ \begin{align} W_m(\varphi^\pm, \varrho )((g_\tau, g_f)) = 2\sqrt{v} {\mathbf{e}}(mu) {\mathbf{e}}(|m|iv) {\mathrm{sgn}}(\lambda_1) \int_{ H_a (\hat{\mathbb{Q}})} \varrho_f(h) (\omega_a (g_f)\varphi_f)(h^{-1}\lambda) dh, \end{align} $$

when $\pm m> 0$ and $\lambda \in V_{a , m}(\mathbb {Q}) $ . Otherwise, it is 0. The integral in (2.66) can be evaluated locally. Notice that it always converges as $\varphi _f(h^{-1} \lambda )$ has compact support as a function of $h \in H_a (\hat {\mathbb {Q}})$ .

2.8 The Deformed theta integral

We recall the real-analytic modular form of weight one constructed in [Reference Charollois and LiCL20] using the notations of Section 2.7. Let $\varrho $ be an odd, continuous character as in (2.64), and $K_\varrho \subset H_a (\hat {\mathbb {Z}})$ the open compact subgroup defined in (2.14). The intersection $H_a(\mathbb {Q})^+ \cap K_\varrho $ , where $H_a(\mathbb {Q})^+ := H_a(\mathbb {Q}) \cap H_a(\mathbb {R})^+$ , is a cyclic subgroup

(2.67) $$ \begin{align} \Gamma_{\varrho} = \langle \varepsilon_{\varrho} \rangle,~ \varepsilon_{\varrho}> 1 > \varepsilon_{\varrho}' > 0 \end{align} $$

of the totally positive units in ${\mathcal {O}}$ . Then we have

(2.68) $$ \begin{align} H_a (\mathbb{A}) = \coprod_{\xi \in C} H_a (\mathbb{Q}) H_a (\mathbb{R})^+ K_\varrho \xi, \end{align} $$

where $C \subset H_a (\hat {\mathbb {Q}})$ is a finite subset of elements representing $H_a (\mathbb {Q})^+ \backslash H_a (\hat {\mathbb {Q}}) / K_\varrho $ . So given $h = (h_f, h_\infty )$ , we can find $\alpha \in H_a (\mathbb {Q}), t \in H_a (\mathbb {R})^+, k_1 \in K_\varrho , \xi \in C$ all depending on h such that

(2.69) $$ \begin{align} h = (\alpha k_1 \xi, \alpha t), \end{align} $$

though the choice is not unique. This gives us the identification

(2.70) $$ \begin{align} H_a (\mathbb{Q}) \backslash H_a (\mathbb{A}) / K_\varrho \cong \coprod_{\xi \in C} \Gamma_{\varrho} \backslash H_a (\mathbb{R})^+ \xi \end{align} $$

by sending $h = (\alpha k_1 \xi , \alpha t) \in H_a (\mathbb {A})$ as in (2.69) to $t \in H_a (\mathbb {R})^+$ in the $\xi $ -component. Just like the decomposition (2.68), this isomorphism depends on the choice of the set of representatives C. Similarly, we have

(2.71) $$ \begin{align} H_a (\hat{\mathbb{Q}}) / K_\varrho \cong \coprod_{\xi \in C} \Gamma_{\varrho} \backslash H_a (\mathbb{Q})^+\xi. \end{align} $$

Using the Fourier coefficient $W_m$ in (2.66) and the decomposition in (2.71), we can write

(2.72) $$ \begin{align} \vartheta_a(g, \varphi^\pm, \varrho) = \mathrm{vol}(K_\varrho) \sum_{\xi \in C} \varrho(\xi) \sum_{\beta \in \Gamma_{\varrho} \backslash V_{a}(\mathbb{Q}),~ \pm Q_a(\beta)> 0} (\omega_a(g) \varphi^{0, \pm}))(\xi^{-1} \beta) \end{align} $$

for $g \in G(\hat {\mathbb {Q}})\times B(\mathbb {R})$ , where $\varphi ^\pm = \varphi _f \varphi _\infty ^\pm $ with $\varphi _f \in {\mathcal {S}}(\hat {V}_a)$ being $K_\varrho $ -invariant and

(2.73) $$ \begin{align} \varphi^{0, \pm} = \varphi_f \varphi_\infty^{0, \pm},~ \varphi^{0, \pm}_\infty(x_1, x_2) := {\mathrm{sgn}}(x_1) e^{\mp 2 \pi x_1 x_2}. \end{align} $$

Although $\varphi ^{0, \pm }_\infty $ is not a Schwartz function on $\mathbb {R}^2$ , the sum above still converges absolutely. For $g \in B(\mathbb {R})$ , the quantity $\omega _a(g) \varphi ^{0, \pm }_\infty $ is defined with the usual formula of the Weil representation, and $\vartheta _a(g, \varphi ^\pm , \varrho )$ is right ${\mathrm {SO}}_2(\mathbb {R})$ -equivariant with weight $\pm 1$ . We also have a left $G(\mathbb {Q})$ -invariant function

(2.74) $$ \begin{align} \begin{aligned} \Theta_{a}(g, \varphi^\pm, \varrho) &:= \int_{H_a(\mathbb{Q})\backslash H_a(\hat{\mathbb{Q}})} \theta_a(g, h, \varphi^\pm) \varrho(h) dh = \mathrm{vol}( {K_\varrho}) \sum_{\xi \in C} \varrho(\xi) \theta_{a}(g, \xi, \varphi^\pm) \end{aligned} \end{align} $$

on $G(\mathbb {A})$ for $\varphi \in {\mathcal {S}}(V_a(\mathbb {A}))$ , which is independent of the choice of C.

We now define a function $\lg _C : H_a (\mathbb {Q}) \backslash H_a (\mathbb {A}) / {K_\varrho } \to [0, 1)$ by

(2.75) $$ \begin{align} \begin{aligned} \lg_C((\alpha k_1 \xi, \alpha t)) &:= 2 \log \varepsilon_{\varrho}\cdot \{\log t / \log \varepsilon_{\varrho}\},\\ \{a\} &:= a - \lim_{\epsilon \to 0} \frac{1}{2}\left ( \lfloor a + \epsilon \rfloor + \lfloor a - \epsilon \rfloor \right ). \end{aligned} \end{align} $$

Note that $\{0\} = \frac {1}{2}$ . Unlike the function considered by Hecke, $\lg _C$ cannot be written as the product of functions on $H_a (\hat {\mathbb {Q}})$ and $H_a (\mathbb {R})$ . Denote

(2.76) $$ \begin{align} \tilde{\varrho}_C := \lg_C \cdot \varrho: H_a (\mathbb{Q}) \backslash H_a (\mathbb{A}) / {K_\varrho} \to \mathbb{C}. \end{align} $$

Given $\varphi = \varphi _f \varphi _\infty \in {\mathcal {S}}(V_a(\mathbb {A}))$ for some ${K_\varrho }$ -invariant $\varphi _f \in {\mathcal {S}}(\hat V_a; \mathbb {C})$ , the deformed theta integral $\vartheta _a (g, \varphi , \tilde {\varrho }_C)$ , where $\vartheta _a $ is defined in (2.63), was studied in [Reference Charollois and LiCL20], To describe its Fourier expansion, denote

$$ \begin{align*} & \vartheta^*_{a}(g, \varphi_f, \varrho) := \int_{H_a(\mathbb{Q})^+\backslash H_a(\hat{\mathbb{Q}})}\varrho(h) \sum_{\substack{\beta \in \Gamma_{\varrho} \backslash V_{a}(\mathbb{Q})\\Q_a(\beta) < 0}} (\omega_a(g) \varphi^{0, *})(h^{-1} \beta) dh\\ & \quad = -\mathrm{vol}({K_\varrho}) \sqrt{v} \sum_{\xi \in C} \varrho(\xi) \sum_{\substack{\beta \in \Gamma_{\varrho} \backslash V_{a}(\mathbb{Q})\\Q_a(\beta) < 0}} (\omega_a(g_f) \varphi_f)(\xi^{-1} \beta) {\mathrm{sgn}}(\beta) {\mathbf{e}}(Q_a(\beta)\tau ) \Gamma(0, 4\pi |Q_a(\beta)| v). \end{align*} $$

for $g = (g_f, g_\tau ) \in G(\mathbb {A})$ with $\varphi ^{0, *} = \varphi _f \varphi ^{0, *}_\infty $ and

(2.77) $$ \begin{align} \varphi^{0, *}_\infty(x_1, x_2) := - {\mathrm{sgn}}(x_1) e^{-2\pi x_1x_2} \Gamma(0, 4\pi |x_1x_2|). \end{align} $$

Note that $\vartheta ^*_a$ is not necessarily left- $G(\mathbb {Q})$ invariant. But it is modular after applying the lowering operator as

$$ \begin{align*}L \left ( \vartheta^*_{a}(g, \varphi_f, \varrho) \right ) = \vartheta_{a}(g, \varphi^-, \varrho). \end{align*} $$

Similarly for $\xi \in H_a(\hat {\mathbb {Q}})$ , define

(2.78) $$ \begin{align} \begin{aligned} \theta^*_a&(g, \xi, \varphi_f) := \sum_{\lambda \in V_{a}(\mathbb{Q})} (\omega_a(g) \varphi^*)(\xi^{-1} \lambda) = \sum_{\lambda \in V_{a}(\mathbb{Q})} (\omega_a(g_f) \varphi_f)(\xi^{-1} \lambda) (\omega_a(g_\tau)\varphi^*)(\lambda) \\ &= - v \sum_{\lambda \in V_{a}(\mathbb{Q})} (\omega_a(g_f) \varphi_f)(\xi^{-1} \lambda) \frac{{\mathrm{sgn}}(\lambda_1 - {\mathrm{sgn}}(a)\lambda_2)}{\sqrt{\pi}} {\mathbf{e}}(Q_a(\lambda)\tau) \Gamma\left ( \frac{1}{2}, \frac{\pi v}{|a|}(\lambda_1 - {\mathrm{sgn}}(a)\lambda_2)^2\right ). \end{aligned} \end{align} $$

Here, we have employed the rapidly decaying function

(2.79) $$ \begin{align} \varphi^*(x_1, x_2) := -e^{-2\pi x_1 x_2} {\mathrm{sgn}}(x_1-x_2)\Gamma\left ( \frac{1}{2}, \pi (x_1-x_2)^2\right ), \end{align} $$

where $B(\mathbb {R})\subset G(\mathbb {R})$ acts via $\omega _a$ and ${\mathrm {SO}}_2(\mathbb {R})$ acts with weight 1. Also, we denote

(2.80) $$ \begin{align} \Theta^*_{a, C}(g, \varphi_f, \varrho) := \mathrm{vol}({K_\varrho}) \sum_{\xi \in C} \varrho(\xi) \theta^*_{a}(g, \xi, \varphi_f), \end{align} $$

which depends on the choice of C and satisfies

(2.81) $$ \begin{align} L \Theta^*_{a, C}(g, \varphi_f, \varrho) = \Theta_{a}(g, \varphi^-, \varrho). \end{align} $$

We recall some results.

