3.1.1 Local Whittaker integrals

Let v be a place of F. For $t\in F_v$ , $\phi \in {\mathcal {S}}( {\mathbb {W}}_v )$ and $g\in G(F_v)$ , define the Whittaker integral

(3.1) $$ \begin{align} W_{v,t} (s, g, \phi) =\int_{N(F_v) } \delta_v(w_vng)^{s }\omega (w_vng) \phi(0) \psi_{-t}(n) dn. \end{align} $$

We immediately have the following equations:

(3.2) $$ \begin{align}\hspace{14.5pt} W_{v,t} (s, gk, \phi)=W_{v,t} (s, g, \omega(k)\phi),\ k\in K^{\max}_v , \end{align} $$

(3.3) $$ \begin{align} W_{v,t}(s, n(b) g , \phi )=\psi _v\left( bt\right) W_{v,t}(s, g , \phi ),\ b\in F_v. \end{align} $$

Since $wn(b)m(a)=m(\overline a^{-1})wn(b{\mathrm {Nm}}(a)^{-1})$ , $ m(\overline a^{-1})n(b)=n(b')m(\overline a^{-1})$ for some $b'$ , and $\chi _{_{\mathbb {W}},v}({\mathrm {Nm}}(a))=1$ , a direct computation gives

(3.4) $$ \begin{align} W_{v,t}(s, m(a)g, \phi)=|a|_{E_v}^{1-\dim {\mathbb{W}}/2-s}\chi_{_{\mathbb{W}},v}(a)W_{v,{\mathrm{Nm}}(a)t}(s, g, \phi), \ a\in E_v^\times. \end{align} $$

Lemma 3.1.1 [Reference IchinoIch04, Proposition 6.2][Reference Yuan, Zhang and ZhangYZZ13, Proposition 2.7 (2)].

Let $t\neq 0$ .

(1) The set $ {\mathbb {W}}_v^t$ is either empty or consists of one orbit of $U( {\mathbb {W}}_v)$ .

(2) If $ {\mathbb {W}}_v^t$ is empty, then $W_{v,t}(0,g,\phi )=0$ . Otherwise, for $x\in {\mathbb {W}}_v^t$ , we have

$$ \begin{align*}W_{v,t}(0,g,\phi)= \kappa \int_{ U( {\mathbb{W}}_v)}\omega(g)\phi(h^{-1} x)dh ,\end{align*} $$

for a nonzero constant $\kappa $ .

(3) If $\dim {\mathbb {W}}=1$ , with the measure fixed in 2.4, $\kappa =\frac {\gamma _{{\mathbb {W}}_v}}{L(1,\eta _v)}$ .

Assume $\dim {\mathbb {W}}=1$ .

We define the following normalization (following [Reference Yuan, Zhang and ZhangYZZ13, 6.1.1]):

(3.5) $$ \begin{align} W_{v,t}^\circ=\gamma_{{\mathbb{W}}_v}^{-1}W_{v,t},\ t\neq 0. \end{align} $$

For $t=0$ , $W_{v,t} (s, g, \phi ) $ has a possible pole at $s=0$ . And we take a different normalization (following [Reference Yuan, Zhang and ZhangYZZ13, 6.1.1], and taking care of the difference between the modulus characters mentioned in 2.6),

(3.6) $$ \begin{align}W_{v,0}^\circ (s, g, \phi):=\gamma_{{\mathbb{W}}_v}^{-1}|{\mathrm{Diff}}_v{\mathrm{Disc}}_v|_{F_v}^{-1/2}\frac{L(2s+1,\eta_v)}{L(2s,\eta_v)}W_{v,0} (s, g, \phi) .\end{align} $$

Here, if v is a finite place, ${\mathrm {Diff}}_v$ is the different of $F_v/{\mathbb {Q}}_v$ and ${\mathrm {Disc}}_v$ is the discriminant of $E_v/F_v$ , and if $v\in \infty $ , ${\mathrm {Diff}}_{v}={\mathrm {Disc}}_v=1$ .

3.1.2 Siegel-Eisenstein series

Now we come back to a general ${\mathbb {W}}$ . For $\phi \in {\mathcal {S}}({\mathbb {W}})$ , we have a Siegel-Eisenstein series of G:

(3.7) $$ \begin{align}E(s,g,\phi)=\sum_{\gamma\in P(F)\backslash G(F)}\delta (\gamma g)^{s } \omega (\gamma g)\phi (0), \end{align} $$

which is absolutely convergent if ${\mathrm {Re}} s>1-\dim {\mathbb {W}}/2$ and has a meromorphic continuation to the entire complex plane [Reference TanTan99]. Moreover, it is holomorphic at $s=0$ [Reference TanTan99, Proposition 4.1].

Let $E_t(s,g,\phi )$ be the $\psi _t$ -Whittaker function of $E(s,g,\phi )$ . Let $W_{t} (s, g, \phi )$ be the global counterpart of the Whittaker integral (3.1) so that if $\phi $ is a pure tensor, then $W_{t} (s, g, \phi )$ is the product of $W_{v,t} (s, g_v, \phi _v)$ over all places of F. By the non-vanishing of $L(1,\eta )$ and Lemma 3.1.3, $W_{0} (s, g, \phi )$ has a meromorphic continuation to the entire complex plane, which is holomorphic at $s=0$ . Then we have

(3.8) $$ \begin{align}E_{t} (s, g, \phi)=W_{t} (s, g, \phi), \ t\neq 0; \end{align} $$

(3.9) $$ \begin{align}E_0(s,g,\phi)=\delta ( g)^{s } \omega ( g)\phi (0)+W_{0} (s, g, \phi). \end{align} $$

By Lemma 3.1.1 (2) and (3.8), for $h\in U({\mathbb {W}})$ , we have

(3.10) $$ \begin{align} E_t(0,g,\omega(h)\phi)=E_t(0,g,\phi). \end{align} $$

For $t\neq 0$ and a pure tensor $\phi $ , define

(3.11) $$ \begin{align} E_t'(0,g,\phi)(v)= W_{v,t} '(0, g_v, \phi_v) \prod_{u\neq v} W_{u,t} (0, g_u, \phi_u).\end{align} $$

Extend the definition to all Schwartz functions by linearity. Then

(3.12) $$ \begin{align} E_t'(0,g,\phi)=\sum_{v } E_t'(0,g,\phi)(v). \end{align} $$

3.1.3 Coherent case: Siegel-Weil formula

Assume that ${\mathbb {W}}=W({\mathbb {A}}_E)$ is coherent. For $\phi \in {\mathcal {S}}(W ({\mathbb {A}}_E))$ and $(g,h)\in G({\mathbb {A}}_F ) \times U({\mathbb {W}} ) $ , we define a theta series, which is absolutely convergent:

$$ \begin{align*}\theta(g,h,\phi)=\sum_{x\in W{{}}} \omega(g,h) \phi ( x).\end{align*} $$

Then $ \theta (g,h,\phi )$ is smooth, slowly increasing and $ G(F)\times U(W) $ -invariant.

Assume that W is anisotropic. Then $E(s,g,\phi )$ is holomorphic at $s=0$ , and the following equation is a special case of the regularized Siegel-Weil formula [Reference IchinoIch04, Theorem 4.2]:

(3.13) $$ \begin{align} E(0,g,\phi)=\frac{\kappa}{{\mathrm{Vol}}([U(W)])} \int _{[U(W)]} \theta(g,h,\phi) dh ,\end{align} $$

where $\kappa =2$ if $ \dim W=1$ and $\kappa =1$ if $ \dim W>1$ .

3.1.4 Incoherent case: derivative

Assume that ${\mathbb {W}}$ is incoherent. By Lemma 3.1.1 (1), for $t\neq 0$ , the summand in (3.12) corresponding to v is nonzero only if t is represented by ${\mathbb {W}}^v$ . By the incoherence, ${\mathbb {W}}_v$ does not represent t. In particular, we have the following lemma.

Assume that $\dim {\mathbb {W}}=1$ . Assume that $\phi $ is a pure tensor.

First, assume that $t\neq 0$ . By the product formula for Hasse invariant and the Hasse principle, if t is represented by ${\mathbb {W}}^v$ , then v is nonsplit in E and t is represented by the v-nearby hermitian space W of ${\mathbb {W}}$ . See 2.2. By Lemma 3.1.1 (2) and Lemma 3.1.4, we have

(3.14) $$ \begin{align} E^{\prime}_t(0,g,\omega(h)\phi)=E^{\prime}_t(0,g, \phi) \end{align} $$

for $h\in U({\mathbb {W}}_v)$ , where v is split in E.

Now we consider the constant term. Let

(3.15) $$ \begin{align}{{\mathfrak{c}}}=4\frac{L'(0,\eta)}{L(0,\eta)}+2\log |{\mathrm{Disc}}_E/{\mathrm{Disc}}_F|,\end{align} $$

where ${\mathrm {Disc}}_E,{\mathrm {Disc}}_F\in {\mathbb {Z}}$ are the discriminants of E and F over ${\mathbb {Q}}$ . By [Reference YuanYZ18, p 586] (note the difference by $2$ between the s-variables in the L-factors in loc. cit. and (3.6)),

(3.16) $$ \begin{align}W_0'(s,g,\phi )=-{{\mathfrak{c}}}\omega(g)\phi(0)-\sum_v \omega(g^v)\phi^v(0){W_{v,0}^\circ} '(0, g_v, \phi_{v}). \end{align} $$

3.2.1 Definition

Let $V^\sharp /E$ be a hermitian space of dimension $n>0$ . Let $\chi _{V^\sharp }$ be a character of $E^\times \backslash {\mathbb {A}}_E^\times $ such that $\chi _{V^\sharp }|_{{\mathbb {A}}_F^\times }=\eta ^{n}$ . Let ${\mathbb {V}}= {\mathbb {W}}\oplus V^\sharp ({\mathbb {A}}_E)$ be the orthogonal direct sum and $\chi _{\mathbb {V}}=\chi _{_{\mathbb {W}}}\chi _{V^\sharp }$ . We have the corresponding Weil representations. Below, we shall use $\omega $ to denote a Weil representation if the hermitian space is indicated in the context – for example, by the function that it acts on.

