\begin{document}

\begin{abstract}
In this article, we prove an explicit Waldspurger formula for the toric Hilbert modular forms. As an application, we construct a class of anticyclotomic \text{$p$-adic } Rankin-Selberg $L$-functions for Hilbert modular forms, generalizing the construction of Bertolini, Darmon, and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient condition when the Iwasawa $\mu$-invariant of this \text{$p$-adic } $L$-function vanishes and prove a result on the non-vanishing modulo $p$ of central Rankin-Selberg $L$-values with anticyclotomic twists. \varepsilonsilonnd{abstract}
\title{

Special values of anticyclotomic Rankin-Selberg $L$-functions}
\tableofcontents

\def\mathbf ZZbox{\mathbf Z_{(\Box)\,}}
\def{\mathcal A}box{{\mathcal A}^{(\Box)}_{K,\mathbf c}}
\def{\mathcal A}boxn{{\mathcal A}^{(\Box)}_{K,\mathbf c,n}}
\def\mathbf Zhatbox{\widehat{\mathbf Z}^{(\Box)}}
\def\tau_{\cK/\cF}{\tau_{E/F}}
\defK{K}
\defSh{Sh}
\defK^n{K^n}
\def{\ell}p{j}
\def{\ell}pp{{algebraically closed ute{e}t}a^{(p)}}
\def{\ell}sgN{K_1^n}
\def\boldsymbol{\omega}{\boldsymbol{\omegaega}}
\defk{k}
\def\vartheta{\vartheta}
\def\mathbb Fv{F}
\defE{E}
\defO_p{\mathcal O_{F_p}}

\def\chi{{\mathbb I}i}
\def\ol{t}{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{t}}
\def\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}
\def\vp{\varphi}
\def\A_{\cF,f}{\mathbf A_{F,f}}
\def\varsigma{\varsigma}
\def\varsigmav{\varsigma_v}
\def\vartheta{\vartheta}
\def\cO_\cK{\mathcal O_E}
\def\boldsymbol{J}{\boldsymbol{J}}
\def\cO_F{\mathcal O_F}
\def\cO_\cKv{\mathcal O_E}
\def\cO_{\cF}{O}
\def\cR{{\mathcal R}}
\def\psi{\psi}
\defE{E}
\defF{F}
\defO_p{O_p}
\def\cU_p{\mathcal U_p}
\def{\ell}oca{\mathrm{l.a.}}
\def\cW{R}
\def\boldsymbol{P}{\boldsymbol{P}}
\defksp{\mathfrak X^{\mathrm{crit}}_{p}}
\def\Gamma^-{\Gamma^-}
\def\mathbf AF{\mathbf A}
\def\mathbf AFf{\mathbf AF_f}
\def\mathbf CMP{\vartheta}
\def\cO_\cKbasis{\boldsymbol{\theta}}
\newcommand{\ell}ocalW[1]{W_{#1}}
\newcommand{\ell}ocalK[1]{\bfa_{#1}}
\newcommand{\ell}ocalP[2]{\boldsymbol{P}(#1,#2)}
\def\mathrm N{\mathrm N}
\def{\varSigma}bar{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{\varSigma}}}
\renewcommand\MX[4]{\begin{pmatrix}#1&#2\\
#3&#4\varepsilonsilonnd{pmatrix}}
\renewcommand\DII[2]{\begin{pmatrix}#1&\\
&#2\varepsilonsilonnd{pmatrix}}
\def\mathbf Cinert{\mathfrak N^-}
\def\mathbf Csplit{{\mathfrak F}}
\def\mathbf Cram{D_{E/F}}
\defk{k}
\def\frakc{{\mathfrak c}}
\defdt{dt}
\def\frakr{\mathfrak r}
\def\mathbf CMring{{{\mathcal W}_p}}
\def\sharp{Sharp}
\section*{Introduction}
The purpose of this article is to (i) construct a class of anticyclotomic Rankin-Selberg \text{$p$-adic } $L$-functions and study the vanishing/non-vanishing of the associated Iwasawa $\mu$-invariant, (ii) prove a result on the non-vanishing modulo $p$ of central Rakin-Selberg $L$-values with anticyclotomic twists.
Let $F$ be a totally real algebraic number field of degree $d$ over $\mathbf Q$ and $E$ be a totally imaginary quadratic extension of $F$. Denote by $z\mapsto \overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{z}$ the nontrivial element in $\Gal(E/F)$. Let $\pi$ be an irreducible cuspidal automorphic representation of $\GL_2(\mathbf A_{F})$ with unitary central character $\omega$. Let $\pi_E$ be a lifting of $\pi$ to $E$ constructed in \cite[Thm.\,20.6]{Jacquet:GLtwoPartII}. Then $\pi_E$ is an irreducible automorphic representation of $\GL_2(\mathbf A_E)$, which is cuspidal if $\pi$ is not obtained from the automorphic induction from $E/F$. Let ${\ell}am:\mathbf AK^{\times}/E^{\times}\to\mathbf C^{\times}$ be a unitary Hecke character of $E^{\times}$ such that
\begin{equation}{\ell}abel{E:self-dual}{\ell}am|_{\mathbf A_{F}^{\times}}=\omega^{-1}.\varepsilonsilonnd{equation}
The automorphic representation $\pi_E\otimes{\ell}am$ is therefore conjugate self-dual, i.e.  $\pi_E^\vee\otimes{\ell}am^{-1}=\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\pi_E\otimes{\ell}am}$. For each place $v$ of $F$, we can associate a local $L$-function $L(s,\pi_{E_v}\otimes{\ell}am_v)$ and a local epsilon factor $\varepsilonsilonp(s,\pi_{E_v}\otimes{\ell}am_v,\psi_v)$ (which depends on a choice of non-trivial character $\psi_v:F_v\to\mathbf C^{\times}$) to the local constituent $\pi_{E_v}\otimes{\ell}am_v$ of $\pi_{E}\otimes{\ell}am$ (\cite[Thm.\,2.18 (iv)]{Jacquet_Langlands:GLtwo}). Let $L(s,\pi_E\otimes{\ell}am)$ be the global $L$-function obtained by the meromorphic continuation of the Euler product of local $L$-functions at all \varepsilonsilonmph{finite} places. In this paper, we study the \text{$p$-adic } variation of the central value $L(\mathbf 1half,\pi_{E}\otimes{\ell}am)$ with anticyclotomic twists under certain hypotheses.

To introduce our hypotheses precisely, we need some notation.  Fix a CM-type ${\varSigma}$ of $E$. Namely, ${\varSigma}$ is a subset of $\Hom(E,\mathbf C)$ such that
\[{\varSigma}\bigsqcup {\varSigma}bar=\Hom(E,\mathbf C)\,;\,{\varSigma}{\mathcal a}p{\varSigma}bar=\varepsilonsilonmptyset.\]

Then ${\varSigma}$ induces an identification $E\otimes_\mathbf Q\mathbf R\simeq\mathbf C^{\varSigma}$. We shall identify ${\varSigma}$ with the set of archimedean places of $F$ via the restriction.  
For each $k=\sum_{\sigma\in{\varSigma}}k_\sigma\sigma\in\mathbf Z[{\varSigma}]$, we write $\Gamma_{\varSigma}(k)=\indent od_{\sigma\in{\varSigma}}\Gamma(k_\sigma)$, and if $x=(x_\sigma)\in (A^{\times})^{\varSigma}$ for an algebra $A$, we let $x^{k}:=\indent od_{\sigma\in{\varSigma}}x_\sigma^{k_\sigma}$. A Hecke character ${\mathbb I}i$ of $E^{\times}$ is of infinity type $(k_1,k_2)$ for $k_1,k_2\in 2^{-1}\mathbf Z[{\varSigma}]$ such that $k_1-k_2\in\mathbf Z[{\varSigma}]$ if \[{\mathbb I}i_\infty(z)=z^{k_1-k_2}(z\bar{z})^{k_2}\text{ for all }z\in(E\otimes_\mathbf Q\mathbf R)^{\times}\simeq(\mathbf C^{\times})^{\varSigma}.\] For each ideal ${\mathfrak a}$ of $F$ (resp. ideal $\mathfrak A$ of $E$),  we have a unique factorization ${\mathfrak a}={\mathfrak a}^+{\mathfrak a}^-$ (resp. $\mathfrak A=\mathfrak A^+\mathfrak A^-$), where ${\mathfrak a}^+$ (resp. $\mathfrak A^+$) is only divisible by primes split in $E$ and ${\mathfrak a}^-$ (resp. $\mathfrak A^-$) is divisible by primes inert or ramified in $E$. Let $\mathfrak n=\mathfrak n^+\mathfrak n^-$ be the conductor of $\pi$. We define the normalized local root number attached to $\pi_{E_v}$ and ${\ell}am_v$ for each place $v$ by
\[\varepsilonsilonp^*(\pi_{E_v},{\ell}am_v):=\varepsilonsilonp(\mathbf 1half,\pi_{E_v}\otimes{\ell}am_v,\psi_v)\cdot \omega_v(-1).\]

We remark that the value $\varepsilonsilonp(\mathbf 1half,\pi_{E_v}\otimes{\ell}am_v,\psi_v)$ does not depend on the choice of $\psi_v$.