Theorem 2.7. Let $\tilde {\varrho }_C$ be as in (2.76) and $\varphi _f \in V_a(\hat {\mathbb {Q}})$ a right- ${K_\varrho }$ invariant function. Then the integral $\vartheta _a(g, \varphi ^+, \tilde {\varrho }_C)$ defines a $G(\mathbb {Q})$ -invariant function in $g \in G(\mathbb {A})$ of weight 1 with respect to ${\mathrm {SO}}_2(\mathbb {R})$ . Furthermore, it has the Fourier expansion

(2.82) $$ \begin{align} \begin{aligned} \vartheta_a(g, \varphi^+, \tilde{\varrho}_C) &= {\vartheta^*_a(g, \varphi_f, \varrho)} + \log \varepsilon_{\varrho} \Theta^*_{a, C}(g, \varphi_f, \varrho)\\ &\quad + \mathrm{vol}({K_\varrho}) \sum_{\substack{\xi \in C\\ \beta \in \Gamma_{\varrho}\backslash V_a(\mathbb{Q})\\ Q_a(\beta)> 0}} \tilde{\varrho}_C( (\xi, \sqrt{|\beta/\beta'|})) (\omega_a(g)\varphi^{0, +})(\xi^{-1} \beta), \end{aligned} \end{align} $$

where $\varphi ^{0, +} = \varphi _f \varphi ^{0, +}_\infty $ is defined in (2.73), and satisfies

(2.83) $$ \begin{align} L \vartheta_a(g, \varphi^+, \tilde{\varrho}_C) = \vartheta_{a}(g, \varphi^-, \varrho) + \log \varepsilon_{\varrho} \Theta_{a}(g, \varphi^-, \varrho). \end{align} $$

Proof. This follows essentially from Proposition 5.5 in [Reference Charollois and LiCL20]. For completeness, we include a different (and slightly shorter) proof here. As in the evaluation of $\vartheta _a(g, \varphi , \varrho )$ in (2.72), we have

$$ \begin{align*} W_m(\varphi, \tilde{\varrho}_C)(g_\tau) &= \mathrm{vol}({K_\varrho}) {\mathbf{e}}(mu) \sqrt{v}\sum_{\xi \in C} \sum_{\beta \in \Gamma_{\varrho} \backslash V_{a, m}(\mathbb{Q})} \varphi_f(\xi^{-1} \beta) J(\beta, v), \\ J(\beta, v) &:=2\log \varepsilon_{\varrho} \int^\infty_0 \varphi^+_\infty(t^{-1}\cdot( \iota_a(\beta) \sqrt{v})) \left\{ \frac{\log t}{\log \varepsilon_{\varrho}} \right\} \frac{dt}{t}\\ &=2\log \varepsilon_{\varrho} {\mathrm{sgn}}(m\beta) \int^\infty_0 \varphi^{{\mathrm{sgn}}(m)}_\infty(\sqrt{|m|v} (t, t^{-1})) \left\{ \frac{\log t}{\log \varepsilon_{\varrho}} + \frac{1}{2} \frac{\log |\beta/\beta'|}{\log \varepsilon_{\varrho}} \right\} \frac{dt}{t}. \end{align*} $$

To verify (2.83), we start with

$$ \begin{align*} L_\tau &{\mathbf{e}}(m u) J(\beta, v) = {\mathbf{e}}( m \tau) v^2 \partial_v \left ( e^{2\pi m v} J(\beta, v) \right )\\ &= {\mathbf{e}}( m \tau) 2\log \varepsilon_{\varrho} {\mathrm{sgn}}(m\beta) \int^\infty_0 v^2 \partial_v e^{2\pi m v} \varphi^{{\mathrm{sgn}}(m)}_\infty(\sqrt{|m|v} (t, t^{-1})) \left\{ \frac{\log t}{\log \varepsilon_{\varrho}} + \frac{1}{2} \frac{\log |\beta/\beta'|}{\log \varepsilon_{\varrho}} \right\} \frac{dt}{t}\\ &= v^{} {\mathbf{e}}(mu ) {\mathrm{sgn}}(m\beta) \log \varepsilon_{\varrho} \int^\infty_0 \partial_t \varphi^{-{\mathrm{sgn}}(m)}_\infty(\sqrt{|m|v}(t, t^{-1})) \left\{ \frac{\log t}{\log \varepsilon_{\varrho}} + \frac{1}{2} \frac{\log |\beta/\beta'|}{\log \varepsilon_{\varrho}} \right\} dt\\ &= - v {\mathbf{e}}(mu ) \left ( {\mathrm{sgn}}(m\beta) \int^\infty_0 \varphi^{-{\mathrm{sgn}}(m)}_\infty(\sqrt{|m|v}(t, t^{-1})) \frac{dt}{t}- \log \varepsilon_{\varrho} \sum_{\varepsilon \in \Gamma_{\varrho}}\varphi^+_\infty(\iota_a(\beta \varepsilon)) \right )\\ &= v {\mathbf{e}}(mu ) \left ( \delta_{m < 0} {\mathrm{sgn}}(\beta) + \log \varepsilon_{\varrho} \sum_{\varepsilon \in \Gamma_{\varrho}}\varphi^+_\infty(\iota_a(\beta \varepsilon)) \right ). \end{align*} $$

Substituting this into the left-hand side of (2.83) proves it.

Now to calculate the Fourier expansion, it suffices prove the claim

(2.84) $$ \begin{align} \lim_{v \to \infty} e^{2\pi m v} J(\beta, v) = {\mathrm{sgn}}(\beta) \lg_C((1, \sqrt{|\beta/\beta'|}). \end{align} $$

For each $\beta \in \Gamma _{\varrho } \backslash V_a(\mathbb {Q})$ , we choose the unique representative $\beta _0 \in \Gamma _{\varrho } \beta $ such that $|\beta _0/\beta _0'| \in [1, \varepsilon _{\varrho }^2)$ . We can then write $J(\beta , v) = J_1(\beta _0, v) + J_2(\beta _0, v)$ , where

$$ \begin{align*} J_1(\beta_0, v) &:=2 \int^\infty_0 \varphi^+_\infty(t^{-1}\cdot( \iota_a(\beta_0) \sqrt{v})) \log t \frac{dt}{t},\\ J_2(\beta_0, v) &:=- 2\log \varepsilon_{\varrho} \int^\infty_0 \varphi^+_\infty(t^{-1}\cdot( \iota_a(\beta_0) \sqrt{v})) \left\lfloor \frac{\log t}{\log \varepsilon_{\varrho}} \right\rfloor \frac{dt}{t}. \end{align*} $$

For $J_1$ , we have

$$ \begin{align*} \lim_{v \to \infty} e^{2\pi m v} J_1(\beta_0, v) &= \log |\beta_0/\beta_0'| {\mathrm{sgn}}(m\beta_0) \lim_{v \to \infty} e^{2\pi m v} \int^\infty_0 \varphi^{{\mathrm{sgn}}(m)}_\infty(\sqrt{|m|v} (t, t^{-1})) \frac{dt}{t}\\ &= \log |\beta_0/\beta_0'| {\mathrm{sgn}}(\beta_0) \delta_{m> 0}. \end{align*} $$

For $J_2$ , the limit vanishes unless $|\beta _0/\beta _0'| = 1$ , in which case

$$ \begin{align*} \lim_{v \to \infty} e^{2\pi m v} J_2(\beta_0, v) &= 2\log \varepsilon_{\varrho} {\mathrm{sgn}}(m\beta_0) \lim_{v \to \infty} e^{2\pi m v} \int^1_{\varepsilon_{\varrho}^{-1}} \varphi^{{\mathrm{sgn}}(m)}_\infty(\sqrt{|m|v} (t, t^{-1})) \frac{dt}{t}\\ &= \log \varepsilon_{\varrho} {\mathrm{sgn}}(\beta_0) \delta_{m> 0}. \end{align*} $$

Putting these together proves claim (2.84).

Finally, we record a result as a direct consequence of Theorem 4.5 in [Reference Charollois and LiCL20] (see also Section 5 in [Reference Li and SchwagenscheidtLS22]).