For $ \phi \in {\mathcal {S}}\left ( {\mathbb {V}}\right )$ , we define a theta-Eisenstein series $ {\theta E}(s,g,\phi )$ on G associated the to the orthogonal decomposition ${\mathbb {V}}= {\mathbb {W}}\oplus V^\sharp ({\mathbb {A}}_E)$ :

(3.17) $$ \begin{align} {\theta E} (s,g,\phi)= \sum_{\gamma\in P(F)\backslash G(F)}\delta (\gamma g)^{s } \sum_{x\in V^\sharp{{}}} \omega (\gamma g)\phi ((0,x)). \end{align} $$

If $\phi =\phi _1\otimes \phi _2$ with $\phi _1\in {\mathcal {S}}\left ( {\mathbb {W}}\right )$ and $\phi _2\in {\mathcal {S}}\left ( V^\sharp ({\mathbb {A}}_E)\right )$ , then

(3.18) $$ \begin{align} {\theta E}(s,g,\phi)= E(s,g,\phi_1) \theta(g, \phi_2). \end{align} $$

For $t\in F$ , let ${\theta E} _t (s,g,\phi )$ be the $\psi _t$ -Whittaker function of ${\theta E} (s,g,\phi )$ .

3.2.2 Coherent case

Assume that ${\mathbb {W}}=W({\mathbb {A}}_E)$ is coherent. The regularized Siegel-Weil formula (3.13) immediately implies the following ‘mixed Siegel-Weil formula’:

(3.19) $$ \begin{align} {\theta E}(0,g,\phi)=\frac{2}{{\mathrm{Vol}}([U(W)])} \int _{[U(W)]} \theta(g,h,\phi) dh.\end{align} $$

(We will prove an arithmetic analog of (3.19) for ${\mathbb {W}}$ being incoherent in 5.2.) Then for $t\in F$ ,

(3.20) $$ \begin{align} {\theta E}_{t} (0,g,\phi) = \frac{2}{ {\mathrm{Vol}}([U(W)]) } \int_{ [U(W)]}\sum_{x\in V ^t } \omega(g)\phi(h^{-1} x)dh. \end{align} $$

For $\phi $ invariant by $U(W(E_v)), v\in \infty $ , the integration in (3.20) is a finite sum.

3.2.3 Incoherent case

Assume that ${\mathbb {W}}$ is incoherent.

For a place v of F nonsplit in E, let W be the v-nearby hermitian space of ${\mathbb {W}}$ . For $\phi =\phi _{1}\otimes \phi _{2}$ with $\phi _{1}\in {\mathcal {S}}\left ( {\mathbb {W}}_v \right )$ , $\phi _2\in {\mathcal {S}}\left ( V^\sharp (E_v)\right )$ and $x=(x_1,x_2)\in V:= W(E_v)\oplus V^\sharp (E_v)$ with $ x_1\neq 0,$ let

(3.21) $$ \begin{align} {W\theta}_{v,x}(s,g,\phi)= \frac{L(1,\eta_v)}{ {\mathrm{Vol}}(U(W(E_v))} W^\circ_{v,q(x_1)}(s,g,\phi_1) \omega (g)\phi_2(x_2). \end{align} $$

This is a local analog of (3.18). Extend this definition to $ {\mathcal {S}}\left ( {\mathbb {W}}_v \right )\otimes {\mathcal {S}}\left ( V^\sharp (E_v)\right )\subset {\mathcal {S}}\left ( {\mathbb {V}}(E_v) \right )$ by linearity. The inclusion is an equality unless $v\in \infty $ . However, this subspace is enough for our purpose. (Besides, there is another definition of ${W\theta }_{v,x}$ for the whole $ {\mathcal {S}}\left ( {\mathbb {V}}(E_v) \right )$ . We will not need it.) For v nonsplit in E and $t\neq 0$ , define

(3.22) $$ \begin{align} {\theta E}^{\prime}_{t} (0,g,\phi) (v) = \frac{2}{ {\mathrm{Vol}}([U(W)]) } \int_{ [U(W)]}\sum_{x\in V ^t -V^\sharp } {W\theta}_{v,h_v^{-1}x} '(0, g_v, \phi_v) \omega(g^v)\phi^v(h^{v,-1} x)dh. \end{align} $$

Note that the analog of (3.12) does not hold.

We study ${W\theta }_{v,x} '(0,g,\phi )$ following [Reference Yuan, Zhang and ZhangYZZ13]. Indeed, the computation is only on the Eisenstein (i.e., Whittaker) part. We remind the reader of the difference between the modulus characters mentioned in 2.6. By (3.3), (3.4) and Lemma 3.1.1, we have the following lemma, which says that under the action of $P(F_v)$ , ${W\theta }_{v,x} '(s_0, g , \phi )$ behaves in the same was as the Weil representation.

Lemma 3.2.2 [Reference Yuan, Zhang and ZhangYZZ13, Lemma 6.6].

The following relations hold:

$$ \begin{align*}{W\theta}_{v,x} '(0, m(a) g , \phi )= \chi_{_{\mathbb{V}},v} (a)|\det a|_{E_v }^{\dim {\mathbb{V}}/2} {W\theta}_{v,ax} '(0, g , \phi ),\ a\in E_v^\times;\end{align*} $$

$$ \begin{align*}{W\theta}_{v,x} '(0, n(b) g , \phi )=\psi _v\left( bq(x)\right) {W\theta}_{v,x} '(0, g , \phi ),\ b\in F_v.\end{align*} $$

3.3.1 Definitions

For $v\in \infty $ , let ${\mathfrak {w}}_v$ be a pair of integers whose sum $|{\mathfrak {w}}_v|$ is $\geq 2$ . For $t\in F_{v,>0}$ , let $W^{{\mathfrak {w}}_v}_{v, t}$ be the standard holomorphic Whittaker function of weight ${\mathfrak {w}}_v$ as in 2.7. Then

$$ \begin{align*}\int_{Z(F_v)N(F_v)\backslash G(F_v)} |W^{{\mathfrak{w}}_v}_{v, t}(h)|^2 dh=(4\pi)^{-|{\mathfrak{w}}_v|+1}\Gamma(|{\mathfrak{w}}_v|-1).\end{align*} $$

For $t\in F_{v,>0}$ , a $\psi _{v,t}$ -Whittaker function W on $G(F_v)$ , and $g\in G(F_v)$ , define

$$ \begin{align*}W_s(g)=\frac{(4\pi)^{|{\mathfrak{w}}_v|-1}}{\Gamma(|{\mathfrak{w}}_v|-1)} W^{{\mathfrak{w}}_v}_{v, t}(g) \int_{Z(F_v)N(F_v)\backslash G(F_v)}\delta(h)^sW(h) \overline{W^{{\mathfrak{w}}_v}_{v, t}(h)} dh.\end{align*} $$

If $W_{s}$ has a meromorphic continuation to $s=0$ , define the quasi-holomorphic projection

$$ \begin{align*}W_{{\mathrm{qhol}}}:=\widetilde{\lim\limits _{s\to 0}}W_{s}\end{align*} $$

of W of weight ${\mathfrak {w}}_v$ . Here, $\widetilde {\lim \limits _{s\to 0}}$ denotes the constant term at $s=0$ .

Let ${\mathfrak {w}}=({\mathfrak {w}}_v)_{v\in \infty }$ , where ${\mathfrak {w}}_v$ is a pair of integers. For a continuous function $f:N(F)\backslash G({\mathbb {A}}_F)\to {\mathbb {C}}$ with $\psi _t$ -Whittaker function $f_t$ for $t\in F_{>0}$ , let $f_{t,{\mathrm {qhol}}}$ be the quasi-holomorphic projection of $f_t$ of weight ${\mathfrak {w}}$ at all infinite places (if it is well defined).

For an automorphic form f on $G({\mathbb {A}}_F)$ , the cuspidal holomorphic projection $f_{{\mathrm {chol}}}$ of weight ${\mathfrak {w}}$ of f is the $L^2$ -orthogonal projection of f to the subspace ${\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})$ of cusp forms – that is, for every cusp form $\phi \in {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}})$ , the Petersson inner product $\langle {f,\phi } \rangle $ equals $\langle {f_{{\mathrm {chol}}},\phi } \rangle $ .

Lemma 3.3.1 [Reference LiuLiu11b, Proposition 6.2][Reference Yuan, Zhang and ZhangYZZ13, Proposition 6.12].

Assume that there exists $\epsilon>0$ such that for $v\in \infty $ and $a \in E_v^\times $ with $|a|_{E_v}\to \infty $ , we have

$$ \begin{align*}f\left( m(a)g \right)=O_{g}\left( |a | _{E_v}^{|{\mathfrak{w}}_v|/2-\epsilon}\right),\end{align*} $$

where $m(a)$ is as in 2.5. Then for $t\in F_{>0}$ , $f_{t,{\mathrm {qhol}}}$ is well defined and $f_{t,{\mathrm {qhol}}}=f_{{\mathrm {chol}},t}$ .