We assume that $\pi$ has infinity type $k=\sum_\sigma k_\sigma\sigma\in\mathbf Z_{>0}[{\varSigma}]$ and ${\ell}am$ has infinity type $(\frac{k}{2},-\frac{k}{2})$. In other words, $\pi_\sigma$ is a discrete series or limit of discrete series of weight $k_\sigma$ with unitary central character at every archimedean place $\sigma$. In particular, this implies $\stt{k_\sigma}_{\sigma\in{\varSigma}}$ have the same parity and the local root number $\varepsilonsilonp^*(\pi_{E_v},{\ell}am_v)=+1$ at every archimedean place. We further assume the following local root number hypothesis for $(\pi,{\ell}am)$:
\begin{hypA}{\ell}abel{H:HypA}The local root number $\varepsilonsilonp^*(\pi_{E_v},{\ell}am_v)=+1$ for each $v\mid\mathfrak n^-$.
\varepsilonsilonnd{hypA}
Let $p$ be an odd rational prime. Fix an embedding $\iota_\infty:\mathbf Qbar\hookrightarrow\mathbf C$ and isomorphism $\iota:\mathbf C\stackrel{\sim}{\to}\mathbf Cp$ once and for all. Each ${\times}i\in\mathbf Z[{\varSigma}\bigsqcup{\varSigma}bar]$ will be regarded as an algebraic character ${\times}i:(E\otimes\mathbf Qp)^{\times}\to\mathbf Cp^{\times}$ via $\iota$. Throughout this article, we make the following assumption \begin{align}
{\ell}abel{ord}\tag{ord}&\text{${\varSigma}$ is $p$-ordinary}.
\varepsilonsilonnd{align} Let ${\varSigma}_p$ be the set of \text{$p$-adic } places of $E$ induced by ${\varSigma}$. Then the ordinary assumption \varepsilonsilonqref{ord} means that ${\varSigma}_p$ and its complex conjugation ${\varSigma}bar_p$ gives a full partition of the set of \text{$p$-adic } places of $E$. Let $E^-_{p^\infty}$ be the maximal anticyclotomic $\mathbf Zp^{[F:\mathbf Q]}$-extension of $E$. Let $\Gamma^-=\Gal(E^-_{p^\infty}/E)$ and let $\Lambda=\mathbf Zbarp\powerseries{\Gamma^-}$ be the Iwasawa algebra of $[F:\mathbf Q]$-variables. If $L$ is a number field, we write $G_L=\Gal(\mathbf Qbar/L)$ for the absolute Galois group. Denote by $\rec_E:\mathbf AK^{\times}\to G_E^{ab}$ the geometrically normalized reciprocity law. To each locally algebraic \text{$p$-adic } character $\widehat\phi:\Gamma^-\to\mathbf Cp^{\times}$ of weight $(m,-m)$, we can associate a Hecke character $\phi:\mathbf AK^{\times}/E^{\times}\to\mathbf C^{\times}$ of infinity type $(m,-m)$ defined dy
\[\phi(a):=\iota^{-1}(\widehat\phi(\rec_E(a))a_p^{-m}\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{a}_p^{m})a_\infty^{m}\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{a}_\infty^{-m},\]
where $a_p\in (E\otimes_\mathbf Q\mathbf Qp)^{\times}$ and $a_\infty\in (E\otimes_\mathbf Q\mathbf R)^{\times}$ are the $p$-component and $\infty$-component of $a$. We say $\widehat\phi$ is the \varepsilonsilonmph{\text{$p$-adic } avatar} of $\phi$. Denote by $ksp$ the set of \varepsilonsilonmph{critical specializations}, consisting of locally algebraic \text{$p$-adic } characters on $\Gamma^-$ of weight $(m,-m)$ with $m\in\mathbf Z_{\geq 0}[{\varSigma}]$ (See \subsecref{SS:padicL}).

Our first result is the construction of the anticyclotomic primitive \text{$p$-adic } $L$-function attached to $\pi_{E}\otimes{\ell}am$. We need more notation. Let $\mathcal D_F$ be the different of $F$ and $\mathcal D_{E/F}$ be the relative different of $E/F$. Let $\mathfrak N$ be the prime-to-$p$ conductor of $\pi_{E}\otimes{\ell}am$. We have a unique decomposition $\mathfrak N=\mathfrak N^+\mathfrak N^-$ and fix a decomposition $\mathfrak N^+=\mathbf Csplit\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathbf Csplit}$ with $(\mathbf Csplit,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathbf Csplit})=1$ such that $\mathfrak N^-$ is only divisible by prime inert or ramified in $E/F$ and $\mathbf Csplit$ is only divisible by primes split in $E/F$ (Such a decomposition is possible because $\pi_E\otimes{\ell}am$ is conjugate self-dual). We choose $\delta\inE$ such that
\begin{itemize}
\item $\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\delta}=-\delta$,
\item  $\Im\sigma(\delta)>0$ for all $\sigma\in{\varSigma}$,
\item The polarization ideal ${\mathfrak c}(\mathcal O_E):=\mathcal D_F^{-1}(2\delta \mathcal D_{E/F})$ is prime to $p\mathfrak N\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathfrak N}$.
\varepsilonsilonnd{itemize}
Let $(\boldsymbol{\omega}ega_\infty,\boldsymbol{\omega}ega_p)$ be the complex and \text{$p$-adic } periods attached to $(E,{\varSigma})$ defined in \cite[(4.4a), (4.4b)]{HidaTilouine:KatzPadicL_ASENS}. For each Hecke character ${\mathbb I}i$ of $E^{\times}$, we define the \text{$p$-adic } multiplier $E_{{\varSigma}_p}(\pi,{\mathbb I}i)$ by
\begin{equation}{\ell}abel{E:coates_modified}E_{{\varSigma}_p}(\pi,{\mathbb I}i):=
\indent od_{\mathfrak P\in{\varSigma}_p,\,{\mathfrak p}=\mathfrak P\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathfrak P}}\varepsilonsilonp(\mathbf 1half,\pi_{\mathfrak p}\otimes{\mathbb I}i_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathfrak P}},\psi_{\mathfrak p})L(\mathbf 1half,\pi_{\mathfrak p}\otimes{\mathbb I}i_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathfrak P}})^{-2}\omega_{\mathfrak p}^{-1}{\mathbb I}i_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathfrak P}}^{-2}(-2\delta).\varepsilonsilonnd{equation}
The shape of $E_{{\varSigma}_p}(\pi,{\mathbb I}i)$ has been suggested by Coates \cite{Coates:motivic-padic-L-function}.
\begin{thmA}{\ell}abel{T:A.W}In addition to \varepsilonsilonqref{ord} and \hypref{H:HypA}, we further assume that
\begin{equation}cd{sf}\mathfrak n^-\text{ is square-free}.\varepsilonsilonnd{equation}cd
Then there exists an element ${\mathcal P}_{\varSigma}(\pi,{\ell}am)\in \Lambda$ such that for every $\widehat\phi\inksp$ of weight $(m,-m)$, we have
\[\frac{\widehat\phi({\mathcal P}_{\varSigma}(\pi,{\ell}am)^2)}{\boldsymbol{\omega}ega_p^{2(k+2m)}}=[\mathcal O_E^{\times}:\mathcal O_F^{\times}]^2\cdot\frac{\Gamma_{\varSigma}(k+m)\Gamma_{\varSigma}(m+1)}{(\Im\delta)^{k+2m}(4\pi)^{k+2m+1\cdot{\varSigma}}}\cdot E_{{\varSigma}_p}(\pi,{\ell}am\phi)\cdot \frac{L(\mathbf 1half,\pi_E\otimes{\ell}am\phi)}{\boldsymbol{\omega}ega_E^{2(k+2m)}}\cdot \phi(\mathbf Csplit)C(\pi,{\ell}am),\]
where $\boldsymbol{\omega}ega_E=(2\pi i)^{-1}\boldsymbol{\omega}ega_\infty$ and $C(\pi,{\ell}am)\in\mathbf Zbar_{(p)}^{\times}$ is an explicit \text{$p$-adic } unit independent of $\phi$, consisting of a product of local epsilon factors outside $p$.
\varepsilonsilonnd{thmA}
In the above expression, $\Im\delta=(\Im\sigma(\delta))_{\sigma\in{\varSigma}}$ and $4\pi$ is considered to be the diagonal element $(4\pi)_{\sigma\in{\varSigma}}$ in $(\mathbf C^{\times})^{\varSigma}$.

In virtue of the above specialization formula, ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ is the \text{$p$-adic } $L$-function that interpolates the square root of central $L$-values. We shall call ${\mathcal L}_{\varSigma}(\pi,{\ell}am):={\mathcal P}_{\varSigma}(\pi,{\ell}am)^2$ the anticyclotomic \text{$p$-adic } $L$-function for $\pi_E\otimes{\ell}am$ with respect to the $p$-ordinary CM type ${\varSigma}$. When $F=\mathbf Q$, $\pi$ is unramified at $p$ and $\mathfrak n^-$ is only divisible by primes ramified in $E$, ${\mathcal L}_{\varSigma}(\pi,{\ell}am)$ is constructed in \cite{BDP}, and a non-primitive version of this \text{$p$-adic } $L$-function is also considered in \cite{Miljan:1} under different hypotheses. We remark that when $\pi$ is obtained from the automorphic induction of a Hecke character of $E^{\times}$, one can show that, up to an explicit unit in $\Lambda$, ${\mathcal L}_{{\varSigma}}(\pi,{\ell}am)$ is a product of two \text{$p$-adic } Hecke $L$-functions for CM fields constructed by Katz \cite{Katz:p_adic_L-function_CM_fields} and Hida-Tilouine \cite{HidaTilouine:KatzPadicL_ASENS} by comparing the interpolation formulas on both sides.