3.1 Quadratic spaces

Let $V_{\pm 1}$ be as in Section 2.2, $\ell ^+, \ell ^-$ be isotropic lines such that $\ell ^+ \oplus \ell ^-$ is a hyperbolic plane and denote

(3.1) $$ \begin{align} V := V_0 \oplus V_1,~ V_0 := \ell^+ \oplus \ell^- \oplus V_{-1}. \end{align} $$

We can realize

$$ \begin{align*} V_0(\mathbb{Q}) &\cong \left\{ \Lambda \in M_2(F): \Lambda^t = \Lambda' \right\}\\ (a, b, \lambda) &\mapsto \begin{pmatrix} a & \lambda\\ \lambda' & b \end{pmatrix} \end{align*} $$

with $ \det $ as the quadratic form and furthermore write

(3.2) $$ \begin{align} V_0 = V_{00} \oplus U_D,~ V_{00} := V_0 \cap M_2(\mathbb{Q}),~ U_D := \sqrt{D}\mathbb{Q}\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} \cong (\mathbb{Q}, Q_D), \end{align} $$

where $Q_D(x) = Dx^2$ . So V has Witt rank 3 and admits the isotropic decomposition

(3.3) $$ \begin{align} V = V^+ + V^-,~ V^\pm := \ell^\pm + (V_{-1} + V_1)^\pm,~ (V_{-1} + V_1)^\pm(\mathbb{Q}) := \{(\lambda, \pm \lambda): \lambda \in F\} \end{align} $$

with $V^\pm $ maximal totally isotropic subspaces. For a $\mathbb {Q}$ -algebra R (e.g., $R \in \{\mathbb {Q}, \mathbb {Q}_p, \mathbb {R}, \hat {\mathbb {Q}} , \mathbb {A}\}$ ), we will use

(3.4) $$ \begin{align} (a, b, \lambda, \mu) \in V(R),~ a, b \in R, \lambda \in R \otimes F \cong V_{-1}(R), \mu \in R \otimes F \cong V_{1}(R) \end{align} $$

to represent elements in $V(R)$ . Define elements $f_j^\pm \in V$ by

(3.5) $$ \begin{align} \begin{aligned} f_1^+ &:= (1, 0, 0, 0),~ f_1^- := (0, 1, 0, 0),~ f_2^+ := (0, 0, 1/2, 1/2),~ f_2^- := (0, 0, 1/2, -1/2),\\ f_3^+ &:= (0, 0, \sqrt{D}/2, \sqrt{D}/2),~ f_3^- := (0, 0, 1/(2\sqrt{D}), -1/(2\sqrt{D})). \end{aligned} \end{align} $$

Then $\{f_j^\pm : j = 1, 2, 3\} \subset V^\pm $ is a $\mathbb {Q}$ -basis of $V^\pm $ . With respect to the ordered basis $(f_1^+, f_2^+, f_3^+, f_1^-, f_2^-, f_3^-)$ , the Gram matrix of Q is $\left (\begin {smallmatrix} 0 & {I_3} \\ I_3 & 0\end {smallmatrix}\right)$ . For $i = 1, 2, 3$ , the following linear transformations

(3.6) $$ \begin{align} w_i(f_j^\pm) := \begin{cases} f_j^\mp, & \text{ if } i = j,\\ f_j^\pm, & \text{ otherwise} \end{cases} \end{align} $$

are easily checked to be in $\mathrm {O}(V)$ . The unimodular lattice

(3.7) $$ \begin{align} V_{\mathbb{Z}} := \{(a, b, \lambda, \mu ) \in V(\mathbb{Q}) \cap (\mathbb{Z}^2 \times (\mathfrak{d}^{-1})^2): \lambda - \mu \in {\mathcal{O}}_F\} \subset V \end{align} $$

provides V with an integral structure. Similarly for $?\in \{00, 0, 1\}$ , the lattice $V_{?, \mathbb {Z}} := V_{\mathbb {Z}} \cap V_? $ in $V_?$ gives it with an integral structure.

For $? \in \{00, 0, 1, -1, \emptyset \}$ , we write

(3.8) $$ \begin{align} {\tilde{H}}_? := \operatorname{GSpin}(V_?),~ H_? := {\mathrm{SO}}(V_?), \end{align} $$

which are subgroups of ${\tilde {H}}$ and H, respectively, by acting trivially on $V^\perp _?$ and have the following exact sequence:

(3.9) $$ \begin{align} 1 \to \mathrm{G}_m \to {\tilde{H}}_? \to H_? \to 1. \end{align} $$

For any commutative ring R, we have explicitly

(3.10) $$ \begin{align} \iota: \operatorname{GSpin}(V_{0, \mathbb{Z}})(R) \cong \{\gamma \in {\mathrm{GL}}_2({\mathcal{O}} \otimes_{\mathbb{Z}} R): \det(\gamma) \in R^\times\}, \end{align} $$

via the action of $\gamma \in {\mathrm {GL}}_2({\mathcal {O}} \otimes _{\mathbb {Z}} R)$ on $V_{\mathbb {Z}, 0}(R)$

(3.11) $$ \begin{align} \Lambda \mapsto \det(\gamma)^{-1} \gamma \Lambda (\gamma')^t. \end{align} $$

For any $\mathbb {Q}$ -algebra R, we also have ${\tilde {H}}_?(R) = \operatorname {GSpin}(V_{?, \mathbb {Z}})(R)$ for $? \in \{00, 0, 1, \emptyset \}$ . Therefore, through $\iota $ , we have

(3.12) $$ \begin{align} G_0 := \mathrm{Spin}(V_0) \cong G_F := \mathrm{Res}_{F/\mathbb{Q}}(G),~ G_{00} := \mathrm{Spin}(V_{00}) \cong G,~ H_{00} \cong \mathrm{PGL}_2 \end{align} $$

and will represent elements in $H_0$ by their preimages in $G_F$ . Denote $T_0 := \iota ^{-1}(T) \subset {\tilde {H}}_0$ Then the relations among these groups can be visualized in the following diagram:

(3.13)

Here, the horizontal and vertical arrows are natural inclusions and surjections of algebraic groups, respectively, and the diagonal arrows are induced by $\iota $ . Let $B_F \subset G_F$ be the standard parabolic subgroup, and $B_0 := \iota ^{-1}(B_F) \subset G_0$ . They can be visualized as

(3.14)

which gives us

(3.15) $$ \begin{align} B_0(\mathbb{Q}) \backslash G_0(\mathbb{Q})\cong B_F(\mathbb{Q}) \backslash G_F(\mathbb{Q}) = B(F)\backslash G(F) \end{align} $$

via $\iota $ . We also denote

(3.16) $$ \begin{align} T^\Delta \subset T \times T_0 \subset {\mathrm{GL}}_2 \times {\tilde{H}}_0 \end{align} $$

the diagonal, which will play a crucial role in the local matching result in Section 3.5.

Now let $P \subset H$ be the Siegel parabolic stabilizing $V^+$ , whose Levi factor is isomorphic to ${\mathrm {GL}}(V^+)$ . Then $P_0 := P \cap H_0 \subset H$ is the subgroup stabilizing the line $\ell ^+$ and acting trivially on $V_1$ . The preimage of $P_0 H_{-1} \subset H_0$ in ${\tilde {H}}_0$ is given by $B_0 T_0$ . Combining with (3.15), we obtain

(3.17) $$ \begin{align} (P_0 H_{-1})(\mathbb{Q}) \backslash H_0(\mathbb{Q}) = (B_0T_0)(\mathbb{Q}) \backslash {\tilde{H}}_0(\mathbb{Q}) = B_0(\mathbb{Q}) \backslash G_0(\mathbb{Q}) \cong B(F)\backslash G(F). \end{align} $$

For $\alpha \in F^\times , \beta \in F$ , we then have $m(\alpha ), n(\beta ) \in G_0(\mathbb {Q}) \subset {\tilde {H}}(\mathbb {Q})$ . It is easy to check that

$$ \begin{align*} ( \omega(m(\alpha)) \varphi)(a, b, \nu, \lambda) &= \varphi(a/{\alpha\alpha'}, \alpha\alpha' b, \alpha'\nu/\alpha, \lambda),\\ ( \omega(n(\beta)) \varphi)(a, b, \nu, \lambda) &= \varphi(a - \beta\nu - \beta'\nu' + \beta\beta'b, \beta, \nu - \beta b, \lambda) \end{align*} $$

for a Schwartz function $\varphi \in {\mathcal {S}}(V(\mathbb {A}))$ .

3.2 Theta integral

Let $\theta _0(g, g_1, \varphi _0)$ denote the theta function on $[G \times H_0]$ associated to $\varphi _0 \in {\mathcal {S}}(V_0(\mathbb {A}))$ . Suppose $\varphi _{0, \infty }$ is in the polynomial Fock space $\mathbb {S}(V_0(\mathbb {R}))$ (see Section 4.1). Using $\mathbb {S}(V_0(\mathbb {R})) = \mathbb {S}(V_{00}(\mathbb {R})) \otimes \mathbb {S}(U_D(\mathbb {R}))$ , we can then restrict $\theta _0$ to the subgroup $[G \times H_{00}]$ , view it as a function on $[G \times G_{00}]$ , and write

(3.18) $$ \begin{align} \theta_0(g, g_{00}, \varphi_0) = \sum_{j \in J} \theta_{00}(g, g_{00}, \varphi_{00, j}) \theta_D(g, \varphi_{D, j}), \end{align} $$

where $\varphi _0= \sum _{j \in J} \varphi _{00, j} \varphi _{D, j}$ with $\varphi _{00, j} \in {\mathcal {S}}(V_{00}(\mathbb {A}))$ and $\varphi _{D, j} \in {\mathcal {S}}(U_D(\mathbb {A}))$ .

We now define

(3.19) $$ \begin{align} I_0(h_0, \varphi_0, f) := \int_{[G]} \theta_0(g, h_0, \varphi_0) f(g) dg \end{align} $$

for f a bounded, integrable function on $[G]$ . Note that the measure $dg$ is normalized so that $[G]$ has volume 1. In particular, for a right $G(\hat {\mathbb {Z}})$ -invariant function $\phi $ on $[G]$ , we have

(3.20) $$ \begin{align} \int_{[G]} \phi(g) dg = \frac{3}{\pi} \int_{\operatorname{SL}_2(\mathbb{Z})\backslash \mathbb{H}} \phi(g_\tau) d\mu(\tau),~ d\mu(\tau) := \frac{dudv}{v^2}. \end{align} $$

When $f(g) = \vartheta _1(g, \varphi _1, \rho )$ for a bounded, integrable function $\rho $ on $H_1(\mathbb {Q})\backslash H_1(\mathbb {A})$ , the integral $I_0$ above becomes

(3.21) $$ \begin{align} \begin{aligned} \mathcal{I}(h_0, \varphi, \rho) &:= \int_{[H_1]} I((h_0, h_1), \varphi) \rho(h_1) dh_1 = I_0(h_0, \varphi_0, \vartheta_1(\cdot, \varphi_1, \rho)),\\ I(h, \varphi) &:= \int_{[G]} \theta(g, h, \varphi) dg \end{aligned} \end{align} $$

with $\varphi = \varphi _0 \otimes \varphi _1$ .