3.3.2 Holomorphic projections of ${\theta E} ' (0,g,\phi )$

Let ${\mathfrak {w}}={\mathfrak {w}}_{\chi _{_{\mathbb {V}}}}$ be as in 2.9. Then $|{\mathfrak {w}}_v|=n+1$ . Holomorphic projections below are of weight ${\mathfrak {w}}$ . Retrieve the notations in 3.2.1. Let ${\theta E}^{\prime }_{{\mathrm {chol}}}(0,g,\phi )$ be the cuspidal holomorphic projection of the derivative ${\theta E}'(0,g,\phi )$ . For $t\in F_{>0}$ , let ${\theta E}^{\prime }_{{\mathrm {chol}},t}(0,g,\phi )$ be its $\psi _t$ -Whittaker function. Let ${\theta E}_t(s,g,\phi )$ be the $\psi _t$ -Whittaker function of ${\theta E}(s,g,\phi )$ . Let ${\theta E}_{t,{\mathrm {qhol}}}'(0,g,\phi )$ be the quasi-holomorphic projection of $ {\theta E}_t'(0,g,\phi )$ if it is well defined. The difference between ${\theta E}^{\prime }_{{\mathrm {chol}},t}(0,g,\phi )$ and ${\theta E}_{t,{\mathrm {qhol}}}'(0,g,\phi )$ is given as follows.

For $\phi =\phi _1\otimes \phi _2\in \overline {\mathcal {S}}\left ( {\mathbb {V}}\right )$ with $\phi _1\in {\mathcal {S}}\left ( {\mathbb {W}}\right )$ and $\phi _2\in \overline {\mathcal {S}}\left ( V^\sharp ({\mathbb {A}}_E)\right )$ , define

$$ \begin{align*}{\theta E}_{00}(s,g,\phi)= \delta(g)^s \omega(g)\phi(0)+ W_0(s,g,\phi_1) \omega(g)\phi_2(0) ,\end{align*} $$

which is the product of $E_0(s,g,\phi _1)$ and the constant term of $\theta (g,\phi _2)$ . See (3.9). The definition extends to general $\phi \in {\mathcal {S}}\left ( {\mathbb {V}}\right )$ by linearity. By (3.3) and (3.4), we can define an Eisenstein series

$$ \begin{align*}J(s,g,\phi)=\sum_{\gamma\in P(F)\backslash G(F)}{\theta E}_{00}(s,\gamma g,\phi). \end{align*} $$

For $t\in F_{>0}$ , let $J_t(s,g,\phi )$ be its $\psi _t$ -Whittaker function. Let $J_{t,{\mathrm {qhol}}}'(0,g,\phi )$ be the quasi-holomorphic projection of the derivative $J_t'(0,g,\phi )$ if it is well defined.

3.3.3 A new Eisenstein series

We introduce a new Eisenstein series in order to compute $J'(0,g,\phi )$ . For $\phi _v=\phi _{v,1}\otimes \phi _{2,v}\in {\mathcal {S}}\left ( {\mathbb {V}}(E_v)\right )$ with $\phi _{v,1}\in {\mathcal {S}}\left ( {\mathbb {W}}_v\right )$ and $\phi _{2,v}\in {\mathcal {S}}\left ( V^\sharp (E_v)\right )$ , define a function on $G(F_v)\times V^\sharp (E_v)$ :

(3.23) $$ \begin{align} c(g,x,\phi_v)={W_{v,0}^\circ} '(0, g, \phi_{1,v}) \omega(g)\phi_{2,v}(x) +\log\delta_v(g)\omega(g)\phi_{v}(x).\end{align} $$

For the moment, we only need

$$ \begin{align*}c(g,\phi_v):=c(g,0,\phi_v).\end{align*} $$

Extend this definition to the whole $ {\mathcal {S}}\left ( {\mathbb {V}}(E_v)\right )$ linearly.

By (3.3), (3.4) and Lemma 3.1.3 (1), a direct computation shows the following lemma.

Thus, we can define the following Eisenstein series in the case that $\phi $ is a pure tensor:

$$ \begin{align*}C(s,g,\phi)(v)=\sum_{\gamma\in P(F)\backslash G(F)}c(\gamma g_v,\phi_v) \omega(\gamma g^v)\phi^v(0).\end{align*} $$

Let

$$ \begin{align*}C(s,g,\phi) =\sum_v C(s,g,\phi)(v),\end{align*} $$

where the sum is over these finite places of F. Let $C_t(s,g,\phi )$ be the $\psi _t$ -Whittaker function of $C(0,g,\phi )$ . The definitions can be obviously extended to the whole $ {\mathcal {S}}\left ( {\mathbb {V}}\right )$ by linearity.

By (3.16), a direct computation shows that

(3.24) $$ \begin{align} J'(0,g,\phi)=2E'(0,g,\phi)-{{\mathfrak{c}}}E(0,g,\phi)-C(0,g,\phi). \end{align} $$

3.3.4 Gaussian functions and holomorphy

Below in this section, assume that ${\mathbb {V}} $ is totally positive definite and $\phi = \phi _\infty \otimes \phi ^\infty \in \overline {\mathcal {S}}\left ( {\mathbb {V}}\right )$ . (So $ \phi _\infty $ is Gaussian. See 2.3.) Let $v\in \infty .$ Then

(3.25) $$ \begin{align}\omega ([k_1,k_2])\phi_v =k_1^{ w_1} k_2^{ w_2}\phi_v \end{align} $$

for $[k_1,k_2]\in K_v^{\max }$ as in 2.5 if ${\mathfrak {w}}_v=(w_1,w_2)$ . (Indeed, first check (3.26) for $g=w_v$ and $g\in K_v^{\max }$ being diagonal, then for $g\in K_v^{\max }$ being anti-diagonal, and finally for general $g\in K_v^{\max }$ .) Then by the Iwasawa decomposition, it is easy to check that for $g\in G(F_v), x\in {\mathbb {V}}(E_v)$ ,

(3.26) $$ \begin{align}\omega (g)\phi_v(x)=W^{{\mathfrak{w}}_v}_{v,q(x)}(g).\end{align} $$

By (3.2) combined with (3.25), (3.3) and (3.4), $W_{v,t} (0, \cdot , \phi _v)$ is a multiple of $ W^{{\mathfrak {w}}_v}_{v, t} $ . Then by [Reference Yuan, Zhang and ZhangYZZ13, Proposition 2.11 (2) (4)], we have

(3.27) $$ \begin{align} W_{v,t} (0, g, \phi_v) &=\gamma_{{\mathbb{V}}(E_v)} \frac{(2\pi)^{n+1}}{\Gamma(n+1)} t^n W^{{\mathfrak{w}}_v}_{v, t}(g), \ t>0, \end{align} $$

(3.28) $$ \begin{align} W_{v,t} (0, g, \phi_v) &=0,\ t\leq 0. \end{align} $$

For $t\in F^\times $ , let $ E_t'(0,g,\phi )(v)$ be as in (3.11) so that we have the decomposition (3.12). Let

(3.29) $$ \begin{align} E^{\prime}_{t,{\mathrm{f}}}(0,g,\phi)=\sum_{v\not\in \infty}E_t'(0,g,\phi)(v).\end{align} $$

By (3.27) and (3.28), if $t\in F_{>0}$ , then $E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$ is a multiple of $W^{{\mathfrak {w}}}_{\infty , t}(g_\infty )$ ; otherwise, $E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )=0$ . We call $E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$ the holomorphic part of the Whittaker function $E_t'(0,g,\phi )$ .

3.3.5 Properties of Eisenstein series

We list some properties of the above Eisenstein series for later use. The reader may skip these properties for the moment.

By [Reference YuanYZ18, Lemma 7.6 (2)] (or its proof) and taking care of the difference between the modulus characters mentioned in 2.6, we have the following lemma.

We omit the routine proof of the following analog of (3.2).

3.3.6 Compute $J_{t,{\mathrm {qhol}}}'(0,g,\phi )$

Let $t\in F_{>0}$ . For $v\in \infty $ , let $E_{t,{\mathrm {qhol}}}'(0,g,\phi )(v)$ be the quasi-holomorphic projection of $ E_t'(0,g,\phi )(v)$ . By (3.24), Lemma 3.3.6 and the discussion below it, to compute $J_{t,{\mathrm {qhol}}}'(0,g,\phi )$ , we only need to compute $E_{t,{\mathrm {qhol}}}'(0,g,\phi )(v)$ .

Consider the quasi-holomorphic projection $W_{v,t,{\mathrm {qhol}}}' (0, g, \phi _v)$ of $W_{v,t}' (0, g, \phi _v)$ . By definition, it is a multiple of $W^{{\mathfrak {w}}_v}_{v, t}(g)$ . Then by (3.27),

$$ \begin{align*}b_{v,t}: = \frac{W_{v,t,{\mathrm{qhol}}}' (0, g, \phi_v)}{ W_{v,t} (0, g, \phi_v)} \end{align*} $$

is a well defined constant $b_{v,t}$ . We define

(3.30) $$ \begin{align} {{\mathfrak{b}}}=b_{v,1}. \end{align} $$

Then $E_{t,{\mathrm {qhol}}}'(0,g,\phi )(v)=({{\mathfrak {b}}} + \log |t|_v) E_t(0,g,\phi ). $ Since both $E(0,g,\phi )$ and $C(0,g,\phi )$ in (3.24) are holomorphic of weight ${\mathfrak {w}}$ , we have

$$ \begin{align*} J_{t,{\mathrm{qhol}}}^{\prime}(0,g,\phi)=( 2{{\mathfrak{b}}}[F:{\mathbb{Q}}]-{{\mathfrak{c}}})E_t(0,g,\phi)-C_t(0,g,\phi)+2\left( E^{\prime}_{t,{\mathrm{f}}}(0,g,\phi)+E_t(0,g,\phi) \log {\mathrm{Nm}}_{F/{\mathbb{Q}}}t\right). \end{align*} $$

Combined with Lemma 3.3.3, we have

(3.31) $$ \begin{align} \begin{aligned} &{\theta E}_{t,{\mathrm{qhol}}}^{\prime}(0,g,\phi)-2\left( E^{\prime}_{t,{\mathrm{f}}}(0,g,\phi) +E_t(0,g,\phi) \log {\mathrm{Nm}}_{F/{\mathbb{Q}}}t\right)\\& \quad ={\theta E}^{\prime}_{{\mathrm{chol}},t}(0,g,\phi) +( 2{{\mathfrak{b}}}[F:{\mathbb{Q}}]-{{\mathfrak{c}}})E_t(0,g,\phi)-C_t(0,g,\phi). \end{aligned} \end{align} $$