Our second theorem is to prove the vanishing of the Iwasawa $\mu$-invariant $\mu^-_{\pi,{\ell}am,{\varSigma}}$ of ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ under certain hypothesis. This in particular implies that the \text{$p$-adic } $L$-function ${\mathcal L}_{\varSigma}(\pi,{\ell}am)$ is non-trivial. Recall that the $\mu$-invariant $\mu^-_{\pi,{\ell}am,{\varSigma}}$ is defined by
 \[\mu^-_{\pi,{\ell}am,{\varSigma}}=\inf\stt{r\in\mathbf Q_{\geq 0}\mid p^{-r}{\mathcal P}_{\varSigma}(\pi,{\ell}am)\not\equiv 0\pmod{\mathfrak m_p\Lambda}},\]where $\mathfrak m_p$ is the maximal ideal of $\mathbf Zbarp$. Let ${\mathfrak c}_{\ell}am$ be the conductor of ${\ell}am$. For each $v\mid{\mathfrak c}_{\ell}am^-$, let $\Delta_{{\ell}am,v}$ be the finite group ${\ell}am(\mathcal O_{E_v}^{\times})$.

\begin{thmA}{\ell}abel{T:B.W}With the assumptions in \thmref{T:A.W}, suppose further that
\begin{mylist}\item $p$ is unramified in $F$,
\item the residual Galois representation \[\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\rho}_p(\pi_E):=\rho_p(\pi)|_{G_E}\pmod{\mathfrak m_p}\text{ is absolutely irreducible},\]
\item $p\nmid \indent od_{v|{\mathfrak c}_{\ell}am^-}\sharp(\Delta_{{\ell}am,v})$.
\varepsilonsilonnd{mylist}Then $\mu^-_{\pi,{\ell}am,{\varSigma}}=0$.

\varepsilonsilonnd{thmA}
The global assumption on the irreducibility of residual Galois representation assures that the new form associated to $\pi$ is not congruent to theta series from $E$ or Eisenstein series, and the local assumption (3) is equivalent to saying that the local residual character ${\ell}am_v\pmod{\mathfrak m_p}$ is ramified for all $v|{\mathfrak c}_{\ell}am^-$. In this situation, our \thmref{T:B.W} roughly suggests that the $\mu$-invariant is essentially contributed by the congruences among primitive but residually non-primitive \text{$p$-adic } $L$-functions. In particular, suppose that $(\mathfrak n,\mathcal D_{E/F})=1$ and $\rho_p(\pi)$ is residually irreducible. Then we always have $\mu^-_{\pi,{\ell}am}=0$ whenever ${\ell}am$ has split conductor over $F$ (i.e.  ${\mathfrak c}_{\ell}am^-=(1)$).
In addition, \thmref{T:B.W} shares the same flavor with Vatsal's $\mu$-invariant formula. In \cite{Vatsal:nonvanishing}, Vatsal obatins the precise $\mu$-invariant formula of a different class of anticyclotomic \text{$p$-adic } Rankin-Selberg $L$-functions associated to elliptic new forms of weight two twisted by \varepsilonsilonmph{finite order} anticyclotomic characters of $p$-power conductors (so the local root number at the archimedean place is $-1$).  In virtue of his formula, the $\mu$-invariant can be positive when we have either Eisenstein congruence or the congruence between forms with opposite signs in the functional equations. Note that the latter congruence does not happen under the assumption (3).

The \text{$p$-adic } $L$-function ${\mathcal L}_{\varSigma}(\pi,{\ell}am)$ is expected to encode the arithmetic information on certain Selmer groups through the main conjecture \`{a} la R. Greenberg \cite{Greenberg:Iwasawa_theory_and_motives}. To introduce the $\Lambda$-adic Selmer groups connected with ${\mathcal L}_{\varSigma}(\pi,{\ell}am)$, we recall that thanks to the works of Deligne, Carayol, Blasius-Rogawski and Taylor et.al (\cite{Blasius_Rogawski:Hilber_modular_forms}, \cite{Taylor:Galois_rep_HMF}, \cite{Jarvis:wtone}), there exists a finite extension $L_\pi$ of $\mathbf Qp$ and a continuous \text{$p$-adic } Galois representation $\rho_p(\pi):G_F\to\GL_2(\mathcal O_{L_\pi})$ such that $\rho_p(\pi)$ is unramified outside $p\mathfrak n$, and for each finite place $v\nmid p\mathfrak n$, \[\iota^{-1}(L(s,\rho_p(\pi)|_{W_{F_v}}))=L(s+\frac{1-k_{mx}}{2},\pi_v^\vee)\quad(k_{mx}=\max_{\sigma\in{\varSigma}}k_\sigma),\]
where $W_{F_v}$ is the Weil group of $F_v$.  Let $\varepsilonsilon_\Lambda:G_E\to\Gamma\hookrightarrow \Lambda^{\times},\,g\mapsto g|_{E_{p^\infty}^-}$ be the universal $\Lambda$-adic Galois character. We consider the rank two $\Lambda$-adic Galois representation:
\[\rho_\Lambda:=\rho_p(\pi)|_{G_E}\otimes{\ell}am\varepsilonsilon_\Lambda:G_E\to\GL_2(\Lambda).\]
and define the local condition for each $w\mid p$ by
\[F^+_w\rho_\Lambda=\begin{cases}
\rho_\Lambda&\text{ if } w\in{\varSigma}_p,\\
\stt{0}&\text{ if } w\in{\varSigma}bar_p.
\varepsilonsilonnd{cases}\]
The triple $(\rho_\Lambda,\stt{F^+_w\rho_\Lambda}_{w|p},ksp)$ satisfies the Panchishkin condition in \cite[\S 4, p.217]{Greenberg:Iwasawa_theory_and_motives}.
Let $\Lambda^*$ be the Pontryagin  dual of $\Lambda$. The associated Greenberg's Selmer group is defined by
\[Sel_{\varSigma}(\pi,{\ell}am):=\ker\stt{H^1(E,\rho_\Lambda\otimes_\Lambda\Lambda^*)\to\indent od_{w\not\in{\varSigma}_p}H^1(I_w,\rho_\Lambda\otimes_\Lambda\Lambda^*)},\]
where $w$ runs over places of $E$ outside ${\varSigma}_p$ and $I_w$ is the inertia group of $w$ in $G_E$. It is known that the Pontryagin  dual $Sel_{\varSigma}(\pi,{\ell}am)^*$ is a finitely generated $\Lambda$-module. Denote by $\mathbf Ch_\LambdaSel_{\varSigma}(\pi,{\ell}am)^*$ the characteristic ideal of $Sel_{\varSigma}(\pi,{\ell}am)^*$. We formulate the anticyclotomic main conjecture for $\pi_E\otimes{\ell}am$ (under the hypotheses in \thmref{T:A.W}).
\begin{conj*}We have the following equality between ideals in $\Lambda$
\[\mathbf Ch_\LambdaSel_{\varSigma}(\pi,{\ell}am)^*=({\mathcal L}_{\varSigma}(\pi,{\ell}am)).\]
\varepsilonsilonnd{conj*}

Let $\varepsilonsilonlll\not =p$ be a rational prime. We next consider the problem of the non-vanishing modulo $p$ of $L$-values twisted by characters of $\varepsilonsilonlll$-power conductor. This problem has been studied extensively in the literature in various settings (\mbox{{\it cf.} } \cite{Washington:nonvanishing}, \cite{Vatsal:nonvanishing}, \cite{Hida:nonvanishingmodp}, \cite{Finis:nonvanishingell}, \cite{Hsieh:Hecke_CM}) and has many arithmetic applications in Iwasawa theory. To state our result along this direction, we introduce some notation. Let $\mathfrak l$ be a prime of $F$ above $\varepsilonsilonlll$ and let $E_{\mathfrak l^\infty}^-$ be the anticyclotomic pro-$\varepsilonsilonlll$ extension in the ray class field over $E$ of conductor $\mathfrak l^\infty$. Let $\Gamma_\mathfrak l^-:=\Gal(E_{\mathfrak l^\infty}^-/E)$ and let $\mathfrak X_\mathfrak l^0$ be the set consisting of finite order characters $\phi:\Gamma_\mathfrak l^-\to\mu_{\varepsilonsilonlll^\infty}$. Let ${\mathbb I}i$ be a Hecke character of infinity type $(\frac{k}{2}+m,-\frac{k}{2}-m)$ and of conductor ${\mathfrak c}_{\mathbb I}i$. We obtain the following theorem.
\begin{thmA}{\ell}abel{T:C.W}
With the same assumptions in \thmref{T:A.W}, we assume that
\begin{mylist}
\item $(p\mathfrak l,\mathfrak n{\mathfrak c}_{\mathbb I}i \mathcal D_{E/F})=1$,
\item $(\mathfrak n,\mathcal D_{E/F})=1$ and $\rho_p(\pi)$ is residually irreducible,
\item $p\nmid \indent od_{v|{\mathfrak c}_{\mathbb I}i^-}\sharp(\Delta_{{\mathbb I}i,v})$.