For our purpose, $\rho = \varrho $ will be an odd, continuous character as in (2.64), and $\varphi = \varphi _f \varphi ^{ ( \epsilon , - \epsilon )}_\infty $ for $\epsilon = \pm 1$ with

(3.22) $$ \begin{align} \begin{aligned} \varphi^{(\epsilon, - \epsilon)}_\infty &:= \varphi^{(\epsilon, -\epsilon)}_{0, \infty} \otimes \varphi^-_\infty,\\ \varphi^{(\epsilon, -\epsilon)}_{0, \infty}(a, b, \nu_1, \nu_2) &:= (\epsilon i (a+b) + (\nu_1 - \nu_2)) e^{-\pi (a^2 + b^2 + \nu_1^2 + \nu_2^2)}, \end{aligned} \end{align} $$

and $\varphi ^\pm _\infty $ defined in (2.65). Here, we have identified $V(\mathbb {R}) = \mathbb {R}^2 \oplus V_{1}(\mathbb {R}) \oplus V_{-1}(\mathbb {R}) \cong (\mathbb {R}^2)^{\oplus 3}$ via (2.11). For any $\theta \in \mathbb {R}, \epsilon = \pm 1$ , we have

$$ \begin{align*}\omega(\kappa(\theta))\varphi^{(\epsilon, -\epsilon)}_\infty = \varphi^{(\epsilon, -\epsilon)}_\infty,~ \kappa(\theta) := \left(\begin{smallmatrix} \cos \theta & \sin \theta\\ -\sin \theta & \cos\theta \end{smallmatrix}\right) \in {\mathrm{SO}}_2(\mathbb{R}) \subset G(\mathbb{R}), \end{align*} $$

where $\omega $ is the Weil representation of $G(\mathbb {R})$ on $V(\mathbb {R})$ . However, for $h(\theta ) = (\kappa (\theta ), 1), h'(\theta ) = (1, \kappa (\theta )) \in H_0(\mathbb {R})$ with any $\theta \in \mathbb {R}$ , it is easy to check that

$$ \begin{align*}\omega(h(\theta))\varphi^{(\epsilon, -\epsilon)}_\infty = e^{\epsilon i\theta} \varphi^{(\epsilon, -\epsilon)}_\infty,~ \omega(h'(\theta))\varphi^{(\epsilon, -\epsilon)}_\infty = e^{-\epsilon i\theta} \varphi^{(\epsilon, -\epsilon)}_\infty. \end{align*} $$

So $\varphi ^{(\epsilon , -\epsilon )}_\infty $ is equivariant of weight $(\epsilon , -\epsilon )$ with respect to the connected component ${\mathrm {SO}}_2(\mathbb {R}) \times {\mathrm {SO}}_2(\mathbb {R})$ of the maximal compact of $H_0(\mathbb {R})$ . Later, we will also consider the following integral

(3.23) $$ \begin{align} \mathcal{I}_f(h_0,\varphi, \varrho) := \int_{H_1(\mathbb{Q})\backslash H_1(\hat{\mathbb{Q}})} \varrho(h_1) \int_{[G]} \theta(g, (h_0, h_1), \varphi) dg dh_1, \end{align} $$

which is similar to $\mathcal {I}(h_0, \varphi , \varrho )$ and well-defined as

$$ \begin{align*}\varrho(-h_1)\theta(g, (h_0, -h_1), \varphi) = \varrho(h_1)\theta(g, (h_0, h_1), \varphi)\end{align*} $$

for all $g, h_0, h_1$ and $\varphi _f \in {\mathcal {S}}(\hat V)$ . When $\varphi = \varphi _0 \otimes \varphi _1$ , we have

(3.24) $$ \begin{align} \mathcal{I}_f(h_0, \varphi, \varrho) = I_0(h_0, \varphi_0, \Theta_1(\cdot, \varphi_1, \varrho)), \end{align} $$

where $\Theta _a$ (with $a = 1$ ) is defined in (2.74).

3.3 Fourier transform and Siegel-Weil formula

We follow [Reference Gan, Qiu and TakedaGQT14] to recall the Siegel-Weil formula needed for our purpose, which goes from the split orthogonal group to the symplectic group. The range we need is in the 1st term range, and was originally proved in [Reference Kudla and RallisKR94]. Let $\varphi = \varphi _\infty \varphi _f \in {\mathcal {S}}(V(\mathbb {A}))$ with $\varphi _\infty $ as in (3.22) above. For $(g, h) \in G(\mathbb {A}) \times H(\mathbb {A})$ , we have the theta function $\theta (g, h, \varphi )$ and are interested in the value of the convergent integral $I(h, \varphi )$ defined in (3.21).

For a rational quadratic space $(V, (,)_V)$ , suppose $V = U^+ + U^- + V_{\circ }$ with $U^+, U^-$ complementary totally isotropic subspaces and $V_{\circ } = (U^+ + U^-)^\perp $ . Let $W = X + Y$ denote the symplectic space of rank 2 over $\mathbb {Q}$ with the symplectic pairing $\langle , \rangle _W$ . The rational vector space $\mathbb {W} := V \otimes W$ is then a symplectic space with respect to the pairing

(3.25) $$ \begin{align} \langle v_1 \otimes w_1, v_2 \otimes w_2 \rangle_{\mathbb{W}} := (v_1, v_2)_V \langle w_1, w_2\rangle_W. \end{align} $$

From this, we have the Fourier transform ${\mathscr {F}}_{U^+}: {\mathcal {S}}(V(\mathbb {A})) \to {\mathcal {S}}(((U^- \otimes W)+ V_{\circ })(\mathbb {A}))$ defined by

(3.26) $$ \begin{align} {\mathscr{F}}_{U^+}(\varphi)(\eta, v_{\circ}) := \int_{U^+(\mathbb{A})} \varphi(u^+, \eta_1, v_{\circ})\psi((u^+, \eta_2)_V) du^+ \end{align} $$

with $\eta = (\eta _1, \eta _2) \in (U^-)^2(\mathbb {A}) \cong (U^- \otimes W)(\mathbb {A})$ and $\eta _i \in U^-(\mathbb {A})$ . Here, $du^+$ is the Usual Haar measure on $U^+(\mathbb {A}) = \mathbb {A}$ . Note that we have $(u^+, \eta _1, v_{\circ }) \in (U^+\otimes X + U^-\otimes X + V_{\circ })(\mathbb {A}) = V(\mathbb {A})$ . Note that on ${\mathcal {S}}((U^- \otimes W + V_{\circ })(\mathbb {A}))$ , the Weil representation $\omega $ acts as

(3.27) $$ \begin{align} \begin{aligned} ( \omega(g, 1)\phi)(\eta, v_{\circ}) &= \omega_{V_{\circ}}(g)\phi(\eta g, v_{\circ}), g \in G(\mathbb{A}),\\ ( \omega(1, a)\phi)(\eta, v_{\circ}) &= |\det(a)| \phi(a^{-1}\eta, v_{\circ}), a \in {\mathrm{GL}}(U^+)(\mathbb{A}), \\ ( \omega(1, u)\phi)(\eta, v_{\circ}) &= \psi(\langle u(\eta), \eta \rangle /2) \phi(\eta, v_{\circ}), u \in N(U^+)(\mathbb{A}) \subset \mathrm{Hom}_{\mathbb{Q}}(U^-, U^+)(\mathbb{A}), \end{aligned} \end{align} $$

which makes ${\mathscr {F}}_{U^+}$ an intertwining map.

For V in (3.1), we can take $U^\pm = V^\pm $ and $V_{\circ }$ trivial with $V^\pm $ defined in (3.3). Another possibility is to take $U^\pm = \ell ^\pm $ and $V_{\circ } = V_1 \oplus V_{-1}$ , which will be used in calculating the Fourier expansion of the theta integral $I_0$ in (3.19). To simplify notations, we write

(3.28) $$ \begin{align} {\mathscr{F}} := {\mathscr{F}}_{V^+},~ {\mathscr{F}}_1 := {\mathscr{F}}_{\ell^+} \end{align} $$

and use them to represent the Fourier transform at the finite and infinite places as well. For example, ${\mathscr {F}}_1$ is given by

(3.29) $$ \begin{align} {\mathscr{F}}_1(\varphi) ((\eta_1, \eta_2), \nu, \lambda) = \int_{\mathbb{A}} \varphi(b, \eta_1, \nu, \lambda)\psi(b\eta_2) db \end{align} $$

for $\varphi \in {\mathcal {S}}(V(\mathbb {A}))$ . As ${\mathscr {F}}_1$ acts as ${\mathscr {F}}_1^{\prime } \otimes \mathrm {id}$ on ${\mathcal {S}}(V(\mathbb {A})) = {\mathcal {S}}(V_0(\mathbb {A})) \otimes {\mathcal {S}}(V_1(\mathbb {A}))$ , we will abuse notation and write ${\mathscr {F}}_1 = {\mathscr {F}}_1^{\prime }$ , which acts on ${\mathcal {S}}(V_0(\mathbb {A}))$ .

For a place $v \le \infty $ of $\mathbb {Q}$ and corresponding local field $k = \mathbb {Q}_v$ , recall we have the Siegel-Weil section

$$ \begin{align*} \Phi_v: {\mathcal{S}}((V^-\otimes W)(k)) &\to I^H_v(0)\\ \phi_v&\mapsto (h \mapsto (\omega_v(h)\phi_v)(0)), \end{align*} $$

where $I^H_v(s) = \operatorname {Ind}^{H(k)}_{P(k)}(|\cdot |^s)$ is the degenerate principal series. The image of $\Phi _v$ is a submodule of $I^H_v(0)$ denoted by $R_v(W)$ . When $v < \infty $ , it is known that (see [Reference Gan, Qiu and TakedaGQT14, Proposition 5.2(ii)])

$$ \begin{align*}I^H_v(0) = R_v(W) \oplus (R_v(W) \otimes {\det}_{H}). \end{align*} $$

It is clear that

(3.30) $$ \begin{align} \Phi_v(\omega(g) \phi_v) = \Phi_v(\phi_v) \end{align} $$

for any $g \in G(k)$ .