3.3.7 Quasi-holomorphic projection of ${\theta E} ' _t (0,g,\phi )$

Assume that ${\mathbb {W}}/{\mathbb {E}}$ is incoherent. For $v\in \infty $ and W the v-nearby hermitian space of ${\mathbb {W}}$ , let $V=W\oplus V^\sharp $ . For $x\in V(E_v)-V^\sharp (E_v)$ , define

$$ \begin{align*}\widetilde {W\theta}_{s}(x)= \frac{\Gamma(s+n)}{\Gamma(n) (4\pi)^{s}}P_s(-{q(x_1)}) , \end{align*} $$

where $x_1$ is the projection of x to $W(E_v)$ (so that $x_1\neq 0$ ), and

(3.32) $$ \begin{align} P_s(t):=\int_{u=1}^\infty\frac{1}{u\left(1+tu\right)^{s+n}}du,\ t>0.\end{align} $$

(For $s \in {\mathbb {C}}$ with $ {\mathrm {Re}} s>-n$ , $P_s(t)$ converges absolutely.) Define

(3.33) $$ \begin{align} {\theta E}^{\prime}_{t,s} (0,g,\phi)(v) = \frac{2 W^{\mathfrak{w}}_{v,t}(g_v) }{ {\mathrm{Vol}}([U(W)]) } \int_{ [U(W)]}\sum_{x\in V {{}} ^t -V^\sharp} \widetilde{W\theta}_{s} (h_v^{-1}x) \omega^v(g^v)\phi^v \left( h^{v,-1} x\right) dh. \end{align} $$

Then the constant term $\widetilde {\lim \limits _{s\to 0}}{\theta E}^{\prime }_{t,s} (0,g,\phi )(v)$ of $ {\theta E}^{\prime }_{t,s} (0,g,\phi )(v)$ at $s=0$ is well defined (and used in the following proposition). We will prove Lemma 3.3.12 after Lemma 6.1.10, by comparing ${\theta E}^{\prime }_{t,s} (0,g,\phi )(v)$ to a Green function with s-variable, which has a meromorphic continuation.

Proposition 3.3.13. Let $t\in F_{>0}$ and let $\phi \in \overline {\mathcal {S}}({\mathbb {V}})$ be a pure tensor.

(1) We have

(3.34) $$ \begin{align} \begin{aligned} {\theta E}_{t,{\mathrm{qhol}}}'(0,g,\phi) &=-\sum_{v \not\in \infty,\ \text{nonsplit}}{\theta E}^{\prime}_t (0,g,\phi)(v)-\sum_{v\in \infty}\widetilde{\lim _{s\to 0}}{\theta E}^{\prime}_{t,s} (0,g,\phi)(v)\\& \quad -\left( 4\frac{L_{\mathrm{f}}'(0,\eta)}{L_{\mathrm{f}}(0,\eta)}+2\log |{\mathrm{Disc}}_E/{\mathrm{Disc}}_F|\right)\sum_{x\in (V^\sharp)^t}\omega(g)\phi(x)\\& \quad -\sum_{v\not\in \infty}\sum_{x\in (V^\sharp)^t}c(g,x,\phi_v)\omega(g^v)\phi^v(x)\\& \quad +\sum_{x\in (V^\sharp)^t}(2\log \delta(g^\infty)+\log|t^\infty|)\omega(g)\phi(x)\end{aligned}. \end{align} $$

Here, $L_{\mathrm {f}}(s,\eta )$ is the finite part of $L(s,\eta )$ .

(2) Assume that $\phi _{{{v}}}$ is supported outside $V^\sharp (E_{v})$ for v in a set S of two places of F and $g\in P ({\mathbb {A}}_{F,S })G({\mathbb {A}}_F^{S })$ . Then we have

(3.35) $$ \begin{align} {\theta E}_{t,{\mathrm{qhol}}}'(0,g,\phi) =&-\sum_{v \not\in \infty, \ \text{nonsplit in }E}{\theta E}^{\prime}_t (0,g,\phi)(v)-\sum_{v\in \infty}\widetilde{\lim _{s\to 0}}{\theta E}^{\prime}_{t,s} (0,g,\phi)(v). \end{align} $$

Lemma 3.3.14. Let $t\in F_{>0}$ and let $\phi $ be a pure tensor such that $\phi ^\infty $ is ${\overline {\mathbb {Q}}}$ -valued. Let u be a finite place of F of residue characteristic p. Let $O\subset {\mathbb {V}}_u$ be an open compact neighborhood of $0$ and $\phi ^O= \phi ^u\otimes (\phi _u 1_{{\mathbb {V}}_u-O})$ . Given $g\in G({\mathbb {A}}_F^{u })P(F_u)$ , for O small enough, we have

$$ \begin{align*}\frac{{\theta E}_{t,{\mathrm{qhol}}}'(0,g,\phi)}{W^{{\mathfrak{w}}}_{\infty, t}(g_\infty)}= \frac{{\theta E} ^{\prime}_{t,{\mathrm{qhol}}} (0,g,\phi^O)} {W^{{\mathfrak{w}}}_{\infty, t}(g_\infty)}(\operatorname{\mathrm{mod}}\overline{{\mathbb{Q}}}\log p).\end{align*} $$

By (3.34), using (3.22) and (3.33), for $h\in U({\mathbb {W}}_v)$ where v is split in E,

(3.36) $$ \begin{align} {\theta E}_{t,{\mathrm{qhol}}}'(0,g,\omega(h)\phi)={\theta E}_{t,{\mathrm{qhol}}}'(0,g,\phi). \end{align} $$

4.1.1 Simple special divisors

Let ${\mathbb {V}}^{\infty }_{>0}\subset {\mathbb {V}}^{\infty }$ be the subset of x’s such that ${q(x)}\in F_{>0}$ . For $x\in {\mathbb {V}}^{\infty }_{>0}$ , let $ x^\perp $ be the orthogonal complement of ${\mathbb {A}}_E ^\infty x $ in ${\mathbb {V}}^\infty $ . Regard $U\left ( x^{\perp } \right )$ as a subgroup of $U({\mathbb {V}}^\infty )$ . Then we have a finite morphism

(4.1) $$ \begin{align} {\mathrm{Sh}}\left( x^{\perp }\right)_{U\left( x^{\perp} \right)\bigcap K }\to{\mathrm{Sh}}({\mathbb{V}})_K, \end{align} $$

explicated in [Reference KudlaKud97a, (2.4)] and [Reference Li and LiuLL21, Definition 4.1]. The proper pushforward defines a divisor $Z(x)_K $ on ${\mathrm {Sh}}({\mathbb {V}})_K$ that is called a simple special divisor. The following observation is trivial.

Let $L_K$ be the Hodge line bundle on ${\mathrm {Sh}}({\mathbb {V}})_K$ . See 4.2.1 for the description in terms of the complex uniformization of ${\mathrm {Sh}}({\mathbb {V}})_K$ . Let $c_1(L_K^\vee )$ be the first Chern class of the dual of $ L_K$ , and $[Z(x)_K]\in {\mathrm {Ch}}^{1}({\mathrm {Sh}}({\mathbb {V}})_K) $ the class of $Z(x)_K$ . For $ \phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$ the subspace of K-invariant functions, define a formal generating series of divisor classes:

$$ \begin{align*}z(\phi)_K=\phi(0)c_1(L_K^\vee)+\sum_{x\in K\backslash {\mathbb{V}}^{ \infty}_{>0}}\phi( x_\infty x) [Z(x)_K] , \end{align*} $$

where $ x_\infty \in {\mathbb {V}}_\infty $ such that $q(x_\infty )={q(x)}\in F_{>0}$ .

Let ${\mathfrak {w}}={\mathfrak {w}}_{\chi _{_{\mathbb {V}}}}$ which is defined in 2.9.

Theorem 4.1.2 [Reference LiuLiu11a, Theorem 3.5].

For every $ \phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K} $ , we have

$$ \begin{align*}z\left(\omega(\cdot )\phi\right)_K\in {\mathcal{A}}_{{\mathrm{hol}}}(G,{\mathfrak{w}}) \otimes {\mathrm{Ch}}^{1}({\mathrm{Sh}}({\mathbb{V}})_K)_{{\mathbb{C}}} .\end{align*} $$

Note that $ \dim {\mathrm {Ch}}^{1}({\mathrm {Sh}}({\mathbb {V}})_K)_{{\mathbb {C}}} <\infty .$

4.1.2 Weighted special divisors

For $\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$ and $t\in F_{>0}$ , define the weighted special divisor

$$ \begin{align*} Z_t(\phi)_K=\sum_{x\in K\backslash {\mathbb{V}}^{ \infty}, \ q(x)=t}\phi ( x_\infty x) Z(x)_K, \end{align*} $$

which is a finite sum. Lemma 4.1.1 implies the following lemma.

Let $\phi = \phi _\infty \otimes \phi ^\infty $ be as in 2.3. We define another weighted special divisor

$$ \begin{align*} Z_t(\phi^\infty)_K=\sum_{x\in K\backslash {\mathbb{V}}^{ \infty}, \ q(x)=t}\phi^\infty( x) Z(x) _K. \end{align*} $$

By (3.26), for $g\in G({\mathbb {A}}_F)$ , we have

(4.2) $$ \begin{align} Z_t(\omega(g)\phi)_K=Z_t(\omega(g^\infty)\phi^\infty) _KW^{{\mathfrak{w}}}_{\infty, t}(g_\infty).\end{align} $$

Then $[Z_t\left ( \omega (\cdot )\phi ^\infty \right )_K] $ is the t-th Fourier coefficient of $z\left (\omega (\cdot )\phi \right )_K$ . See 2.7.