\varepsilonsilonnd{mylist}
Then for almost all characters $\phi\in\mathfrak X_\mathfrak l^0$, we have
\[[\mathcal O_E^{\times}:\mathcal O_F^{\times}]^2\cdot\frac{\Gamma_{\varSigma}(k+m)\Gamma_{\varSigma}(m+1)}{(\Im\delta)^{k+2m}(4\pi)^{k+2m+1\cdot{\varSigma}}}\cdot \frac{L(\mathbf 1half,\pi_E\otimes{\mathbb I}i\phi)}{\boldsymbol{\omega}ega_E^{2(k+2m)}}\not\equiv 0\pmod{\mathfrak m_p}.\]
Here almost all means "except for finitely many $\phi\in\mathfrak X_\mathfrak l^-$" if $\dim_{\mathbf Q_\varepsilonsilonlll} F_\mathfrak l=1$ and "for $\phi$ in a Zariski dense subset of $\mathfrak X_\mathfrak l^0$" if $\dim_{\mathbf Q_\varepsilonsilonlll}F_\mathfrak l>1$ $($\cite[p.737]{Hida:nonvanishingmodp}$)$.
\varepsilonsilonnd{thmA}

In our forthcoming work \cite{Hsieh:Namikawa}, we will apply \thmref{T:C.W} to study the non-vanishing (modulo p) of certain theta lifts constructed in \cite{Murase:FJcoeff_Kudla} and \cite{MuraseNarita:Arakawa}.

The proof of \thmref{T:A.W} is based on an explicit Waldspurger formula, which connects toric period integrals of automorphic forms in $\pi$ and central values of $\pi_E\otimes{\ell}am$ \cite{Waldsupurger:Central_value}. To obtain the optimal $p$-integrality of cental $L$-values, we will consider holomorphic \varepsilonsilonmph{toric} cusp forms and calculate explicitly their period integrals. To be precise, let ${\mathbb I}i$ be a Hecke character of $E^{\times}$ such that ${\mathbb I}i|_{\mathbf AF^{\times}}=\omega^{-1}$ and let $\mathcal T\subset \mathbf AK^{\times}$ be the subgroup consisting of ideles $z=(z_v)\in\indent od_vE_v^{\times}$ with $z_v/\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{z_v}\in\mathcal O_{E_v}^{\times}$ for all primes $v$ split in $E$. Fixing an embedding $\iota:E^{\times}\hookrightarrow \GL_2(F)$, we say an automorphic form $\varphi:\GL_2(F)\backslash\GL_2(\mathbf A_{F})\to\mathbf C$ in $\pi$ is a toric form of character ${\mathbb I}i$ if
\[\varphi(g\iota(t))={\mathbb I}i^{-1}(t)\varphi(g)\text{ for all }t\in\mathcal T.\]
In other words, $\varphi$ belongs to the space $\Hom_\mathcal T(\mathbf C,\pi\otimes{\mathbb I}i)$.

The construction of ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ is outlined as follows.
 \begin{mylist}
\item Construct a toric Hilbert modular form $\varphi_{{\ell}am\phi}$ of character ${\ell}am\phi$ for each $\widehat\phi\inksp$ as above by a careful choice of toric local Whittaker functions in local Whittaker models of $\pi$ (See \defref{D:toric}, {\ell}mref{L:4.W}).
\item Make an explicit calculation of the Fourier expansion of $\varphi_{{\ell}am\phi}$.
\item Via the \text{$p$-adic } interpolation of the Fourier expansion, construct a \text{$p$-adic } family of toric forms $\stt{\varphi_{{\ell}am\phi}}_{\widehat\phi\inksp}$. The \text{$p$-adic } $L$-function ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ is obtained by a weighted sum of the evaluation of this family at a finite set of CM points.
\varepsilonsilonnd{mylist}

The evaluation of $\varphi_{\mathbb I}i$ with ${\mathbb I}i={\ell}am\phi$ at CM points in the step (3) is actually the toric period integral $P_{{\mathbb I}i}(\varphi_{{\mathbb I}i})$ given by
\[P_{\mathbb I}i(\varphi_{\mathbb I}i)=\int_{\mathbf AF^{\times}E^{\times}\backslash\mathbf AK^{\times}}\varphi_{\mathbb I}i(\iota(t)){\mathbb I}i(t)dt.\]
To prove the formula in \thmref{T:A.W}, we have to express the square $P_{\mathbb I}i(\varphi_{\mathbb I}i)^2$ in terms of the central $L$-value $L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)$. This is usually referred to as an explicit Waldspurger formula. Such a formula has been exploited widely in the literature based on either explicit theta lifts (\cite{Murase:WaldI}, \cite{Murase:WaldII}, \cite{XH:central_value}, \cite{Hida:central_value} and \cite{BDP}) or the technique of relative trace formula (\cite{Kimble_W:centrals}). In this paper we adopt a different approach, making use of a formula of Waldspurger which is indeed proved but not stated explicitly in \cite{Waldsupurger:Central_value}. This formula decomposes the square $P_{\mathbb I}i(\varphi)$ of the global period toric integral into a product of local period integrals involving local Whittaker functions of $\varphi$. Explicit computation of these local integrals shows that $P_{\mathbb I}i(\varphi)^2$ is essentially equal to the central value of the $L$-function $L(s,\pi_E\otimes{\ell}am)$. We emphasize that this explicit formula does not depend on the choices of explicit Bruhat-Schwartz  functions in the classical approach of theta lifting but on choices of local Whittaker functions which reflect the arithmetic of modular forms directly via the Fourier expansion. We make a few remarks on our assumptions. The restriction \varepsilonsilonqref{sf} is due to the computational difficulty on the local period integrals and the local Fourier coefficients, and it is expected to be unnecessary. We hope to come back to the removal of this in the future. However, \hypref{H:HypA} is fundamental, the failure of which makes the period integral $P_{\mathbb I}i(\varphi_{\mathbb I}i)$ vanish by a well-known theorem of Saito-Tunnell (\cite{Saito:SaitoTunnell}, \cite{Tunnell:SaitoTunnell}).

The proofs of \thmref{T:B.W} and \thmref{T:C.W} are based on the ideas of Hida in \cite{Hida:nonvanishingmodp} and \cite{Hida:mu_invariant}. Thanks to Hida's theorems on the linear independence of modular forms applied by transcendental automorphisms of the local moduli of CM points in Hilbert modular varieties modulo $p$ \mbox{{\it loc.cit.} }, the vanishing/non-vanishing modulo $p$ properties of the algebraic part of $L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)$ with anticyclotomic twists can be deduced from the vanishing/non-vanishing of the Fourier expansions of the toric cusp form $\varphi_{\ell}am$ at cusps $({\mathfrak a},{\mathfrak b})$ such that ${\mathfrak a}{\mathfrak b}^{-1}$ is the polarization of abelian varieties with CM by $\mathcal O_E$. A new ingredient of this paper is the explicit computation of Fourier coefficients of toric new forms $\varphi_{\ell}am$ and a study on their non-vanishing modulo $p$ property in \subsecref{SS:localFourier}. Exploiting the connection between the Fourier coefficients of Hilbert modular forms and the trace of Frobenius of the associated Galois representations, we deduce from the vanishing modulo $p$ of Fourier coefficients of $\varphi_{\ell}am$ at these cusps that the trace of residual Galois representation $\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\rho}_p(\pi)$ is vanishing on the coset $G_F-G_E$. A simple lemma ({\ell}mref{L:3.W}) on modular representations shows that $\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\rho}_p(\pi)|_{G_E}$ is reducible.

This paper is organized as follows. After fixing notation and definitions in \secref{S:notation.W}, we derive a key formula of Waldspurger on the decomposition of global toric period integrals into local toric period integrals (\indent opref{P:Waldformula.W}) in \secref{S:Wald}. The bulk of this article is \secref{S:toric}, where we give the choices of local toric Whittaker functions $W_{{\mathbb I}i,v}$ and calculate explicitly these local period integrals attached to $W_{{\mathbb I}i,v}$. The explicit Waldspurger formula is proved in \thmref{T:main1.W}, and a non-vanishing modulo $p$ of these toric Whittaker functions is proved in \indent opref{P:4.W}. After reviewing briefly theory of complex and geometric Hilbert modular forms in \secref{S:Hilbert.W}, we prove \thmref{T:A.W} in \subsecref{S:anticy_padic_L}. The key ingredient is  \indent opref{P:3.W}, the construction of a \text{$p$-adic } measure ${\EuScript F}_{{\ell}am,{\mathfrak c}}$ on $\Gamma^-$ with values in the space of \text{$p$-adic } modular forms, and the \text{$p$-adic } $L$-function ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ is thus obtained by evaluating ${\EuScript F}_{{\ell}am,{\mathfrak c}}$ at suitable CM points. The precise evaluation formula of ${\mathcal P}_{\varSigma}(\pi,{\ell}am)^2$ is established in \thmref{T:padicL.W}. In \secref{S:mu_invariant}, we study the $\mu$-invariant of ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ and prove \thmref{T:B.W} in \thmref{T:mu-invariant}. Finally, the non-vanishing of central $L$-values modulo $p$ is considered in \secref{S:NV} and \thmref{T:C.W} is proved in \thmref{T:main3.W}.