Given any $\phi = \otimes _v \phi _v \in {\mathcal {S}}((V^-\otimes W)(\mathbb {A}))$ , we denote $\Phi _s(\phi ) \in I^H(s)$ the standard section satisfying $\Phi _0(\phi ) = \otimes _{v} \Phi _v(\phi _v)$ . We can then form the Eisenstein series

$$ \begin{align*}E^H_P(s, \phi)(h) := \sum_{\gamma \in P(\mathbb{Q}) \backslash H(\mathbb{Q})} \Phi_s(\phi)(\gamma h), \end{align*} $$

which has meromorphic continuation to $s \in \mathbb {C}$ and is holomorphic at $s = 0$ . The regularized Siegel-Weil formula by Kudla-Rallis gives then the following equality (see [Reference Gan, Qiu and TakedaGQT14, Theorem 7.3(ii)]):

(3.31) $$ \begin{align} 2 I(h, \varphi) = E^H_P(0, {\mathscr{F}}(\varphi))(h). \end{align} $$

As a special case of the proposition in Section 2 of [Reference MœglinMœg97], following an argument in [Reference Gelbart, Piatetski-Shapiro and RallisGPSR87], we have the following lemma.

Lemma 3.1. For any $h \in H(\mathbb {A})$ , we have

(3.32) $$ \begin{align} E^H_P(s, \phi)(h) = \sum_{\gamma_0 \in B(F) \backslash G(F),~ \gamma_1 \in H_1(\mathbb{Q})} \Phi_s(\phi)((\gamma_0, \gamma_1 )h). \end{align} $$

Proof. We will show that $ P(\mathbb {Q})\backslash H(\mathbb {Q})\cong (B(F)\backslash G(F) ) \times H_1(\mathbb {Q})$ with the map induced by (3.11). First, we have $P(\mathbb {Q})\backslash H(\mathbb {Q}) = (P \cap (H_0 \times H_1))(\mathbb {Q})\backslash (H_0 \times H_1)(\mathbb {Q})$ . Let $H_{-1} \subset H_0$ denote the image of ${\mathrm {SO}}(V_{-1})$ , which is isomorphic to $H_1$ , and $P_0 := P \cap H_0$ . Then $P \cap (H_0 \times H_1) = P_0P_1^\Delta $ with $P_1^\Delta \cong H_1$ the image of the diagonal embedding of $H_1$ into $H_{-1} \times H_1$ . From this, we obtain

$$ \begin{align*}(P \cap (H_0 \times H_1)) (\mathbb{Q})\backslash (H_0\times H_1)(\mathbb{Q}) = (P_0P_1^\Delta) (\mathbb{Q})\backslash (H_0\times H_1)(\mathbb{Q}) = (((P_0 H_{-1}) \backslash H_0) \times H_1 )(\mathbb{Q}). \end{align*} $$

Equation (3.17) then finishes the proof.

Suppose $\varrho = \otimes _{p \le \infty } \varrho _p$ is an odd character of $H_1(\mathbb {A})/H_1(\mathbb {Q})$ and

(3.33) $$ \begin{align} \chi:= \varrho \circ {\mathrm{Nm}}^- = \otimes_{v \le \infty} \chi_v \end{align} $$

a totally odd character of $\mathbb {A}_F^\times /F^\times $ , which can be viewed as a character on $B_0(\mathbb {A})$ . Denote

(3.34) $$ \begin{align} I^{G_0}(\chi) := \mathrm{Ind}_{B_0(\mathbb{A})}^{G_0(\mathbb{A})} \chi,~ I^{G_0}_{ p}(\chi_p) := \mathrm{Ind}_{B_0(\mathbb{Q}_p)}^{G_0(\mathbb{Q}_p)} \chi_p,~ \chi_p := \bigotimes_{v \mid p} \chi_v. \end{align} $$

From (3.14), we see that

(3.35) $$ \begin{align} I^{G_0}(\chi) = I(0, \chi),~ I^{G_0}_p(\chi_p) = \bigotimes_{v \mid p} I_{v}(0, \chi_v) \end{align} $$

with $I(s, \chi )$ and $I_v(s, \chi _v)$ defined in (2.52). Using the formula (3.31) and Lemma 3.1, we can rewrite the function $\mathcal {I}(g_0, \varphi , \varrho )$ in (3.21) as

$$ \begin{align*} 2\mathcal{I}(g_0, \varphi, \varrho) &= \mathrm{CT}_{s=0} \int_{[H_1]} \varrho(h_1) E^H_P(s, {\mathscr{F}}(\varphi))(g_0, h_1) dh_1 = E^{G_0}_{B_0}(0, F_{\varphi, \varrho})(g_0), \end{align*} $$

for $g_0 \in G_0(\mathbb {A})$ , where $E^{G_0}_{B_0}(s', F_{\varphi , \varrho })$ is the Eisenstein series for the standard section associated to

(3.36) $$ \begin{align} \begin{aligned} F_{\varphi, \varrho}(g_0) &:= F_{\varphi, \varrho, 0}(g_0) \in \operatorname{Ind}^{G_0}_{B_0} \chi,\\ F_{\varphi, \varrho, s}(g_0) &:= \int_{H_1(\mathbb{A})}\Phi_s({\mathscr{F}}(\varphi))(g_0, h_1) \varrho(h_1) dh_1. \end{aligned} \end{align} $$

Note that $F_{\varphi , \varrho , s}$ is not a standard section (i.e., it depends on s when restricted to any open compact subgroup of $G_0(\hat {\mathbb {Q}})$ ).

If $\varrho = \otimes _{p \le \infty } \varrho _p$ and $\varphi = \otimes _{p \le \infty } \varphi _p$ , then $F_{\varphi , \varrho , s}$ is a product of local integrals.

(3.37) $$ \begin{align} F_{\varphi_p, \varrho_p, s}(g_{0, p}) := \int_{H_1(\mathbb{Q}_p)}\Phi_s({\mathscr{F}}(\varphi_p))(g_{0, p}, h_1) \varrho(h_1) dh_1,~ F_{\varphi_p, \varrho_p} := F_{\varphi_p, \varrho_p, 0} \in I_{0, p}(\chi_p). \end{align} $$

Recall that $dh_1$ is normalized so that the maximal compact subgroup of $H_1(\mathbb {Q}_p)$ has volume 1. We have explicitly

(3.38) $$ \begin{align} F_{\varphi_p, \varrho_p}(g) = \int_{H_1(\mathbb{Q}_p) \times F_p \times \mathbb{Q}_p} \varphi_p((g, h_1)^{-1} (x, 0, \lambda, \lambda)) \varrho_p(h_1) dx\, d\lambda\, dh_1 \end{align} $$

with $d\lambda $ the self-dual measure on $F_p$ such that $\int _{{\mathcal {O}}_{F_p}} d\lambda = |D|_p^{1/2}$ . From this, we see that

(3.39) $$ \begin{align} \sigma_a^{}(|D|_p^{1/2}F_{\varphi_p, \varrho_p}(g)) = |D|_{p}^{1/2} F_{ \sigma_a^{}(\varphi_p), \varrho_p}(g),~ F_{\varphi_p, \varrho_p}(t_0g) = F_{ \varphi_p, \varrho_p}(g) \end{align} $$

for all $a \in \mathbb {Z}_p^\times $ and $t_0 \in T_0(\mathbb {Z}_p)$ . At all but finitely many cases, the function $F_{\varphi _p, \varrho _p, s}$ is given explicitly as follows.

Lemma 3.2. Suppose p is unramified in E and $\varphi _p$ is the characteristic function of the maximal lattice $V_{\mathbb {Z}} \otimes \mathbb {Z}_p \subset V_p$ . Then

(3.40) $$ \begin{align} F_{\varphi_p, \varrho_p, s}(g_p) = (1-p^{-2-2s})\prod_{v \mid p} L(1 + s, \chi_v) \end{align} $$

for all $g_p \in G_0(\mathbb {Z}_p)$ .

Proof. Since $\varphi _p$ is $G_0(\mathbb {Z}_p)$ -invariant, we can suppose $g_p = 1$ .

If p is inert in F, then

$$ \begin{align*}\Phi_s({\mathscr{F}}(\varphi_p))(1, h_1) = \Phi_0({\mathscr{F}}(\varphi_p))(1, h_1) = 1 = \varrho(h_1)\end{align*} $$

for all $h_1 \in H_1(\mathbb {Q}_p) = H_1(\mathbb {Z}_p) = {\mathcal {O}}_{F_p}^1 \subset {\mathcal {O}}_{F_p}^\times $ , and $ F_{\varphi _p, \varrho _p, s}(g_p) = \int _{H_1(\mathbb {Z}_p)} dh_1 = 1$ .

If p is split in F, we have $F_p \cong \mathbb {Q}_p^2$ , $H_1(\mathbb {Q}_p) = \{(\alpha , \alpha ^{-1}) \in F_p: \alpha \in \mathbb {Q}_p^\times \} \cong \mathbb {Q}_p^\times $ and $\chi _v = \chi _{v'}$ is a character of $\mathbb {Q}_p^\times $ . Straightforward (though involved) calculations show that

$$ \begin{align*}\Phi_s({\mathscr{F}}(\varphi_p))(1, h_1) = \Phi_0({\mathscr{F}}(\varphi_p))(1, h_1) \min\{|h_1|_v, |h_1|_{v'}\}^s. \end{align*} $$

For $h_1 = (\alpha , \alpha ^{-1})$ with $o(\alpha ) = m$ , we have

(3.41) $$ \begin{align} \begin{aligned} \Phi_0({\mathscr{F}}(\varphi_p))(1, h_1) &= \int_{\mathbb{Q}_p^3} \operatorname{Char}(\mathbb{Z}_p^6)(a, 0, \lambda_1, \lambda_2, \alpha^{-1} \lambda_1, \alpha \lambda_2) da d\lambda_1 d\lambda_2\\ &= \int_{p^{\max\{0, m\}}\mathbb{Z}_p}d\lambda_1 \int_{p^{\max\{0, -m\}}\mathbb{Z}_p} d\lambda_2 = p^{-|m|}. \end{aligned} \end{align} $$

Since p is unramified in E, we have $\varrho _p((\alpha , \alpha ^{-1})) = \epsilon ^{o(\alpha )}$ with $\epsilon := \varrho _p((p, p^{-1})) = \chi _v(p) = \chi _{v'}(p)$ . Putting these together then gives us

(3.42) $$ \begin{align} \begin{aligned} F_{\varphi_p, \varrho_p, v, s}(1) &=\int_{\mathbb{Q}_p^\times} \min\{|\alpha|_p, |\alpha^{-1}|_p\}^{1+s} \epsilon^{o(\alpha)} |\alpha|_p^s d^\times \alpha\\ &= \sum_{m \in \mathbb{Z}} \epsilon^m p^{-|m|(1+s)} = L(1 + s, \chi_v)L(1 + s, \chi_{v'})(1-p^{-2-2s}). \end{aligned} \end{align} $$

This finishes the proof.