4.1.3 Change level

For $K\subset K'$ , let $\pi _{_K,_{K'}}:{\mathrm {Sh}}({\mathbb {V}})_{K}\to {\mathrm {Sh}}({\mathbb {V}})_{K'}$ be the natural projection.

Below, let $L=L_K$ , $Z(x)=Z(x)_K$ , $Z_t(\phi )=Z_t(\phi )_K$ and $z(\phi )=z(\phi )_K$ for simplicity if K is clear from the context.

4.2.1 Complex uniformization

For nonnegative integers $p,q$ , let ${\mathbb {C}}^{p,q}$ be the $p+q$ dimensional hermitian space associated to the hermitian matrix ${\mathrm {diag}}(-1_p,1_q)$ . Let $U\left ({\mathbb {C}}^{1,n}\right )$ be the unitary group of ${\mathbb {C}}^{1,n}$ (so of signature $(n,1)$ in the usual convention). Let ${\mathbb {B}}_n$ be the complex open unit ball of dimension n. Embed ${\mathbb {B}}_n$ in ${\mathbb {P}}({\mathbb {C}}^{1,n})$ as the set of negative lines as follows: $[z_1,\ldots ,z_n]\in {\mathbb {B}}_{n}$ , where $\sum _{i=1}^{n}|z_i|^2<1$ , is the line containing the vector $(1,z_1\ldots ,z_n) $ . Then $U\left ({\mathbb {C}}^{1,n}\right )$ acts on ${\mathbb {B}}_n$ naturally and transitively. Let $\Omega $ be the tautological bundle of negative lines on ${\mathbb {B}}_n$ , and $\overline \Omega $ the hermitian line bundle with the metric induced from the negative of the hermitian form. The Chern form of $\overline \Omega $ is a $U\left ({\mathbb {C}}^{1,n}\right )$ -invariant Kähler form

(4.3) $$ \begin{align}c_1(\overline\Omega)= \frac{1}{2\pi i } \partial \bar\partial \log (1-\sum_{i=1}^{n}|z_i|^2).\end{align} $$

For an arithmetic subgroup $\Gamma \subset U({\mathbb {C}}^{1,n})$ , on (the orbifold) $\Gamma \backslash {\mathbb {B}}_n $ , we have the descent $\overline \Omega _\Gamma $ of $\overline \Omega $ whose (orbifold) Chern curvature form is the descent of $c_1(\overline \Omega )$ . Define the degree

$$ \begin{align*}\deg(\overline\Omega_\Gamma)=\int_{\Gamma\backslash{\mathbb{B}}_n} c_1(\overline\Omega_ \Gamma)^n.\end{align*} $$

If $\Gamma \backslash {\mathbb {B}}_n$ is compact, this degree is the usual degree via intersection theory.

For $x\in {\mathbb {C}}^{1,n}$ , let ${\mathbb {B}}_x\subset {\mathbb {B}}_n$ be the subset of negative lines perpendicular to x. Later, we will use the following function on ${\mathbb {B}}_n $ measuring ‘the distance to ${\mathbb {B}}_x$ ’:

$$ \begin{align*}R_x(z)=-\frac{|\langle {x,\widetilde z} \rangle|^2}{\langle {\widetilde z,\widetilde z} \rangle},\end{align*} $$

where $\widetilde z $ is a nonzero vector contained in the line z. In particular, $R_x>0$ outside ${\mathbb {B}}_x$ . If $q(x)<0$ , then $x^\perp $ is of signature $(n,0)$ so that ${\mathbb {B}}_x=\emptyset $ . Below, until 4.2.3, assume ${q(x)}>0$ . Then $x^\perp $ is of signature $(n-1,1)$ . So ${\mathbb {B}}_x$ is a complex unit ball of dimension $n-1$ . Assume that $\Gamma \cap U (x^\perp )$ is an arithmetic subgroup of $U (x^\perp )$ . Let $C(x,\Gamma )$ be the pushforward of the fundamental cycle by

$$ \begin{align*}\left( \Gamma \cap U(x^\perp)\right) \backslash {\mathbb{B}}_x \to \Gamma \backslash {\mathbb{B}}_n.\end{align*} $$

Define

$$ \begin{align*}\deg_{\overline\Omega_\Gamma}(C(x,\Gamma))=\int_{C(x,\Gamma)} c_1(\overline\Omega_ \Gamma)^{n-1}.\end{align*} $$

Now we can uniformize Shimura varieties and special divisors. Let v be an infinite place of F. Let V be the unique hermitian space over E such that $V({\mathbb {A}}_E^v){\ \simeq \ } {\mathbb {V}}^v$ and $V(E_v){\ \simeq \ } {\mathbb {C}}^{1,n}$ . By [Reference Li and LiuLL21, Lemma 5.5], we have the complex uniformization:

(4.4) $$ \begin{align}{\mathrm{Sh}}({\mathbb{V}})_{K,E_v} {\ \simeq \ } U(V)\backslash( {\mathbb{B}}_n \times U({\mathbb{V}}^\infty)/K). \end{align} $$

Then the Hodge bundle $L_{E_v}$ is the descent of $\Omega \times 1_{U({\mathbb {V}}^\infty )/K}$ . Let $\overline {L}_{E_v}$ be the descent of $\overline \Omega \times 1_{U({\mathbb {V}}^\infty )/K}$ . Let $h_1,.. .,h_m$ be a set of representatives of $U(V)\backslash U({\mathbb {V}}^\infty )/K$ . Let $\Gamma _{h_j}=U(V)\cap h_jKh_j^{-1}$ . Then (4.4) decomposes as the disjoint union of $\Gamma _{h_j} \backslash {\mathbb {B}}_n$ ’s. Then for $t\in F_{>0}$ , we have (see [Reference KudlaKud97a, PROPOSITION 5.4])

(4.5) $$ \begin{align} Z_t(\phi^\infty)_{E_v}=\sum_{j=1}^m\sum_{x\in \Gamma_{h_j}\backslash V^t} \phi^\infty(h_j^{-1}x)C(x,\Gamma_{h_j}).\end{align} $$

4.2.2 Admissible Green functions

Admissible Green functions are Green functions with harmonic curvatures. See Appendix A.3. Admissible Green functions for special divisors are constructed by Bruinier [Reference BruinierBru02, Reference BruinierBru12], Oda and Tsuzuki [Reference Oda and TsuzukiOT03]. For $F={\mathbb {Q}}$ and $n=1$ , it appeared in the work of Gross and Zagier [Reference Gross and ZagierGZ86] for $n=1$ . We learned the following explicit computation from S. Zhang.

First, we start by working on ${\mathbb {B}}_n$ . Let $x_0=(0,\ldots ,0,1)\in {\mathbb {C}}^{1,n}$ . Then ${\mathbb {B}}_{x_0}\subset {\mathbb {B}}_n$ consists of points $ (z_1,\ldots ,z_{n-1},0) $ ’s. We want a $U\left ( x_0 ^\perp \right )$ -invariant smooth function G on ${\mathbb {B}}_n-{\mathbb {B}}_{x_0}$ such that $G(z_1,\ldots ,z_n)+\log |z_n|^2 $ extends smoothly to ${\mathbb {B}}_n$ , $\lim _{|z|\to 1}G(z) =0,$ and G is a solution of the following Laplacian equation:

(4.6) $$ \begin{align}\left(\frac{i }{2\pi } \partial \bar\partial G \right) {c_1(\Omega)}^{n-1}=\frac{s(s+n) }{2}G \cdot {c_1(\Omega)}^{n} .\end{align} $$

The quotient of ${\mathbb {B}}_n-{\mathbb {B}}_{x_0}$ by $U\left ( x_0 ^\perp \right )$ is isomorphic to $(1,\infty )$ via

$$ \begin{align*}z=[z_1,\ldots,z_n]\mapsto t(z):= 1+ R_{ x_0 }\left( z\right)=\frac{1-\sum_{i=1}^{n-1}|z_i|^2}{1-\sum_{i=1}^{n}|z_i|^2} .\end{align*} $$

Thus, we look for $G=Q(t(z))$ , where Q is a smooth function on $(1,\infty )$ such that $ Q(t)+ \log ( t-1) $ extends to a smooth function on ${\mathbb {R}}$ . By (4.3) and the $U\left ( x_0 ^\perp \right )$ -invariance, (4.6) is reduced to the following hypergeometric differential equation

$$ \begin{align*} \left( t-t^2\right) \frac{d^2Q}{dt^2} +\left( n-(n+1)t\right)\frac{dQ}{dt} +s(s+n) Q=0,\ t\in (1,\infty). \end{align*} $$

For ${\mathrm {Re}} s>-1$ , there is a unique solution $Q_s$ such that $Q_s(t)+\log (t-1)$ extends to a smooth function on ${\mathbb {R}}$ and $\lim _ {t\to \infty }Q_s(t)=0$ :

(4.7) $$ \begin{align} Q_s(t)=\frac{\Gamma(s+n)\Gamma(s+1)}{\Gamma(2s+n+1)t^{s+n}} F\left( s+n,s+1,2s+n+1;\frac{1}{t},\right), \ t>1\end{align} $$

where F is the hypergeometric function. (See also [Reference Oda and TsuzukiOT03, 2.5.3]. When $n=1$ , our $Q_s$ is the Legendre function of the second kind in [Reference Gross and ZagierGZ86, 238] up to shifting s by 1).