\section{Notation and definitions}{\ell}abel{S:notation.W}
\subsection{Measures on local fields}We fix some general notation and conventions on local fields through this article. Let $\psi_\mathbf Q:\mathbf A_\mathbf Q/\mathbf Q\to \mathbf C^{\times}$ be the additive character such that $\psi_\mathbf Q(x_\infty)=\varepsilonsilonxp(2\pi i x_\infty)$ with $x_\infty\in\mathbf R$. Let $q$ be a place of $\mathbf Q$ and let $F$ be a finite extension of $\mathbf Q_q$. Let $\psi_q$ be the local component of $\psi$ at $q$ and let $\psi_F:=\psi_q\circ{\mathrm T}_{F/\mathbf Q_q}$, where ${\mathrm T}_{F/\mathbf Q_q}$ is the trace from $F$ to $\mathbf Q_q$. Let $dx$ be the Haar measure on $F$ self-dual with respect to the pairing $(x,x')\mapsto \psi_F(xx')$. Let $\mathbf Abs_F$ be the absolute value of $F$ normalized by $d(ax)=abelian variety s{a}_Fdx$ for $a\in F^{\times}$. We often simply write $\mathbf Abs=\mathbf Abs_F$ if it is clear from the context without possible confusion. We recall the definition of the local zeta function $\zeta_F(s)$. If $F$ is non-archimedean, let $\varpi_F$ be a uniformizer of $F$ and let \[\zeta_F(s)=\frac{1}{1-abelian variety s{\varpi_F}_F^s}.\] If $F$ is archimedean, then \[\zeta_\mathbf R(s)=\pi^{-s/2}\Gamma(s/2);\,\zeta_\mathbf C(s)=2(2\pi)^{-s}\Gamma(s).\]
The Haar measures $d^\x x$ on $F^{\times}$ is normalized by
\[d^\x x=\zeta_{F}(1)abelian variety s{x}_F^{-1}dx.\]
In particular, if $F=\mathbf R$, then $dx$ is the Lebesgue measure and $d^\x x=abelian variety s{x}_\mathbf R^{-1}dx$, and if $F=\mathbf C$, then $dx$ is twice the Lebesgue measure on $\mathbf C$ and $d^\x x=2\pi^{-1}r^{-1}drd\theta\,\,(x=re^{i\theta})$.

 Suppose that $F$ is non-archimedean. Let $\mathcal O_F$ be the ring of integers of $F$ and let $\mathcal D_F$ be the absolute different of $F$. Then $\mathcal D_F^{-1}$ is the Pontryagin  dual of $\mathcal O_F$ with respect to $\psi_F$, and $\vol(\mathcal O_F,dx)=abelian variety s{\mathcal D_F}_F^\mathbf 1half$. If $\mu:F^{\times}\to\mathbf C^{\times}$ is a character of $F^{\times}$, define the local conductor $a(\mu)$ by
\[a(\mu)=\inf\stt{n\in \mathbf Z_{\geq 0}\mid
\mu(x)=1\text{ for all }x\in (1+\varpi_F^n \mathcal O_F){\mathcal a}p \mathcal O_F^{\times}}.\]

\subsection{}
If $L$ is a number field, the ring of integers of
$L$ is denoted by $\mathcal O_L$, $\mathbf A_L$ is the adele of $L$ and $\mathbf A_{L,f}$
is the finite part of $\mathbf A_L$. For $a\in\mathbf A_L^{\times}$, we put
\[\mathfrak i\frakl_L(a):=a(\mathcal O_L\otimes\mathbf Zhat){\mathcal a}p L.\]
Denote by $G_L$ the absolute Galois group and by $\rec_L:\mathbf A_L^{\times}\to G^{ab}_{L}$ the geometrically normalized reciprocity law. 
 We define $\psi_L:\mathbf A_L^{\times}/L\to\mathbf C^{\times}$ by
$\psi_L(x)=\psi_\mathbf Q\circ{\mathbb T}r_{L/\mathbf Q}(x)$.

Let $v_p$ be the \text{$p$-adic } valuation on $\mathbf Qbarp$ normalized so that $v_p(p)=1$. We regard $L$ as a subfield in $\mathbf C$ (resp. $\mathbf C_p$) via
$\iota_\infty:\mathbf Qbar\hookrightarrow\mathbf C$ (resp. $\iota_p=\iota^{-1}\circ\iota_\infty:\mathbf Qbar\hookrightarrow\mathbf Cp$) and $\Hom(L,\mathbf Qbar)=\Hom(L,\mathbf C_p)$.

Let $\mathbf Zbar$ be the ring of algebraic integers of $\mathbf Qbar$ and let $\mathbf Zbarp$ be the \text{$p$-adic } completion of $\mathbf Zbar$ in $\mathbf Qbarp$. Let $\mathbf Zbar$ be the ring of algebraic integers of $\mathbf Qbar$ and let $\mathbf Zbarp$ be the \text{$p$-adic } completion of $\mathbf Zbar$ in $\mathbf Qbarp$ with the maximal ideal $\mathfrak m_p$. Let $\mathfrak m=\iota_p^{-1}(\mathfrak m_p)$.

\subsection{Local $L$-functions}
Let $F$ be a non-archimedean local filed. Let $\mu,\nu:F^{\times}\to\mathbf C^{\times}$ be two characters of $F^{\times}$. Denote by $I(\mu,\nu)$ the space consisting of smooth and $\GL_2(\mathcal O_F)$-finite functions $f:\GL_2(F)\to\mathbf C$ such that
\[f(\MX{a}{b}{0}{d}g)=\mu(a)\nu(d)abelian variety s{\frac{a}{d}}^\mathbf 1half f(g).\]
Then $I(\mu,\nu)$ is an admissible representation of $\GL_2(F)$. Denote by $\pi(\mu,\nu)$ the unique infinite dimensional subquotient of $I(\mu,\nu)$.  We call $\pi(\mu,\nu)$ a principal series if $\mu\nu^{-1}\not=\mathbf Abs^\pm$ and a special representation if $\mu\nu^{-1}=\mathbf Abs^\pm$.

Let $E$ be a quadratic extension of $F$ and let ${\mathbb I}i:E^{\times}\to\mathbf C^{\times}$ be a character. We recall the definition of local $L$-functions $L(s,\pi_{E}\otimes{\mathbb I}i)$ (\cite[\S 20]{Jacquet:GLtwoPartII}) when $\pi=\pi(\mu,\nu)$ is a subrepresentation of $I(\mu,\nu)$. If $E=F\oplus F$, then we write ${\mathbb I}i=({\mathbb I}i_1,{\mathbb I}i_2):F^{\times}\oplus F^{\times}\to\mathbf C^{\times}$ and put
\begin{align*}L(s,\pi_{E}\otimes{\mathbb I}i)=&\begin{cases}L(s,\pi\otimes{\mathbb I}i_1)L(s,\pi\otimes{\mathbb I}i_2)&\text{ if $\mu\nu^{-1}\not =\mathbf Abs^\pm$},\\
L(s,\mu{\mathbb I}i_1)L(s,\mu{\mathbb I}i_2)&\text{ if $\mu\nu^{-1}=\mathbf Abs$}.\varepsilonsilonnd{cases}
\intertext{If $E$ is a field, then} L(s,\pi_{E}\otimes{\mathbb I}i)=&\begin{cases}L(s,\mu'{\mathbb I}i )L(s,\nu'{\mathbb I}i)&\text{ if $\mu\nu^{-1}\not =\mathbf Abs^\pm$},\\
L(s,\mu'{\mathbb I}i )&\text{ if $\mu\nu^{-1}=\mathbf Abs$}.\varepsilonsilonnd{cases}\varepsilonsilonnd{align*}
Here $\mu'=\mu\circ\mathrm N_{E/F},\,\nu'=\nu\circ\mathrm N_{E/F}$ are characters of $E^{\times}$.
\subsection{Whittaker and Kirillov models}
Let $F$ be a local field. Let $\pi$ be an irreducible admissible representation of $\GL_2(F)$ and let $\psi:F\to\mathbf C^{\times}$ be a non-trivial additive character. We let ${\mathcal W}(\pi,\psi)$ be the Whittaker model of $\pi$. Recall that ${\mathcal W}(\pi,\psi)$ is a subspace of smooth functions $W:\GL_2(F)\to\mathbf C$ such that
\begin{mylist}
\item $W(\MX{1}{x}{0}{1}g)=\psi(x)W(g)$ for all $x\in F$.
\item If $v$ is archimedean, $W(\DII{a}{1})=O(abelian variety s{a}^N)$ for some positive number $N$.
\varepsilonsilonnd{mylist}
(\mbox{{\it cf.} }\cite[Thm.\,6.3]{Jacquet_Langlands:GLtwo}). Let ${\mathcal a}lK(\pi,\psi)$ be the Kirillov model of $\pi$. If $F$ is non-archimedean, then ${\mathcal a}lK(\pi,\psi)$ is a subspace of smooth $\mathbf C$-valued functions on $F^{\times}$, containing all Bruhat-Schwartz  functions on $F^{\times}$. A function in ${\mathcal a}lK(\pi,\psi)$ shall be called a \varepsilonsilonmph{local Fourier coefficient} of $\pi$. In addition, it is well known that we have the following $\GL_2(F)$-equivariant isomorphism
\begin{equation}{\ell}abel{E:KisoW}\begin{aligned}
{\mathcal W}(\pi,\psi)&\stackrel{\sim}{\to} {\mathcal a}lK(\pi,\psi)\\
W&\mapsto {\times}i_W(a):=W(\DII{a}{1}).\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}