3.4 Matching global sections

The function $\mathcal {I}(g_0, \varphi ^{(k, k')}, \varrho )$ is a Hilbert modular form of weight $(k, k')$ . We want to suitably choose $\varrho $ and $\varphi _f$ and compare this function to a coherent Eisenstein series.

Let $\chi = \chi _{E/F}$ be a Hecke character associated to a quadratic extension $E/F$ with $E/\mathbb {Q}$ biquadratic, and $\varrho : \mathbb {A}_F^\times /F^\times \to \mathbb {C}^\times $ the character satisfying (2.13), whose kernel in $H_1(\hat {\mathbb {Z}})$ is denoted by $K_\varrho $ . Let $\alpha \in F^\times , W_\alpha $ be the same as in Section 2.6. For our purpose, we will choose $\phi ^{(k, k')}_\infty \in {\mathcal {S}}(W_\alpha (F \otimes \mathbb {R}))$ to be eigenfunctions of $K_\infty = \operatorname {SL}_2(\mathbb {R})^2$ with weight $(k, k')$ and normalized to have

$$ \begin{align*}\phi^{(k, k')}_\infty(0) = 1. \end{align*} $$

The matching result we will prove is the following.

Theorem 3.3. For $\alpha \in F^\times $ with ${\mathrm {Nm}}(\alpha ) < 0$ , given any $\phi _f \in {\mathcal {S}}(\hat W_{\alpha })$ , there exists $\varphi _f \in {\mathcal {S}}(\hat V; \mathbb {Q}^{\mathrm {ab}})$ such that $\omega _f(-1)\varphi _f = - \varphi _f$ for $-1 \in H_1(\hat {\mathbb {Q}})$ , it is invariant with respect to the compact subgroup $G(\hat {\mathbb {Z}}) T^\Delta (\hat {\mathbb {Z}})K_\varrho \subset G(\mathbb {A}) \times H_{}(\mathbb {A})$ and satisfies

(3.43) $$ \begin{align} \frac{\pi}{3} F_{\varphi, \varrho} = 2 \Lambda(1, \chi) \lambda_\alpha(\phi) \in I(0, \chi). \end{align} $$

Here, $\varphi = \varphi _f \varphi ^{(\epsilon , -\epsilon )}_{\infty }$ with $\epsilon := {\mathrm {sgn}}(\alpha _1) = -{\mathrm {sgn}}(\alpha _2)$ and $\varphi ^{(\pm 1, \mp 1)}_{ \infty }$ defined in (3.22), and $\phi = \phi _f \phi ^{(\epsilon , -\epsilon )}_\infty $ . In particular, we have the equality

(3.44) $$ \begin{align} \frac{\pi}{3} \mathcal{I}(g, \varphi, \varrho) = E^*(g, \phi). \end{align} $$

Remark 3.5. For $L \subset W_{\alpha }(\hat {\mathbb {Q}})$ a lattice and $\mu \in L^\vee /L$ , suppose $\varphi _\mu \in {\mathcal {S}}(\hat V; \mathbb {Q}^{\mathrm {ab}})$ satisfies (3.43) with $\phi _f = \phi _{L + \mu }$ . Then it is easy to see that

$$ \begin{align*}\sum_{\mu \in L^\vee/L} \mathcal{I}(g^\Delta_\tau, \varphi_\mu, \varrho) \mathfrak{e}_\mu: \mathbb{H} \to \mathbb{C}[L^\vee/L] \end{align*} $$

is a (non-holomorphic) vector-valued modular form of weight 0 on $\operatorname {SL}_2(\mathbb {Z})$ with representation $\overline {\rho _L}$ .

Remark 3.6. If we decompose $V_{0} = U \oplus U^\perp $ with $U = \ell ^+ + \ell ^-$ the hyperbolic plane, then it is easy to see that $T_0 \subset {\mathrm {SO}}(U) \subset H_0$ . Therefore, for any $\varphi \in {\mathcal {S}}(\hat V; \mathbb {Q}^{\mathrm {ab}})^{T^\Delta (\hat {\mathbb {Z}})}$ , we can write it as

$$ \begin{align*}\varphi = \sum_{j \in J} \varphi_{U, j} \otimes \varphi_{U^\perp, j} \end{align*} $$

such that $\varphi _{U, j} \in {\mathcal {S}}(\hat U; \mathbb {Q}^{\mathrm {ab}})^{T^\Delta (\hat {\mathbb {Z}})}$ and $\varphi _{U^\perp , j} \in {\mathcal {S}}(\hat U^\perp; \mathbb {Q}^{\mathrm {ab}})^{T^\Delta (\hat {\mathbb {Z}})}$ for all $j \in J$ . This in particular implies that $\varphi _{U^\perp , j}$ is $T(\hat {\mathbb {Z}})$ -invariant (i.e., it is $\mathbb {Q}$ -valued by (2.30)).

Proof of Theorem 3.3.

Suppose $\phi = \otimes _{v \le \infty }\phi _v$ . By Theorem 3.10, there exists $\varphi _p \in {\mathcal {S}}(V_p; \mathbb {Q}(\zeta _{p^\infty }))^{}$ invariant with respect to $G(\mathbb {Z}_p) T^\Delta (\mathbb {Z}_p)$ and satisfying (3.52). Furthermore, $\varphi _p$ is the characteristic function of the maximal lattice in $V_p$ for all but finitely many p. Therefore, $\varphi _f := \bigotimes _{p < \infty } \varphi _p$ is in ${\mathcal {S}}(\hat V; \mathbb {Q}^{\mathrm {ab}})^{(G\cdot T^\Delta )(\hat {\mathbb {Z}})}$ and satisfies

$$ \begin{align*}F_{\varphi_f, \varrho_f} = \zeta(2)^{-1} {L(1, \chi)}{\sqrt{D_E/D}} \lambda_\alpha(\phi_f) = 6 \Lambda(1, \chi) \lambda_\alpha(\phi_f). \end{align*} $$

Since $\varrho _f(-1) = {\mathrm {sgn}}(-1) = -1$ , the function $\omega _f(-1)\varphi _f$ with $-1 \in H_1(\hat {\mathbb {Q}})$ also satisfies these conditions, and we can replace $\varphi _f$ by $(\varphi _f - \omega _f(-1)\varphi _f)/2$ so that $\omega _f(-1) \varphi _f = - \varphi _f$ . Furthermore, we have $F_{\omega _f(h) \varphi _f, \varrho _f} = F_{\varphi _f, \varrho _f}$ for all $h \in K_\varrho $ , and can therefore average over $K_\varrho $ to ensure that $\varphi _f$ is $K_\varrho $ -invariant.

To prove (3.44), it suffices to check that $F_{\varphi ^{(\epsilon , -\epsilon )}_\infty , \varrho _\infty }(g) = \pi ^{-1} \lambda _\alpha (\phi _\infty ^{(\epsilon , -\epsilon )})(g)$ for $g = (g_{\tau _1}, g_{\tau _2})$ . Using

$$ \begin{align*} & \Phi_\infty({\mathscr{F}}(\varphi^{(\epsilon, -\epsilon)}_\infty))(g, t) \\&\quad = {\mathscr{F}}(\omega(g, t)\varphi^{(\epsilon, -\epsilon)}_\infty)(0) = \int_{\mathbb{R}^3} (\omega(g, t)\varphi^{(\epsilon, -\epsilon)}_\infty)(a, 0, \lambda_1, \lambda_2, \lambda_1, \lambda_2) da d\lambda_1 d\lambda_2 \\ &\quad = \int_{\mathbb{R}^3} \varphi^{(\epsilon, -\epsilon)}_\infty \left ( \frac{a - \lambda_1 u_1 - \lambda_2 u_2}{\sqrt{v_1v_2}}, 0, v_1\lambda_1/\sqrt{v_1v_2}, v_2\lambda_2/\sqrt{v_1v_2}, t^{-1}\lambda_1, t\lambda_2\right ) da d\lambda_1 d\lambda_2 \\ &\quad = \int_{\mathbb{R}^2} (v_1\lambda_1 - v_2\lambda_2) (t^{-1}\lambda_1 - t^{}\lambda_2) e^{-\frac{\pi}{v_1v_2} ((v_1\lambda_1 - v_2\lambda_2)^2 + v_1v_2((t^{-1}\lambda_1 + t\lambda_2)^2))} d\lambda_1 d\lambda_2\\ &\quad = \int_{\mathbb{R}^2} x \frac{2x + (t^{-1}v_2 - tv_1)y}{v_1t + v_2t^{-1}} e^{-\frac{\pi}{v_1v_2} (x^2 + v_1v_2y^2))} \frac{dx dy}{v_1t + v_2t^{-1}} \\ &\quad = \pi^{-1} 2 {(v_1v_2)^{3/2}}{} \frac{t^2}{(v_1t^2 + v_2)^2}, \end{align*} $$

where we have used the change of variable $x = v_1\lambda _1 - v_2\lambda _2, y = t^{-1} \lambda _1 + t \lambda _2$ , we obtain

$$ \begin{align*} F_{\varphi^{(\epsilon, -\epsilon)}_\infty, \varrho_\infty}(g) &= \int_{0}^\infty \Phi_\infty({\mathscr{F}}(\varphi_\infty))(g, t) \frac{dt}{t} =\pi^{-1} 2 {(v_1v_2)^{3/2}}{} \int_{0}^\infty \frac{t dt}{(v_1t^2 + v_2)^2} =\pi^{-1} \sqrt{ v_1v_2}. \end{align*} $$

However, we have

$$ \begin{align*}\lambda_{\alpha}(\phi^{(\epsilon, -\epsilon)}_\infty)(g) = \sqrt{v_1v_2}. \end{align*} $$

This finishes the proof.