For a general x with $q(x)>0$ , we have the following Green function for ${\mathbb {B}}_x$ :

$$ \begin{align*} G_{x,s}(z)=Q_s\left( 1+R_x(z)\right),\ z \in {\mathbb{B}}_n- {\mathbb{B}}_x .\end{align*} $$

We will need the following explicit formula later: if $ x=(x_1,x_2)\in {\mathbb {C}}^{1,n}= {\mathbb {C}}^{1,0}\oplus {\mathbb {C}}^{0,n} $ , then

(4.8) $$ \begin{align}G_{x,s}([0,\ldots,0]) =Q_s\left( 1-{q(x_1)}\right).\end{align} $$

Second, we define Green functions for $ C( x ,\Gamma )$ on arithmetic quotient of ${\mathbb {B}}_n.$ Let $\Gamma $ be an arithmetic subgroup of $U\left ({\mathbb {C}}^{1,n}\right )$ . Let

$$ \begin{align*}g_s=\sum_{\gamma\in \Gamma}\gamma^{*}G_{ x ,s}.\end{align*} $$

It is easy to see that $g_s$ with ${\mathrm {Re}} s>0$ is a Green function for $ C( x ,\Gamma )$ .

Recall that $\widetilde {\lim \limits _{s\to 0}}$ denotes taking the constant term at $s=0$ .

By [Reference Oda and TsuzukiOT03, Proposition 3.1.2] and taking care of the differences in the remark, we have

$$ \begin{align*}\int_{\Gamma\backslash {\mathbb{B}}_n} g_s {c_1(\overline \Omega)}^n=\frac{n\deg_{\overline\Omega_\Gamma}(C( x ,\Gamma))}{s(s+n)}.\end{align*} $$

Thus,

(4.9) $$ \begin{align} \int_{\Gamma\backslash {\mathbb{B}}_n}(\widetilde{\lim\limits _{s\to 0}}g_s ) {c_1(\overline \Omega)}^n=-\frac{\deg_{\overline\Omega_\Gamma}(C( x ,\Gamma))}{n}. \end{align} $$

Finally, we define Green functions for $Z_t(\phi )$ , $t\in F_{>0}$ . Let $v\in \infty $ and V as in 4.2.1. For $s \in {\mathbb {C}}$ with ${\mathrm {Re}} s> 0$ , consider the following formal sum for $(z,h)\in {\mathbb {B}}_n \times U({\mathbb {V}}^\infty )$ :

$$ \begin{align*} {\mathcal{G}}_{Z_t(\phi^\infty)_{E_v},s}(z,h):=\sum_{x\in V^t } \phi^\infty\left( h^{-1}x\right) G_{x,s}(z). \end{align*} $$

By (4.5) and Lemma 4.2.1, if $(z,h)$ is not over $Z_t(\phi ^\infty )_{E_v}$ via (4.4), the formal sum is defined and absolutely convergent. Thus, $ {\mathcal {G}}_{Z_t(\phi ^\infty )_{E_v},s}$ descends to ${\mathrm {Sh}}({\mathbb {V}})_{K,E_v}-Z_t(\phi ^\infty )_{E_v} $ via (4.4), which we still denote by ${\mathcal {G}}_{Z_t(\phi ^\infty )_{E_v},s}$ . By (4.5) and Theorem 4.2.2, ${\mathcal {G}}_{Z_t(\phi ^\infty )_{E_v},s} $ is a Green function for $Z_t(\phi ^\infty )_{E_v}$ . For $g\in G({\mathbb {A}}_F)$ , let

$$ \begin{align*} {\mathcal{G}}_{Z_t(\omega(g)\phi)_{E_v},s}= {\mathcal{G}}_{Z_t(\omega(g^\infty)\phi^\infty)_{E_v},s} W^{{\mathfrak{w}}}_{\infty, t}(g_\infty),\end{align*} $$

which is a Green function for $Z_t(\omega (g)\phi )_{E_v}$ by (4.2). Define the automorphic Green functions for $Z_t(\phi ^\infty )_{E_v}$ and $Z_t(\omega (g)\phi )_{E_v}$ to be

(4.10) $$ \begin{align} {\mathcal{G}}^{{\mathrm{aut}}}_{Z_t(\phi^\infty)_{E_v}}=\widetilde{\lim_{s\to 0}} {\mathcal{G}}_{Z_t(\phi^\infty)_{E_v},s},\ {\mathcal{G}}^{{\mathrm{aut}}}_{Z_t(\omega(g)\phi)_{E_v}}=\widetilde{\lim_{s\to 0}} {\mathcal{G}}_{Z_t(\omega(g)\phi)_{E_v},s}, \end{align} $$

respectively, which are admissible by Theorem 4.2.2.

Let $ {\mathcal {G}}^{\overline L_{E_v}}_{Z_t(\phi )_{E_v}}$ be the normalized admissible Green function for $Z_t(\phi )_{E_v}$ with respect to $\overline L_{E_v}$ as in Definition A.3.3. In particular, its integration against $c_1(\overline L_{E_v})^{n}$ is 0.

Lemma 4.2.4. (1) We have

$$ \begin{align*}{\mathcal{G}}^{{\mathrm{aut}}}_{Z_t(\omega(g)\phi)_{E_v}}- {\mathcal{G}}^{\overline L_{E_v}}_{Z_t(\omega(g)\phi)_{E_v}}=-\frac{1}{n} \frac{c_1(L)^{n-1}\cdot Z_t(\omega(g)\phi) }{c_1(L)^{n} }.\end{align*} $$

Both sides are independent of K.

(2) We have

$$ \begin{align*}\frac{1}{n}\omega(g)\phi(0)+\sum_{t\in F_{>0}} \left( {\mathcal{G}}^{{\mathrm{aut}}}_{Z_t(\omega(g)\phi)_{E_v}}- {\mathcal{G}}^{\overline L_{E_v}}_{Z_t(\omega(g)\phi)_{E_v}}\right)\in {\mathcal{A}}_{{\mathrm{hol}}}(G,{\mathfrak{w}}).\end{align*} $$

Remark 4.2.5. The automorphic form in Lemma 4.2.4 (2) can be made explicit:

(4.11) $$ \begin{align} c_1(L)^{n-1}\cdot z\left(\omega(g)\phi\right) =-c_1(L)^{n} E(0,g, \phi). \end{align} $$

This is a geometric version of the Siegel-Weil formula (3.13). It is stated in [Reference KudlaKud97a, COROLLARY 10.5] for the orthogonal case; the proof carries over to the unitary case.

4.2.3 Kudla’s Green function

We recall Kudla’s Green functions for special divisors [Reference KudlaKud97b], following [Reference LiuLiu11a, 4C] in the unitary case. We consider simple special divisors $Z(x)$ ’s, instead of weight special divisors $Z_t(\phi )$ ’s. We extend the definition of special divisors as follows. For $x\in {\mathbb {V}}^{\infty } $ such that ${q(x)}\in F ^\times -F_{>0}$ , let $Z(x)= 0$ .

First, we work on ${\mathbb {B}}_n$ . For $v\in \infty $ , V as in the end of 4.2.1 with respect to v, $g\in G(F_v)$ and $x\in V$ , define

$$ \begin{align*}G^{\mathrm{Kud}}(x,g)(z)=-{\mathrm{Ei}}(-2\pi \delta_v(g) R_x(z)), \ z \in {\mathbb{B}}_n\backslash {\mathbb{B}}_x ,\end{align*} $$

where the exponential integral ${\mathrm {Ei}}(t)=\int _{-\infty }^t\frac {e^s}{s}ds$ on $t\in (-\infty ,0)$ has a log-singularity at 0. If $q(x)\neq 0$ so that ${\mathbb {B}}_x$ is either empty or a complex unit ball of dimension $n-1$ , $G^{\mathrm {Kud}}(x,g)$ is a Green function for ${\mathbb {B}}_x$ . If $q(x)<0$ , equivalently ${\mathbb {B}}_x$ is empty, then $G^{\mathrm {Kud}}$ is smooth.

Now we work on ${\mathrm {Sh}}({\mathbb {V}})_K$ . Let $x\in {\mathbb {V}}^\infty $ with $q(x)\in F^\times $ . For $v\in \infty $ , if $u(q(x))>0$ for every $u\in \infty -\{v\}$ , by the Hasse-Minkowski theorem and Witt’s theorem, there exists $h\in U({\mathbb {V}}^\infty )$ and $x^{(v)}\in V-\{0\}$ , where V is as in the last paragraph, such that $ x=h^{-1}x^{(v)}$ . Define

(4.12) $$ \begin{align}{\mathcal{G}}^{\mathrm{Kud}}_{Z( x)_{E_v}}(g)=\sum_{j=1}^m\sum_{y\in U(V)x^{(v)}\cap h_jK h^{-1}x^{(v)}} G^{\mathrm{Kud}}(y,g) ,\end{align} $$

where $h_1,.. .,h_m$ is a set of representatives of $U(V)\backslash U({\mathbb {V}}^\infty )/K$ . By the decomposition analogous to (4.5) (see the second equation on [Reference KudlaKud97a, p 56]) and [Reference LiuLiu11a, Proposition 4.9]), ${\mathcal {G}}^{\mathrm {Kud}}_{Z( x)_{E_v}}(g)$ is absolutely convergent and descends to ${\mathrm {Sh}}({\mathbb {V}})_{K,E_v}$ via (4.4). And it is a Green function for $Z(x)_{E_v}$ .

Besides Kudla’s Green functions, we will need their projections to the constant function $1$ to modify the normalized Green function. See 4.4.3.

Definition 4.2.6. For $x\in {\mathbb {V}}^\infty $ with $q(x)\in F^\times $ , $g\in G({\mathbb {A}}_{F,\infty })$ and $v\in \infty $ , let

$$ \begin{align*}{\mathfrak{k}}(x,g_v)=\frac{1}{\deg (\overline{L}_{E_v})}\int_{{\mathrm{Sh}}({\mathbb{V}})_{K,E_v} }{\mathcal{G}}^{\mathrm{Kud}}_{Z(x)_{E_v}}(g_v) c_1(\overline{L}_{E_v})^n.\end{align*} $$

Note that if $u(q(x))<0$ for some $u\in \infty -\{v\}$ , then ${\mathcal {G}}^{\mathrm {Kud}}_{Z(x)_{E_v}}(g)=0$ . Thus, if $q(x) $ is negative at more than one infinite places, then ${\mathcal {G}}^{\mathrm {Kud}}_{Z(x)_{E_v}}(g)=0$ and $ {\mathfrak {k}}(x,g_v ) =0$ for every $v\in \infty $ .