\section{Waldspurger formula}{\ell}abel{S:Wald}
Let $F$ be a number field and $E$ be a quadratic field extension of $F$. Let $\mathbf AF=\mathbf A_F$. Let $G=\GL_2{}_{/F}$. Let $\pi$ be an irreducible cuspidal automorphic representation of $G(\mathbf AF)$ with unitary central character $\omega$. Denote by ${\mathcal A}(\pi)$ the realization of $\pi$ in the space ${\mathcal A}_0(G)$ of cusp forms on $G(\mathbf AF)$. Let ${\mathbb I}i$ be a unitary Hecke character of $E^{\times}$ such that ${\mathbb I}i|_{\mathbf AF^{\times}}=\omega^{-1}$. Let $\pi_E$ be the quadratic base change of $\pi$ to the quadratic extension $E/F$. The existence of $\pi_E$ is established in \cite{Jacquet:GLtwoPartII}. The goal of this section is to deduce from results in \cite{Waldsupurger:Central_value} a formula (\indent opref{P:Waldformula.W}) which expresses the central value $L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)$ in terms of a product of local toric period integrals of Whittaker functions.

Let $\psi:=\psi_F:\mathbf AF/F\to\mathbf C^{\times}$ be the standard non-trivial additive character. For a place $v$ of $F$, we let $G_v=G(F_v)$ and let ${\mathbb I}i_v:E_v^{\times}\to\mathbf C^{\times}$ (resp. $\psi_v:F_v\to\mathbf C^{\times}$) denote the local constituent of ${\mathbb I}i$ (resp. $\psi$).
\subsection{}{\ell}abel{SS:notation1}For $x\inE$, let ${\mathrm T}(x):=x+\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{x}$ and $\mathrm N(x)=x\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{x}$.
Let $\stt{1,\mathbf CMP}$ be a basis of $E$ over $F$.  We let $\iota:E\to M_2(F)$ be the embedding attached to $\mathbf CMP$ given by
\begin{equation}{\ell}abel{E:imb.W}\iota(a\mathbf CMP+b)=\MX{a{\mathrm T}(\mathbf CMP)+b}{-a\mathrm N(\mathbf CMP)}{a}{b}\quad(a,b\inF).\varepsilonsilonnd{equation}
Put
\[\boldsymbol{J}:=\MX{-1}{{\mathrm T}(\mathbf CMP)}{0}{1}.\]Then $M_2(F)=\iota(E)\oplus\iota(E)\boldsymbol{J}$. It is clear that $\boldsymbol{J}^2=1$ and $\iota(t)\boldsymbol{J}=\boldsymbol{J}\iota(\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{t})$ for all $t\in E$.
\subsection{The local bilinear form and toric integral} For each place $v$ of $F$, denote by $\pi_v$ (resp. $\psi_v$) the local constituent of $\pi$ (resp. $\psi$) at $v$.
Define a $\mathbf C$-bilinear form $\bfb_v:{\mathcal W}(\pi_v,\psi_v){\times}{\mathcal W}(\pi_v,\psi_v)\to\mathbf C$ by
\begin{align*}\bfb_v(W_{1},W_{2}):=&\sum_{n=-\infty}^\infty\int_{\varpi^n\mathcal O_F^{\times}}W_{1}(\DII{a}{1})W_{2}(\DII{-a}{1})\omega^{-1}(a)d^\x a\quad (W_1,W_2\in {\mathcal W}(\pi_v,\psi_v))\\
=&\int_{F_v^{\times}}W_{1}(\DII{a}{1})W_{2}(\DII{-a}{1})\omega^{-1}(a)d^\x a.
\varepsilonsilonnd{align*}
It is known that this series converges absolutely as $\pi_v$ is a local constituent of a unitary cuspidal automorphic representation. Moreover, the pairing $\bfb_v$ enjoys the property:
\begin{equation}{\ell}abel{E:bilinear.W}\bfb_v(\pi(g)W_1,\pi(g)W_2)=\omega(\det g)\bfb_v(W_1,W_2).\varepsilonsilonnd{equation}
The pairing $\bfb_v$ thus gives rise to an isomorphism between the contragredient representation $\pi^\vee$ and $\pi\otimes\omega^{-1}$.

The local toric period integral for $W_1,W_2\in{\mathcal W}(\pi_v,\psi_v)$ is given by
\[P(W_{1},W_{2},{\mathbb I}i_v):=\int_{E_v^{\times}/F_v^{\times}}\bfb_v(\pi(\iota(t))W_{1},\pi(\boldsymbol{J})W_{2}){\mathbb I}i_v(t)dt\cdot\frac{L(1,\tau_{E_v/F_v})}{\zeta_{F_v}(1)}.\]
The above integral converges as ${\mathbb I}i_v$ is unitary (\cite[LEMME 7]{Waldsupurger:Central_value}).
\subsection{A formula of Waldspurger}
Let $\Lambda(s,\pi_{E}\otimes{\mathbb I}i)$ be the completed $L$-function of $\pi_{E}\otimes{\mathbb I}i$ given by
\[\Lambda(s,\pi_{E}\otimes{\mathbb I}i):=\indent od_v L(s,\pi_{E_v}\otimes{\mathbb I}i_v)=L(s,\pi_{E}\otimes{\mathbb I}i)\cdot\indent od_{v|\infty}L(s,\pi_{E_v}\otimes{\mathbb I}i_v).\]
It is well known that $\Lambda(s,\pi_{E}\otimes{\mathbb I}i)$ converges absolutely for $\mathbf Re s\gg 0$ and has meromorphic continuation to all $s\in\mathbf C$. Moreover, it satisfies the functional equation
\[\Lambda(s,\pi_{E}\otimes{\mathbb I}i)=\varepsilonsilonp(s,\pi_E\otimes{\mathbb I}i)\Lambda(s,\pi_{E}\otimes{\mathbb I}i).\]
Here we have used the self-duality condition ${\mathbb I}i|_{\mathbf AF^{\times}}=\omega^{-1}$. The global toric period integral for $\varphi\in{\mathcal A}(\pi)$ is defined by
\[P_{\mathbb I}i(\varphi):=\int_{E^{\times}\mathbf AF^{\times}\backslash \mathbf AK^{\times}}\varphi(\iota(t)){\mathbb I}i(t)dt.\]
The following proposition connects the global toric periods and central $L$-values of $\pi_E\otimes{\mathbb I}i$.
\begin{prop}[Waldspurger]{\ell}abel{P:Waldformula.W}Let $\varphi_1,\varphi_2\in {\mathcal A}(\pi)$ and let $W_{\varphi_1},W_{\varphi_2}$ be the associated global Whittaker functions. We suppose that $W_{\varphi_i}=\indent od_vW_{i,v}$, where  $W_{i,v}\in{\mathcal W}(\pi_v,\psi_v)$ such that $W_{i,v}(1)=1$ for almost $v$ $(i=1,2)$. Then there exists a finite set $S_0$ of places of $F$ including all archimedean places such that for every finite set $S\supset S_0$, we have
\[P_{\mathbb I}i(\varphi_1)P_{\mathbb I}i(\varphi_2)=\Lambda(\mathbf 1half,\pi_E\otimes{\mathbb I}i)\cdot\indent od_{v\in S}\frac{1}{L(\mathbf 1half,\pi_{E_v}\otimes{\mathbb I}i_v)}\cdot P(W_{1,v},W_{2,v},{\mathbb I}i_v).
\]
\varepsilonsilonnd{prop}
\begin{proof}The proof is the combination of various formulae established in \cite{Waldsupurger:Central_value}. We first recall some important local integrals. Let $D=G{\times} G$. For each place $v$ of $F$, let ${\mathcal S}_v={\mathcal S}(M_2(F_v))\otimes{\mathcal S}(F_v^{\times})$ and let $D_v=G_v{\times} G_v$.
Let $r=r'{\times} r'':G_v{\times} D_v\to \End{\mathcal S}_v$ be the Weil representation of $G_v{\times} D_v$ defined in \cite[\S I.3 p.178]{Waldsupurger:Central_value}