The requirement that $\varphi _f$ in Theorem 3.3 is invariant with respect to $T^\Delta (\hat {\mathbb {Z}})$ will be important to deduce important rationality results in Section 4.3. We give a taste of such results in the following lemma.

Lemma 3.7. If $\varphi _0 \in {\mathcal {S}}(V_0; \mathbb {Q}^{\mathrm {ab}})$ is invariant with respect to $T^\Delta (\hat {\mathbb {Z}}) \subset ({\mathrm {GL}}_2 \times H_0)(\hat {\mathbb {Z}})$ , then ${\mathscr {F}}_1(\varphi _0) \in {\mathcal {S}}(((\ell ^- \otimes W) + V_{-1})(\hat {\mathbb {Q}}); \mathbb {Q}^{\mathrm {ab}})$ satisfies

(3.45) $$ \begin{align} \sigma_a \left ( {\mathscr{F}}_1(\varphi_0) ((\eta_1, \eta_2), \nu) \right ) = {\mathscr{F}}_1(\varphi_0) ((a^{-1} \eta_1, \eta_2), \nu) \end{align} $$

for any $\sigma _a \in \mathrm {Gal}(\mathbb {Q}^{\mathrm {ab}}/\mathbb {Q})$ associated to $a \in \hat {\mathbb {Z}}^\times $ as in section 2.3. In particular, we have

(3.46) $$ \begin{align} {\mathscr{F}}_1(\varphi_0) ((0, r), \nu) \in\mathbb{Q} \end{align} $$

for all $r \in \hat {\mathbb {Q}}, \nu \in \hat F$ .

Proof. Using the expression for ${\mathscr {F}}_1$ in (3.29), we can write

$$ \begin{align*} \sigma_a & \left ( {\mathscr{F}}_1(\varphi_0) ((\eta_1, \eta_2), \nu) \right ) = \sigma_a \left ( \int_{\hat{\mathbb{Q}}} \varphi_0(b, \eta_1, \nu)\psi_f(b\eta_2) db \right ) = \int_{\hat{\mathbb{Q}}} \sigma_a(\varphi_0(b, \eta_1, \nu))\psi_f(a b\eta_2) db\\ &= \int_{\hat{\mathbb{Q}}} \omega((t(a), 1))(\varphi_0)(b, \eta_1, \nu)\psi_f(a b\eta_2) db = \int_{\hat{\mathbb{Q}}} \omega((1, \iota(t(a^{-1}))))(\varphi_0)(b, \eta_1, \nu)\psi_f(a b\eta_2) db\\ &= \int_{\hat{\mathbb{Q}}} \varphi_0(ab, a^{-1}\eta_1, \nu)\psi_f(a b\eta_2) db = {\mathscr{F}}_1(\varphi_0) (( a^{-1}\eta_1,\eta_2), \nu). \end{align*} $$

For the second step, we moved $\sigma _a$ inside the integral as $\varphi _0$ is a Schwartz function and the integral is a finite sum. The third and fourth steps used (2.29) and the invariance of $\varphi _0$ under $(t, \iota (t)) \in T^\Delta (\hat {\mathbb {Z}})$ , respectively. Equation (3.46) now follows from (3.45) via (2.29).

3.5 Matching local sections I

The goal of this section is to prove Theorem 3.10, the non-archimedean local counterpart of the matching result 3.3. For this purpose, we fix a prime $p < \infty $ throughout this section. The main input to Theorem 3.10 is the following surjectivity result.

Proposition 3.8. Let $\varrho _p$ and $\chi _p$ be as in (3.33). Then the following map

(3.47) $$ \begin{align} \begin{aligned} \beta: {\mathcal{S}}(V_{p}; \mathbb{C})^{G(\mathbb{Z}_p)} \subset {\mathcal{S}}(V_p; \mathbb{C}) &\to I_p^{G_0}(\chi_p)\\ \varphi &\mapsto F_{\varphi, \varrho_p} \end{aligned} \end{align} $$

is surjective. Furthermore, if $\Phi \in I_p^{G_0}(\chi _p)$ is valued in $\mathbb {Q}(\zeta _{p^\infty })$ , then there exists $\varphi \in {\mathcal {S}}(V_p; \mathbb {Q}(\zeta _{p^\infty }))^{G(\mathbb {Z}_p)}$ satisfying $\beta (\varphi ) = \Phi $ . Here, $\mathbb {Q}(\zeta _{p^\infty }) \subset \mathbb {Q}^{\mathrm {ab}}$ is the subfield defined in (2.26).

Proof. Using (3.35), we can suppose $\Phi = \otimes _{v \mid p} \Phi _v$ with $\Phi _v \in I_v(0, \chi _v)$ . Since $F_{\omega (g)\varphi , \varrho _p} = F_{\varphi , \varrho _p}$ for all $g \in G(\mathbb {Z}_p)$ and $\varphi \in {\mathcal {S}}(V_p; \mathbb {C})$ , it suffices to prove the surjectivity of $\beta $ on ${\mathcal {S}}(V_p; \mathbb {C})$ . To do this, we will use the m-th Fourier coefficient of $\Phi _v \in I_v(0, \chi _v)$ for $m \in F_v$ , which is defined by

(3.48) $$ \begin{align} W_m(\Phi_v) := \int_{F_v} \Phi_v(w n(b)) \psi_v(-mb) db \end{align} $$

with $\psi _v$ an additive character of $F_v$ . For $m = (m_v)_{v \mid p} \in F_p$ and $\varphi \in {\mathcal {S}}(V_p; \mathbb {C})$ , we denote

(3.49) $$ \begin{align} \begin{aligned} W_m(\varphi) &:= \prod_{v \mid p} W_{m_v}( (F_{\varphi, \varrho_p})_v)\\ &= \int_{F_p\times H_1(\mathbb{Q}_p)} (\omega_p(wn(b), h_1) {\mathscr{F}}(\varphi))(0)\psi_p(-mb) \varrho_p(h_1) dh_1 \, db \end{aligned} \end{align} $$

with $\psi _p := \prod _{v \mid p} \psi _v$ . Now, the $G(F_v)$ -module $I_v(0, \chi _v)$ can be written as

$$ \begin{align*}I_v(0, \chi_v) = \oplus_{\alpha \in F_v^\times / {\mathrm{Nm}}(E_v^\times)} R(W_\alpha) \end{align*} $$

with $R(W_\alpha )$ the image of $\lambda _{\alpha , v}$ and irreducible. So $I_v(0, \chi _v)$ is irreducible if and only if $\chi _v$ is trivial. Otherwise, it is the direct sum of two irreducible submodules. Furthermore, for $\Phi \in R(W_\alpha )$ , the coefficient $W_m(\Phi )$ is zero unless $m/\alpha \in \operatorname {N}_{E_{v}/F_{v}}E_{v}^\times $ . We then have two cases to consider, depending on whether $\chi _v = \chi _{v'}$ is trivial or not.

When $\chi _v = \chi _{v'}$ is trivial, Lemma 3.12 gives us $\varphi $ such that $W_m( \varphi ) \neq 0$ for some $m \in F_p^\times $ . So for $v \mid p$ , the restriction of $\mathrm {im}(\beta ) \subset I_p^{G_0}(\chi _p)$ to $G(F_v)$ gives a nonzero section in $I_v(0, \chi _v)$ and generates a nontrivial, irreducible sub $G(F_v)$ -module. As $I_v(0, \chi _v)$ is irreducible, the map $\beta $ is surjective. When $\chi _v = \chi _{v'}$ is nontrivial, we again apply Lemma 3.12 to obtain a submodule $R \subset I_v(0, \chi _v) = R(W_{\alpha _0}) \oplus R(W_{\alpha _1})$ from $\mathrm {im}(\beta )$ such that $\pi _i(R)$ is nontrivial with $\pi _i: I_v(0, \chi _v) \to R(W_{\alpha _i})$ the projection. As $R(W_{\alpha _i})$ is irreducible, we have $\pi _i(R) = \pi _i(I_v(0, \chi _v))$ . Consider $R_i := \ker \pi _i \cap R$ as a submodule of the irreducible module $\ker \pi _i$ . As $R(W_{\alpha _0})$ and $R(W_{\alpha _1})$ are not isomorphic [Reference Kudla and RallisKR92, Proposition 3.4], $R_i$ cannot be trivial for both $i = 0, 1$ , otherwise, $R \cong \pi _i(R) = R(W_{\alpha _i})$ . Thus, $R_i = \ker \pi _i \subset R$ for an i, which implies $R = I_v(0, \chi _v)$ and proves surjectivity.

When $\Phi = \otimes _{v \mid p} \Phi _v$ has value in $\mathbb {Q}(\zeta _{p^\infty })$ , we apply the surjectivity of $\beta $ and the discussion in Section 2.3 to choose $\varphi _j \in {\mathcal {S}}(V_p; \mathbb {Q}(\zeta _{p^\infty }))^{G(\mathbb {Z}_p)}$ and $c_j \in \mathbb {C}$ such that

$$ \begin{align*}\varphi := \sum_{j = 1}^J c_j \varphi_j \in {\mathcal{S}}(V_p; \mathbb{C}) \end{align*} $$

satisfies $\beta (\varphi ) = \Phi $ and J is minimal. Therefore, $F_{\varphi , \varrho _p} = \sum _{j = 1}^J c_j F_{\varphi _j, \varrho _p}$ is valued in $\mathbb {Q}(\zeta _{p^\infty })$ . By the minimality of J, the section $F_{\varphi _j, \varrho _p}$ is not identically zero for all j. Therefore, the set $\{1, c_1, \cdots , c_J\} \subset \mathbb {C}$ is linearly dependent over $\mathbb {Q}(\zeta _{p^\infty })$ . The minimality of J then implies that $J = 1$ and $c_1 \in \mathbb {Q}(\zeta _{p^\infty })$ , and hence, $\varphi \in {\mathcal {S}}(V_p; \mathbb {Q}(\zeta _{p^\infty }))^{G(\mathbb {Z}_p)}$ .

Using this proposition, we can match any continuous function on $G_0(\mathbb {Z}_p)$ via the map $\beta $ . Furthermore, we can incorporate Galois action to obtain the following result.