The following can be read from [Reference Garcia and SankaranGS19, (1.12), Theorem 1.2, (1.18), (1.19), Proposition 5.9].

Theorem 4.2.7. Let $ E_t'(0,g,\phi )(v)$ be as in (3.11). For $t\in F_{>0}$ and $v\in \infty $ , we have

$$ \begin{align*} &-W^{{\mathfrak{w}}}_{\infty, t}(g_\infty) \sum_{x\in K\backslash {\mathbb{V}}^{ \infty}, \ q(x)=t}\omega(g^\infty)\phi^\infty ( x) {\mathfrak{k}}(x,g_v)\\& \quad = E' _t(0,g,\phi)(v) -E_t(0,g,\phi) \left( \log \pi-(\log\Gamma)'(n+1) + \log v(t)\right). \end{align*} $$

For $t\in F^\times $ with $v(t)<0$ for exactly one infinite place v, we have

$$ \begin{align*}-W^{{\mathfrak{w}}}_{\infty, t}(g_\infty) \sum_{x\in K\backslash {\mathbb{V}}^{ \infty}, \ q(x)=t}\omega(g^\infty)\phi ^\infty( x) {\mathfrak{k}}(x,g_v)= E' _t(0,g,\phi)(v) .\end{align*} $$

And for $t=0$ , we have

$$ \begin{align*}\omega(g)\phi (0 ) \log \delta_\infty(g_\infty)= E' _0(0,g,\phi).\end{align*} $$

We remind the reader that $\infty $ is the set of infinite places of F. And our formulas differ from [Reference Garcia and SankaranGS19] by a factor $1/2$ since in loc. cit., the authors use the set of infinite places of E.

4.2.4 Modularity of difference of Green functions

We need a more general notion of modular forms.

Clearly, if V is the topological direct sum of $V_1, V_2$ and $f_i\in {\mathcal {A}}(G,{\mathfrak {w}}, V_i) $ , then $f_1+f_2\in {\mathcal {A}}(G,{\mathfrak {w}}, V) $ .

Now we define the formal generating series of Green functions. Recall that $\infty $ is the set of infinite places of F, of cardinality $[F:{\mathbb {Q}}]$ . Let

$$ \begin{align*}E_\infty:=E\otimes_{\mathbb{Q}}{\mathbb{R}}{\ \simeq \ } \prod_{v\in \infty}E_v,\end{align*} $$

which is the product of $[F:{\mathbb {Q}}]$ many copies of ${\mathbb {C}}$ . Then

$$ \begin{align*}{\mathrm{Sh}}({\mathbb{V}})_{K,E_\infty}: = {\mathrm{Sh}}({\mathbb{V}})_{K} \otimes_E E_\infty = {\mathrm{Sh}}({\mathbb{V}})_{K}\otimes_{\mathbb{Q}}{\mathbb{R}}\end{align*} $$

is the (disconnected) complex manifold that is the disjoint union of all base changes of ${\mathrm {Sh}}({\mathbb {V}})_{K}$ to $E_v$ ’s (each base change itself may not be connected either!).

Let

$$ \begin{align*}{\mathcal{G}}^{{\overline{\mathcal{L}}}}(g,\phi)=\sum_{t\in F_{>0}} \sum_{v\in \infty} {\mathcal{G}}^{\overline L_{E_v}}_{Z_t(\omega(g)\phi)_{E_v}},\end{align*} $$

$$ \begin{align*} {\mathcal{G}}^{\mathrm{Kud}}(g,\phi)= \sum_{x\in {\mathbb{V}}^\infty,\ q(x)\in F^\times} \omega(g^\infty)\phi^\infty ( x) W^{{\mathfrak{w}}}_{\infty, q(x)}(g_\infty) \sum_{v\in \infty} {\mathcal{G}}^{\mathrm{Kud}}_{Z( x)_{E_v}}(g) , \end{align*} $$

which are formal generating series of smooth functions on ${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$ with logarithmic singularities along the same formal generating series of special divisors. Then $ {\mathcal {G}}^{{\overline {\mathcal {L}}}}(g,\phi )- {\mathcal {G}}^{\mathrm {Kud}}(g,\phi )$ is a formal generating series valued in ${C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }),$ the space of smooth ${\mathbb {C}}$ -valued functions on ${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$ .

Let $E_1:={\mathcal {O}}_{{\mathrm {Sh}}({\mathbb {V}})_{K}}\left ( {\mathrm {Sh}}({\mathbb {V}})_{K}\right )$ , which is a finite field extension of E (since ${\mathrm {Sh}}({\mathbb {V}})_{K}$ is connected). Then we have a morphism ${\mathrm {Sh}}({\mathbb {V}})_{K }\to \operatorname {\mathrm {Spec}} E_1$ . By Stein factorization, ${\mathrm {Sh}}({\mathbb {V}})_{K }$ , as a variety over $E_1$ , is geometrically connected. So the connected components of ${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty } ={\mathrm {Sh}}({\mathbb {V}})_{K}\otimes _{\mathbb {Q}}{\mathbb {R}} $ are exactly indexed by the underlying set of $\operatorname {\mathrm {Spec}} E_1\otimes _{\mathbb {Q}}{\mathbb {R}}$ , equivalently, the set of conjugate pairs of infinite places of $E_1$ (which, as a finite field extension of the CM field E, has only complex embeddings but no real embeddings). Let $ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$ be the space of locally constant functions on ${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$ . Then we have the canonical isomorphism from $ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$ to the product copies of ${\mathbb {C}}$ indexed by the set of conjugate pairs of infinite places of $E_1$ . Now we can embed $ {\mathcal {O}}_{E_1}^\times $ in ${\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$ via this isomorphism and the Dirichlet regulator map, so that the ${\mathbb {C}}$ -span of the image of $ {\mathcal {O}}_{E_1}^\times $ , denoted by ${\mathbb {C}} {\mathcal {O}}_{E_1}^\times $ is of codimension 1 in $ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$ by Dirichlet’s unit theorem. Let ${\mathbb {C}} {\mathcal {O}}_{E_1}^\times $ be this span. Let

$$ \begin{align*}\overline{C^\infty}({\mathrm{Sh}}({\mathbb{V}})_{K,E_\infty})=C^\infty({\mathrm{Sh}}({\mathbb{V}})_{K,E_\infty})/{\mathbb{C}} {\mathcal{O}}_{E_1}^\times.\end{align*} $$

Equip $\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $ with the quotient of the $L^\infty $ -topology. Define an embedding

(4.13) $$ \begin{align} {\mathbb{C}}{\ \simeq \ }{\mathrm{LC}}({\mathrm{Sh}}({\mathbb{V}})_{K,E_\infty})/ {\mathbb{C}} {\mathcal{O}}_{E_1}^\times \subset \overline{C^\infty}({\mathrm{Sh}}({\mathbb{V}})_{K,E_\infty}) \end{align} $$

by mapping $a\in {\mathbb {C}}$ to the constant function a on ${\mathrm {Sh}}({\mathbb {V}})_{K,E_v}$ for some $v\in \infty $ (rather than ${\mathrm {Sh}}({\mathbb {V}})_{K,E_{1,w}}$ for some infinite place w of $E_1$ ). Below, by a complex number in $ \overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $ , we understand it as the image by (4.13).

Theorem 4.2.10. Let $E^{\prime }_{t,{\mathrm {f}}}(0,g,\phi )$ be as in (3.29). For $g\in G({\mathbb {A}})$ , the generating series of $\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $ -valued functions on $G({\mathbb {A}})$

$$ \begin{align*} {\mathcal{D}}(g)&:= \sum_{t\in F_{>0}}\left( E^{\prime}_{t,{\mathrm{f}}}(0,g,\phi) +E_t(0,g,\phi ) \log {\mathrm{Nm}}_{F/{\mathbb{Q}}}t\right) + \left( {\mathcal{G}}^{{\overline{\mathcal{L}}}}(g,\phi)- {\mathcal{G}}^{\mathrm{Kud}}(g,\phi) \right) \\& \quad - \omega(g)\phi (0 ) \left( -\log \delta_\infty(g_\infty)+ [F:{\mathbb{Q}}]\left( \log \pi-(\log\Gamma)'(n+1)\right)\right) \end{align*} $$

pointwise converges to an element in $ {\mathcal {A}}\left ( G,{\mathfrak {w}}, \overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })\right ) .$

Proof. We follow [Reference Kudla, Rapoport and YangKRY06] and [Reference Mihatsch and ZhangMZ21]. Let $C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ $ be the $L^2$ -orthogonal complement of ${\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$ in $C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })$ , endowed with $L^\infty $ -topology. Then $\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $ is the topological direct sum of $C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ $ and $ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) /{\mathbb {C}} {\mathcal {O}}_{E_1}^\times $ .