Let $\varphi\in{\mathcal A}(\pi)$ be an automorphic form in the automorphic realization of $\pi$. Recall that the global Whittaker function of $\varphi$ is defined by
\[W_\varphi(g)=\int_{F\backslash\mathbf A_F}\varphi(\MX{1}{x}{0}{1}g)\psi(-x)dx.\]
Write ${\mathcal W}_v={\mathcal W}(\pi_v,\psi_v)$. We further assume that $W_\varphi$ has the factorization $W_\varphi=\indent od _vW_{\varphi,v}\in \otimes_v'{\mathcal W}_v$ such that $W_{\varphi,v}(1)=1$ for almost $v$. For each $v$, let $U:{\mathcal S}_v\to {\mathcal W}_v\otimes{\mathcal W}_v,\,f_v\to U_{f_v}$ be the $G_v{\times} G_v$-equivaraint surjective morphism associated to $W_v$ introduced in \cite[COROLLAIRE, p.187]{Waldsupurger:Central_value}. Define the following local integrals:
\begin{align*}C(f_v):=&\int_{F_v^{\times}}U_{f_v}(\DII{a}{1},\DII{-a}{1}))\omega^{-1}(a)d^\x a,\\
B(f_v,1):=&\int_{Z_v\backslash G_vN_v}\int_{F_v^{\times}}W_{\varphi,v}(\sigma)r'(\sigma)f_v(x,x^{-2})dxd\sigma,\\
P(f_v,{\mathbb I}i_v,\mathbf 1half):=&\int_{F^{\times}_v\backslash E^{\times}_v}B(r''(\iota(t),1)f_v,1){\mathbb I}i_v(t)dt.
\varepsilonsilonnd{align*}
The convergence and analytic properties of these local integrals are studied in \cite[LEMME 2, LEMME 3, LEMME 5]{Waldsupurger:Central_value}.
Moreover, we have
\[B(f_v,1)=C(f_v)\cdot\frac{1}{\zeta_{F_v}(1)}.\]
For each $v$, we take a special test function $f_v\in{\mathcal S}_v$  such that
\begin{equation}{\ell}abel{E:1.W}U_{f_v}=W_{1,v}\otimes \pi(\boldsymbol{J})W_{2,v}.\varepsilonsilonnd{equation}
Note that $f_v$ can be chosen to be the spherical test function $f_v^0:=\mathbb I_{M_2(\mathcal O_{F_v})}\otimes\mathbb I_{\mathcal O_{F_v}^{\times}}$ for all but finitely many $v$.
With this particular choice of $f_v$, we have
\begin{equation}{\ell}abel{E:2.W}\begin{aligned}
P(f_v,{\mathbb I}i_v,\mathbf 1half)&=\int_{F^{\times}_v\backslash E^{\times}_v}C(r''(\iota(t),1)f_v)dt\cdot\frac{1}{\zeta_{F_v}(1)}\\
&=\int_{F^{\times}_v\backslash E^{\times}_v}\bfb_v(\pi(\iota(t))W_1,\pi(\boldsymbol{J})W_2){\mathbb I}i_v(t)dt\cdot\frac{1}{\zeta_{F_v}(1)}\\
&=P(W_{1,v},W_{2,v},{\mathbb I}i_v)\cdot \frac{1}{L(1,\tau_{E_v/F_v})}.
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}

Let ${\mathcal S}=\otimes {\mathcal S}_v$ be the restricted product with respect to spherical test functions $\stt{f_v^0}_v$.
Define the theta kernel for $f:=\otimes f_v\in {\mathcal S}$ by
\[\theta_f(\sigma,g):=\sum_{(x,u)\in M_2(F){\times}F^{\times}}r(\sigma,g)f(x,u),\,\sigma\in G(\mathbf AF)\quad(g\in D(\mathbf AF)=G(\mathbf AF){\times} G(\mathbf AF)),\]
and define the automorphic form $\theta(f,\varphi,g)$ on $G(\mathbf AF){\times} G(\mathbf AF)$ by
\[\theta(f,\varphi,g)=\int_{G(F)\backslash G(\mathbf AF)}\varphi(\sigma)\theta_f(\sigma,g)d\sigma.\]
Note that according to \varepsilonsilonqref{E:1.W}, we have \[\theta(f,\varphi,g_1,g_2)=\varphi_1(g_1)\varphi_2(g_2\boldsymbol{J}).\]
We define the toric period integral $P(f,{\mathbb I}i)$ by
\[P(f,{\mathbb I}i):=\int_{[E^{\times}\mathbf AF^{\times}\backslash \mathbf AK^{\times}]^2}\theta(f,\varphi,\iota(t_1),\iota(t_2)){\mathbb I}i(t_1){\mathbb I}i(\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{t}_2)dt_1dt_2.\]
By the relation $\boldsymbol{J} \iota(t_2)\boldsymbol{J}=\iota(\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{t}_2)$ and the automorphy of $\varphi_2$, we find that
\[P(f,{\mathbb I}i)=P_{\mathbb I}i(\varphi_1)P_{\mathbb I}i(\varphi_2).\]
Let $S_0$ be a finite set of places of $F$ such that $W_{\varphi,v}$, $W_{i,v}$ and $f_v$ are spherical for all $v\not\in S_0$.
From \cite[Prop.\,4, p.196 and LEMME 7, p.219]{Waldsupurger:Central_value}, we deduce the following formula for every finite set $S\supset S_0$:
\begin{align*}P_{\mathbb I}i(\varphi_1)P_{\mathbb I}i(\varphi_2)=\Lambda(\mathbf 1half,\pi_E\otimes{\mathbb I}i)\cdot \indent od_{v\in S}  P(f_v,{\mathbb I}i_v,\mathbf 1half)\cdot \frac{L(1,\tau_{E_v/F_v})}{L(\mathbf 1half,\pi_{E_v}\otimes{\mathbb I}i)}.\varepsilonsilonnd{align*}
We thus establish the desired formula in virtue of \varepsilonsilonqref{E:2.W}.
\varepsilonsilonnd{proof}

\section{Toric period integrals}{\ell}abel{S:toric}
\subsection{Notation}
Throughout we suppose that $F$ is a totally real number field and $E$ is a totally imaginary quadratic extension of $F$. We retain the notation in the introduction and \subsecref{SS:notation1}. Let ${\varSigma}$ be a fixed CM type of $E$. Let $\pi$ be an irreducible automorphic cuspidal representation of $\GL_2(\mathbf AF)$. Let $\mathfrak n$ be the conductor of $\pi$. Suppose that $\pi$ has infinity type $k=\sum_{\sigma\in{\varSigma}}k_\sigma\sigma\in\mathbf Z_{\geq 1}[{\varSigma}]$. Let $m=\sum_\sigma m_\sigma\sigma\in\mathbf Z_{\geq 0}[{\varSigma}]$ and let ${\mathbb I}i$ be a Hecke character of infinity type $(k/2+m,-k/2-m)$ such that ${\mathbb I}i|_{\mathbf AF^{\times}}=\omega^{-1}$. Let $\mathbf h$ be the set of finite places of $F$. Recall that the set of infinite places of $F$ is identified with the CM-type ${\varSigma}$..

In this section, we will choose a special local Whittaker function at each place $v$ of $F$ in \subsecref{SS:Whittaker.W} and calculate their associated local toric period integrals in \subsecref{SS:toricintegralI.W} and \subsecref{SS:toricII.W}. Finally, we prove in \subsecref{SS:localFourier} a non-vanishing modulo $p$ result of these local Whittaker functions. This result plays an important role in the later application to the calculation of the $\mu$-invariant.

Let ${\mathfrak C}_{\mathbb I}i$ (resp. ${\mathfrak c}_\omega$) be the conductor of ${\mathbb I}i$ (resp. $\omega$).  Let ${\mathfrak c}_{\mathbb I}i={\mathfrak C}_{\mathbb I}i{\mathcal a}pF$. We further decompose $\mathfrak n^-=\mathfrak n^-_s\mathfrak n^-_r$, where $\mathfrak n^-_s$ is prime to ${\mathfrak c}_\omega$ and $\mathfrak n^-_r$ is only divisible by prime factors of ${\mathfrak c}_\omega$. Put \begin{equation}{\ell}abel{E:alv.W}\begin{aligned}c_v({\mathbb I}i)=&\inf\stt{n\in\mathbf Z_{\geq 0}\mid {\mathbb I}i=1\text{ on }(1+\varpi^n\mathcal O_E)^{\times}},\\
m_v({\mathbb I}i,\pi)=&c_v({\mathbb I}i)-v(\mathfrak n^-).\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
It is clear that $c_v({\mathbb I}i)=v({\mathfrak c}_{\mathbb I}i)$. We put
\begin{align*}
A({\mathbb I}i)=&\stt{v\in\mathbf h\mid\text{$E_v$ is a field, $\pi_v$ is special and $c_v({\mathbb I}i)=0$}}.
\varepsilonsilonnd{align*}

Let $p>2$ be a rational prime satisfying \varepsilonsilonqref{ord}. The assumption \varepsilonsilonqref{ord} in particular implies that every prime factor of $p$ in $F$ splits in $E$. Let ${\varSigma}_p$ be the \text{$p$-adic } places induced by ${\varSigma}$ via $\iota_p$. Thus ${\varSigma}_p$ and its complex conjugation ${\varSigma}bar_p$ give a partition of the places of $E$ above $p$. Let $\mathfrak N$ be the prime-to-$p$ conductor of $\pi_{E}\otimes{\mathbb I}i$. We fix a decomposition $\mathfrak N^+=\mathbf Csplit\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathbf Csplit}$ such that $(\mathbf Csplit,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathbf Csplit})=1$.