Proposition 3.9. In the setting of Proposition 3.8, given any continuous function $\Phi : G_0(\mathbb {Z}_p) \to \mathbb {C}$ satisfying

(3.50) $$ \begin{align} \Phi(m(a)n(b)k) = \chi(a) \Phi(k),~ \end{align} $$

for all $m(a), n(b) \in B_0(\mathbb {Z}_p), k \in G_0(\mathbb {Z}_p)$ , there exists $\varphi \in {\mathcal {S}}(V_p; \mathbb {C})^{G(\mathbb {Z}_p)}$ such that $F_{\varphi , \varrho _p}(g) = \Phi (g)$ for all $g \in G_0(\mathbb {Z}_p)$ . Furthermore, if $\Phi $ takes values in $\mathbb {Q}(\zeta _{p^\infty })$ and satisfies

(3.51) $$ \begin{align} \sigma_a(|D|_p^{-1/2} \Phi(t_0^{-1} g t_0)) = |D|_p^{-1/2} \Phi(g),~ \end{align} $$

with $ t_0 = \iota (t(a)) \in {\tilde {H}}_0(\mathbb {Z}_p), t(a) = \left (\begin {smallmatrix} a & \\ & 1 \end {smallmatrix}\right ) \in T \subset {\mathrm {GL}}_2(\mathbb {Z}_p)$ for all $a \in \mathbb {Z}_p^\times $ and $g \in G_0(\mathbb {Z}_p)$ , then $\varphi \in {\mathcal {S}}(V; \mathbb {Q}(\zeta _{p^\infty }))^{G(\mathbb {Z}_p)}$ can be chosen to be $T^\Delta (\mathbb {Z}_p)$ -invariant.

Proof. A continuous function $\Phi $ on $G_0(\mathbb {Z}_p)$ satisfying (3.50) can be uniquely extended to a section $\tilde \Phi \in I^{G_0}_p(\chi _p)$ by setting

$$ \begin{align*}\tilde\Phi(g) := \chi_p(a) \Phi(k) \end{align*} $$

with $g = m(a)n(b)k$ the Iwasawa decomposition of g. Therefore, the first claim is a direct consequence of Proposition 3.8.

For the second claim, we take any $\varphi \in {\mathcal {S}}(V_p; \mathbb {Q}(\zeta _{p^\infty }))^{G(\mathbb {Z}_p)}$ and observe that

$$ \begin{align*} \frac{F_{\omega_p(t, t_0)\varphi_p, \varrho_p}(g)}{ |D|_p^{1/2}} &= \frac{ F_{\omega_p(t)\varphi_p, \varrho_p}(gt_0)}{|D|_p^{1/2} } = \frac{F_{\sigma_a(\varphi_p), \varrho_p}(t^{-1}gt_0)}{|D|_p^{1/2} } = \sigma_a^{}\left ( \frac{ F_{\varphi_p, \varrho_p}(t^{-1}gt)}{ |D|_p^{1/2}}\right )\\ & = \sigma_a(|D|_p^{-1/2} \Phi(t_0^{-1} g t_0)) = |D|_p^{-1/2} \Phi(g) \end{align*} $$

for any $(t, t_0) \in T^\Delta (\mathbb {Z}_p)$ with $t = t(a), t_0 = \iota (t(a))$ and $g \in G_0(\mathbb {Z}_p)$ . Here, we used (3.39) for the first line. By averaging $\varphi $ over $T^\Delta (\mathbb {Z}_p)$ , we can suppose that it is $T^\Delta (\mathbb {Z}_p)$ -invariant. This finishes the proof.

We are now ready to state and prove the local matching result. This is just the Kudla matching principle [Reference KudlaKud03] in some sense.

Theorem 3.10. For any $\phi _v \in {\mathcal {S}}(W_\alpha (F_v))$ with $v \mid p$ , there exists $\varphi _p \in {\mathcal {S}}(V_p; \mathbb {Q}(\zeta _{p^\infty }))^{(G\cdot T^\Delta )(\mathbb {Z}_p)}$ such that

(3.52) $$ \begin{align} F_{\varphi_p, \varrho_p} = (1 - p^{-2})|D/D_E|_p^{1/2} \prod_{v \mid p} L(1, \chi_v) \lambda_{\alpha, v}(\phi_v). \end{align} $$

In addition, if p is unramified in E and co-prime to $\alpha $ , and $\phi _v$ is the characteristic function of the maximal lattice in $W_\alpha (F_v)$ , then we can choose $\varphi _p$ to be the characteristic function of the maximal lattice in $V_p$ .

Proof. Suppose p and $\alpha $ are co-prime, $E_p/\mathbb {Q}_p$ is unramified and $\phi _v = \operatorname {Char}({\mathcal {O}}_{E_v}), \varphi _p = \operatorname {Char}({\mathcal {O}}_{F_p} \times \mathbb {Z}_p^2 \times {\mathcal {O}}_{F_p})$ . Then it is easy to check that $F_{\varphi _p, \varrho _p}$ and $\prod _{v \mid p} \lambda _{\alpha , v}(\phi _v)$ are both right $G({\mathcal {O}}_{F_p})$ -invariant. Since they are both in $\prod _{v \mid p} I(0, \chi _v)$ , we only need to check that

$$ \begin{align*}F_{\varphi_p, \varrho_p}(1) = (1-p^{-2})\prod_{v \mid p} L(1, \chi_v) \lambda_{\alpha, v}(\phi_v)(1) \end{align*} $$

by the Iwasawa decomposition of $G(F_p)$ . This is given precisely by Lemma 3.2 and proves (3.52) for all but finitely many places.

When $\phi _v$ is $\mathbb {Q}$ -valued, we can use (2.29) to check that

$$ \begin{align*} \sigma_a^{}\left ( \prod_{v \mid p} \lambda_{\alpha, v}(\phi_v)(t^{-1}gt)\right ) &= \prod_{v \mid p} \sigma_a^{} \left (\omega_{\alpha, v}(t^{-1}gt)(\phi_v)(0)\right ) = \prod_{v \mid p} \left (\omega_{\alpha, v}(g)(\sigma_a(\phi_v))(0)\right )\\ & = \prod_{v \mid p} \left (\omega_{\alpha, v}(g)(\phi_v)(0)\right ) = \prod_{v \mid p} \lambda_{\alpha, v}(\phi_v)(g) \end{align*} $$

for any $(t, t_0) \in T^\Delta (\mathbb {Z}_p)$ with $t = t(a), t_0 = \iota (t(a))$ and $g \in G_0(\mathbb {Z}_p)$ . Proposition 3.9 combined with Remark 3.4 then completes the proof.

Finally, we record the two local calculation lemmas used in proving Proposition 3.8.

Lemma 3.11. Suppose $F_p/\mathbb {Q}_p$ is non-split with valuation ring ${\mathcal {O}}_p$ , uniformizer $\varpi $ , residue field size q, and a nontrivial additive character $\psi $ . For a character $\varrho $ of $H_1(\mathbb {Q}_p) = F_p^1 \subset {\mathcal {O}}_p^\times $ , let

$$ \begin{align*}n(\psi) :=\min\{n: \psi(\varpi^n {\mathcal{O}}_p) =1\},~ n(\varrho) :=\min\{n\ge 0: \varrho(K_n) =1\} \end{align*} $$

be the conductors of $\psi $ and $\varrho $ , respectively, where $K_n:=F_p^1 \cap (1+ \varpi ^n {\mathcal {O}}_p)$ . Then

$$ \begin{align*}\int_{F_p^1} \varrho(x) \psi(mx) dx \ne 0 \end{align*} $$

for some $m \in {\mathcal {O}}_p^\times $ if and only if $n(\varrho ) \le n(\psi )$ .

Proof. Let

$$ \begin{align*}f(m) =\begin{cases} \int_{F_p^1} \varrho(x) \psi(mx) dx &\operatorname{if } m \in {\mathcal{O}}_p^\times \\ 0 &\mbox{otherwise}. \end{cases} \end{align*} $$

Then $f \in {\mathcal {S}}(F_p)$ and its Fourier transformation with respect to $\psi $ is

$$ \begin{align*} \hat f(m) &= \int_{F_p} f(n) \psi(-nm) dn =\int_{F_p^1} \varrho(x) \int_{{\mathcal{O}}_p^\times} \psi(n(x-m) dn dx \\ &=\int_{F_p^1} \varrho(x) (\operatorname{Char}(m+\varpi^{n(\psi)}{\mathcal{O}}_p)(x) -q^{-1}\operatorname{Char}(m+\varpi^{n(\psi)-1}{\mathcal{O}}_p)(x) ) dx. \end{align*} $$

First, assume that there is some $h_0\in F_p^1$ such that $h_0-m \in \varpi ^{n(\psi )}{\mathcal {O}}_p$ . Then

$$ \begin{align*} \hat f(m) &= \varrho(h_0) \left(\int_{K_{n(\psi)}} \varrho(x) dx -q^{-1} \int_{K_{n(\psi)-1}} \varrho(x) dx\right) \\ &=\begin{cases} 0 &\operatorname{if } n(\varrho)> n(\psi), \\ \varrho(h_0) \mathrm{vol}(K_{n(\psi)}) &\operatorname{if } n(\varrho) = n(\psi), \\ \varrho(h_0)(\mathrm{vol}(K_{n(\psi)}) - q^{-1} K_{n(\psi)-1}) &\operatorname{if } n(\varrho) < n(\psi). \end{cases} \end{align*} $$

Next, we assume that there no $h_0\in F_p^1$ such that $h_0-m \in \varpi ^{n(\psi )}{\mathcal {O}}_p$ but some $h_0 \in F_p^1$ with $h_0-m \in \varpi ^{n(\psi )-1}{\mathcal {O}}_p$ . Then

$$ \begin{align*}\hat f(m) = \begin{cases} 0 &\operatorname{if } n(\varrho) \ge n(\psi), \\ -q^{-1} \varrho(h_0)\mathrm{vol}(K_{n(\psi)-1}) &\operatorname{if } n(\varrho) < n(\psi). \end{cases} \end{align*} $$

Finally, if there is no $h_0 \in F_p^1$ with $h_0-m \in \varpi ^{n(\psi )-1}{\mathcal {O}}_p$ , then $\hat f (m) =0$ . Now the lemma is clear.