Let the generating series ${\mathcal {D}}^\circ $ be the projection of ${\mathcal {D}}$ to $C^\infty ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ $ . By [Reference Mihatsch and ZhangMZ21, Lemma 3.8]Footnote ² and the same argument in the proof [Reference Kudla, Rapoport and YangKRY06, Theorem 4.4.4],Footnote ³ for every $g\in G({\mathbb {A}})$ , ${\mathcal {D}}^\circ (g)$ converges in $L^\infty \left ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right ) $ . Particularly, ${\mathcal {D}}^\circ (g)$ converges in $L^2\left ({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right )$ since ${\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$ is compact. So we can consider ${\mathcal {D}}^\circ $ (identified as its limit) as a function on $G({\mathbb {A}}) \times {\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }$ . Then by the argument in the proof of [Reference Mihatsch and ZhangMZ21, Theorem 3.9], ${\mathcal {D}}^\circ \in C^\infty \left ( G({\mathbb {A}}) \times {\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right )$ . Also note that for every $g\in G({\mathbb {A}})$ , ${\mathcal {D}}^\circ (g)\in C^\infty \left ( {\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right )^\circ $ . Thus, ${\mathcal {D}}^\circ $ is a smooth $C^\infty \left ( {\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }\right )^\circ $ -valued function on $G({\mathbb {A}})$ . See [Reference TrévesTré67, Theorem 40.1 and Corollary]. Then the argument in the proof of either [Reference Kudla, Rapoport and YangKRY06, Theorem 4.4.4] or [Reference Mihatsch and ZhangMZ21, Theorem 3.9] shows that ${\mathcal {D}}^\circ \in {\mathcal {A}}\left ( G,{\mathfrak {w}}, \overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty })^\circ \right ) .$

Consider the projection from $\overline {C^\infty }({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) $ to $ {\mathrm {LC}}({\mathrm {Sh}}({\mathbb {V}})_{K,E_\infty }) /{\mathbb {C}} {\mathcal {O}}_{E_1}^\times {\ \simeq \ } {\mathbb {C}}$ – that is,

$$ \begin{align*}f\mapsto \sum_{v\in \infty}\frac{1}{\deg (\overline{L}_{E_v})}\int_{{\mathrm{Sh}}({\mathbb{V}})_{K,E_v} } f\cdot c_1(\overline{L}_{E_v})^n. \end{align*} $$

By Theorem 4.2.7 and that $E_0(0,g,\phi )= \omega ( g)\phi (0)$ (see Lemma 3.3.9 (1)), the projection of ${\mathcal {D}}$ (defined on each of its terms) pointwise converges to an element in $ {\mathcal {A}}\left ( G,{\mathfrak {w}} \right ).$ Since ${\mathcal {D}}$ is the sum of this projection and ${\mathcal {D}}^\circ $ , the theorem follows.

4.3.1 Preliminaries

A (regular) integral model of ${\mathrm {Sh}}({\mathbb {V}})_{K}$ over an integral domain R with fraction field E is a (regular) Deligne-Mumford stack proper flat over $\operatorname {\mathrm {Spec}} R$ with a fixed isomorphism of its generic fiber to $ {\mathrm {Sh}}({\mathbb {V}})_{K }.$ An isomorphism between integral models is an isomorphism over $\operatorname {\mathrm {Spec}} R$ that respects the fixed isomorphisms to $ {\mathrm {Sh}}({\mathbb {V}})_{K }.$

Let ${\mathcal {X}}_K$ be a regular integral model of ${\mathrm {Sh}}({\mathbb {V}})_{K}$ over $\operatorname {\mathrm {Spec}} {\mathcal {O}}_E$ . Let $\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}({\mathcal {X}}_K) $ be the Chow group of arithmetic divisors with ${\mathbb {C}}$ -coefficients. See Definition A.3.4. In particular, we have an isomorphism

$$ \begin{align*}\deg: \widehat {\mathrm{Ch}}^1_{{\mathbb{C}}}\left( \operatorname{\mathrm{Spec}} {\mathcal{O}}_{E} \right){\ \simeq \ } {\mathbb{C}}\end{align*} $$

by taking degrees (see Remark A.3.5), and an arithmetic intersection pairing

$$ \begin{align*}\widehat{\mathrm{Ch}}^1_{\mathbb{C}}({\mathcal{X}}_K)\times Z_1({\mathcal{X}}_K)_{\mathbb{C}} \to {\mathbb{C}},\ (\widehat z,Y) \mapsto \widehat z\cdot Y. \end{align*} $$

See Appendix A.4. Here, we recall that $Z_1({\mathcal {X}}_K)$ is the group of 1-cycles on ${\mathcal {X}}_K$ .

Let $ {\mathcal {L}}= {\mathcal {L}}_K$ be an extension of $L=L_K $ to ${\mathcal {X}}_K$ , which we allow to be a line bundle. Let $ \overline {\mathcal {L}} $ be ${\mathcal {L}}$ equipped with a hermitian metric. Let $c_1(\overline {\mathcal {L}}^\vee )\in \widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K) $ be the first arithmetic Chern class of the dual of $\overline {\mathcal {L}}$ . See Example A.3.7.

4.3.2 Kudla’s problem

We consider the following modularity problem of Kudla [Reference KudlaKud02, Reference KudlaKud03, Reference KudlaKud04]: find an arithmetic divisor $\widehat {\mathcal {Z}}(x)$ on ${\mathcal {X}}$ extending $Z(x)$ , explicitly and canonically, such that

$$ \begin{align*} \omega(g)\phi(0)c_1(\overline {\mathcal{L}}^\vee)+\sum_{x\in K\backslash {\mathbb{V}}^{ \infty}_{>0}} \omega(g)\phi( x_\infty x) \widehat {\mathcal{Z}}(x) ,\\[-20pt] \end{align*} $$

where $ x_\infty \in {\mathbb {V}}_\infty $ such that $q(x_\infty )={q(x)}\in F_{>0}$ lies in $ {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}}) \otimes \widehat {\mathrm {Ch}}^{1}_{{\mathbb {C}}}({\mathcal {X}}_K) $ . The existence of such $\widehat {\mathcal {Z}}(x)$ is obvious, by choosing a section of the natural surjection $\widehat {\mathrm {Ch}}^1_{{\mathbb {C}}}({\mathcal {X}}_K)\to {\mathrm {Ch}}^1({\mathrm {Sh}}({\mathbb {V}})_K)_{\mathbb {C}}$ . However, it is only defined at the level of divisor classes, and not explicit.

4.3.3 Admissible extensions

We consider the above modularity problem for admissible extensions. In particular, we assume Assumption A.1.1 for ${\mathcal {X}}_K$ . (Also recall that ${\mathcal {X}}_K$ is connected, as ${\mathrm {Sh}}({\mathbb {V}})_K$ is.) Assume that $ {\mathcal {L}}$ is ample. Let $ \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) \subset \widehat {\mathrm {Ch}}^1_{\mathbb {C}}({\mathcal {X}}_K)$ be the subgroup of arithmetic divisors that are admissible with respect to $\overline {\mathcal {L}}$ . See Definition A.3.4. By Lemma A.3.6, the natural map

(4.14) $$ \begin{align} \widehat {\mathrm{Ch}}^{1}_{{\overline{\mathcal{L}}},{\mathbb{C}}}({\mathcal{X}}_K) \to {\mathrm{Ch}}^1({\mathcal{X}}_{K,E })_{{\mathbb{C}}} \end{align} $$

is surjective, and the kernel is the image of the pullback

(4.15) $$ \begin{align} \widehat {\mathrm{Ch}}^1_{\mathbb{C}}\left( \operatorname{\mathrm{Spec}} {\mathcal{O}}_{E} \right){\ \simeq \ } \widehat {\mathrm{Ch}}^1_{\mathbb{C}}\left( \operatorname{\mathrm{Spec}} {\mathcal{O}}_{{\mathcal{X}}_K}({\mathcal{X}}_K)\right)_{\mathbb{C}}\hookrightarrow \widehat{\mathrm{Ch}}^1_{{\overline{\mathcal{L}}},{\mathbb{C}}}({\mathcal{X}}_K). \end{align} $$

In particular, $ \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) $ is finite dimensional.

Let ${\widehat {Z}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K)$ the group of admissible arithmetic divisors. See Definition A.3.1. For a divisor Z on ${\mathrm {Sh}}({\mathbb {V}})_{K}{\ \simeq \ } {\mathcal {X}}_{K,E}$ , let

(4.16) $$ \begin{align} Z^{\overline{\mathcal{L}}}\in {\widehat{Z}}^1_{{\overline{\mathcal{L}}},{\mathbb{C}}}({\mathcal{X}}_K) \end{align} $$

be the normalized admissible extension of Z with respect to $\overline {\mathcal {L}}$ . See Definition A.3.3. Let $[ Z(x)^{\overline {\mathcal {L}}}] $ be its class in $\widehat {\mathrm {Ch}}^1_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K)$ . Then a preiamge of $[ Z(x)] $ via (4.14) is of the form $[ Z(x)^{\overline {\mathcal {L}}}] +e(x)$ for some $e(x)\in {\mathbb {C}}$ . Note that $c_1(\overline {\mathcal {L}}^\vee )\in \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) $ . See Example A.3.7.

Problem 4.3.3. Find $a\in {\mathbb {C}}$ and $e=\{e(x) \in {\mathbb {C}}\}_{x\in K\backslash {\mathbb {V}}^{ \infty }_{>0}}$ explicitly such that for every $\phi \in \overline {\mathcal {S}}\left ( {\mathbb {V}} \right )^{K}$ , the generating series

(4.17) $$ \begin{align}z(\omega(g)\phi)_{e,a} ^{\overline{\mathcal{L}}}:=\omega(g)\phi(0)\left( c_1(\overline {\mathcal{L}}^\vee)+a\right)+\sum_{x\in K\backslash {\mathbb{V}}^{ \infty}_{>0}}\omega(g)\phi( x_\infty x) \left([ Z(x)^{\overline{\mathcal{L}}}]+e(x)\right), \end{align} $$

where $ x_\infty \in {\mathbb {V}}_\infty $ such that $q(x_\infty )={q(x)}\in F_{>0}$ , lies in $ {\mathcal {A}}_{{\mathrm {hol}}}(G,{\mathfrak {w}}) \otimes \widehat {\mathrm {Ch}}^{1}_{{\overline {\mathcal {L}}},{\mathbb {C}}}({\mathcal {X}}_K) $ . Moreover, $e(x)$ should to be naturally decomposed into a sum of ‘local components’ such that the v-component should be 0 at all but finitely many places and $\infty $ -component should be independent of the choice of the regular integral model.