\subsection{Galois representation attached to $\pi$}{\ell}abel{SS:Galois.W}
Let $\rho_p(\pi):G_F\to\GL_2(\mathcal O_{{L_\pi}})$ be the \text{$p$-adic } Galois representation associated to $\pi$ as in the introduction. Let $v\nmid p$ and let $W_{F_v}$ be the local Weil group at $v$. Suppose that $\pi_v=\pi(\mu_v,\nu_v)$ is a subquotient of the induced representations. 
By the local-global compatibility (\cite{Carayol:GaloisHMF}, \cite{Taylor:Galois_rep_HMF} and \cite{Jarvis:wtone}), we have
\begin{equation}{\ell}abel{E:Galois.W}\rho_p(\pi)|_{W_{F_v}}\simeq\MX{\mu_v^{-1}\mathbf Abs^\frac{1-k_{mx}}{2}}{*}{0}{\nu_v^{-1}\mathbf Abs^\frac{1-k_{mx}}{2}}\quad(k_{mx}=\max_{\sigma}k_\sigma).\varepsilonsilonnd{equation}
In particular, this implies that $\mu_v(\varpi_{F_v})$ and $\nu_v(\varpi_{F_v})$ are \text{$p$-adic } units in $\mathcal O^{\times}_{L_\pi}$.

\subsection{Open-compact subgroups}
For each finite place $v$, we put
\begin{align*}K^0_v=&\stt{g=\MX{a}{b}{c}{d}\in G_v\mid a,d\in \mathcal O_{F_v},\,b\in\mathcal D_{F_v}^{-1},\,c\in\mathcal D_{F_v},\,\det g\in\mathcal O_{F_v}^{\times}},
\intertext{and for an integral ideal ${\mathfrak a}$ of $F$, we put}
K^0_v({\mathfrak a})=&\stt{g=\MX{a}{b}{c}{d}\in K^0_v\mid c\in{\mathfrak a}\mathcal D_{F_v},\,a-1\in{\mathfrak a}},\\
U_v({\mathfrak a})=&\stt{g\in \GL_2(\mathcal O_{F_v})\mid g\equiv 1\pmod{{\mathfrak a}}}.
\varepsilonsilonnd{align*}
Let $K^0=\indent od_{v\in\mathbf h}K^0_v$ and $U({\mathfrak a})=\indent od_{v\in\mathbf h} U_v({\mathfrak a})$ be open-compact subgroups of $\GL_2(\mathbf AFf)$.
\subsection{The choices of $\mathbf CMP$ and $\varsigma_v$}{\ell}abel{SS:choiceofcmpt}
 Fix an integral ideal $\frakr\subset {\mathfrak c}_{\mathbb I}i\mathfrak n \mathcal D_{E}^2$ of $F$. For each finite place, let $d_{F_v}$ be a generator of the absolute different $\mathcal D_{F_v}$. We choose $\mathbf CMP\inE$ such that
\begin{itemize}
\item[(d1)] $\Im \sigma(\mathbf CMP)>0$ for all $\sigma\in{\varSigma}$,
\item[(d2)]$\stt{1,d_{F_v}^{-1}\mathbf CMP}$ is an $\mathcal O_{F_v}$-basis of $\mathcal O_{E_v}$ for all $v\mid p\frakr$,
\item[(d3)]$d_{F_v}^{-1}\mathbf CMP$ is a uniformizer of $E_v$ for every $v$ ramified in $E$.\varepsilonsilonnd{itemize}
Then $\mathbf CMP$ is a generator of $E$ over $F$ which determines an embedding $E\hookrightarrow M_2(F)$ in \varepsilonsilonqref{E:imb.W}. Let
\[\delta=2^{-1}(\mathbf CMP-\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathbf CMP})\in E^{\times}.\]

For each $v$ split in $E$, we shall fix a place $w$ of $E$ above $v$ throughout, and decompose $E_v:=E\otimes_FF_v=F_ve_w\oplus F_v e_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}$, where $e_w$ and $e_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}$ are the idempotents attached to $w$ and $\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}$. If $v|p\mathfrak N^+$, we further require that $w|\mathbf Csplit{\varSigma}bar_p$, i.e.  $w|\mathbf Csplit$ or $w\in{\varSigma}bar_p$. We identify $\delta\in E_w=F_v$ and write $\mathbf CMP_v=\mathbf CMP_we_w+\mathbf CMP_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}e_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}$ for split $v$.

For each finite place $v$, we fix a uniformizer $\varpi_v=\varpi_{F_v}$ of $F_v$. By (d2), we fix a choice of $d_{F_v}$ as follows.
\[d_{F_v}=\begin{cases}2\delta &\text{ if } v\mid p\frakr\text{ is split },\\
\varpi_v^{v(\mathcal D_{F})}&\text{ otherwise }.\varepsilonsilonnd{cases}\] We also fix an $\mathcal O_{F_v}$-basis $\stt{1,\cO_\cKbasis_v}$ of $\mathcal O_{E_v}$ such that $\cO_\cKbasis_v=\mathbf CMP$ except for finitely many $v$ and \[\cO_\cKbasis_v=d_{F_v}^{-1}\mathbf CMP\text{ for }v|p\frakr.\] Write $\cO_\cKbasis_v=a_v\mathbf CMP+b_v$ with $a_v,b_v\inF_v$.

For each place $v$, we define $\varsigmav\in \GL_2(F_v)$ as follows:
\begin{equation}{\ell}abel{E:cmptv.W}\begin{aligned}
\varsigmav=&\MX{\Im\sigma(\mathbf CMP)}{\mathbf Re \sigma(\mathbf CMP)}{0}{1}\text{ for }v=\sigma\in{\varSigma},\\
\varsigmav=&(\mathbf CMP_w-\mathbf CMP_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}})^{-1}\MX{d_{F_v}\mathbf CMP_w}{\mathbf CMP_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}}{d_{F_v}}{1}\text{ for split }v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w},\\
\varsigmav=&\MX{d_{F_v}}{-b_v}{0}{a_v}\text{ for non-split finite $v$.}
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation} 
For $t\in E_v$, we put
\[\iota_{\varsigmav}(t):=\varsigmav^{-1}\iota(t)\varsigmav.\]
It is straightforward to verify that if $v=\sigma\in{\varSigma}$ is archimedean and $t=x+iy\in\mathbf C^{\times}$, then
\begin{align}
{\ell}abel{E:cm1.W}\iota_{\varsigma_\sigma}(t)=&\MX{x}{-y}{y}{x},\,\intertext{ and if $v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}$ is split and $t=t_1e_w+t_2e_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}$, then }
{\ell}abel{E:cm2.W}\iota_{\varsigmav}(t)=&\DII{t_1}{t_2}.
\varepsilonsilonnd{align}
Moreover, we note that for all finite places $v$\[\iota_{\varsigmav}(\mathcal O_{E_v}^{\times})=\iota_{\varsigmav}(E_v^{\times}){\mathcal a}p K^0_v.\]

\subsection{Running assumptions}
In this section, we will assume \hypref{H:HypA} for $(\pi,{\mathbb I}i)$ and
\begin{equation}cd{sf} \mathfrak n^-\text{ is square-free}.\varepsilonsilonnd{equation}cd
The assumption \varepsilonsilonqref{sf} implies that $\pi_v$ is an unramified special representation if $v|\mathfrak n^-_s$ and $\pi_v$ is a ramified principal series if $v|\mathfrak n^-_r$. In particular, for every place $v$ inert or ramified in $E$, $\pi_v$ is a sub-quotient of induced representations and the local $L$-function $L(s,\pi_v)\not =1$. We shall write $\pi_v=\pi(\mu_v,\nu_v)$ such that $L(s,\pi_v)=L(s,\mu_v)$ for $v|\mathfrak n^-$. Moreover, by the local root number formulas \cite[Prop.\,3.5,Thm.\,2.18]{Jacquet_Langlands:GLtwo}, under the assumption \varepsilonsilonqref{sf} \hypref{H:HypA} on the sign of local root numbers is equivalent to the following condition:
\begin{equation}cd{R1}
\begin{aligned}\text{ For each }v\in A({\mathbb I}i),\,v\text{ is ramified in $E$ and }\mu_v'{\mathbb I}i_v(\varpi_{E_v})=-abelian variety s{\varpi}^\mathbf 1half\quad(\mu'_v=\mu_v\circ\mathrm N_{E_v/F_v}).\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}cd

\def\mathbb Fv{F}
\def\cK{E}

In what follows, we fix a place $v$ of $F$. Let $F=F_v$ and $E=E_v$. Let $\mathcal O=\mathcal O_F$ and $\varpi=\varpi_v$ if $v$ is finite. We shall suppress the subscript $v$ and write $\pi=\pi_v$, ${\mathbb I}i={\mathbb I}i_v$, $\varsigma=\varsigmav$ and $\psi=\psi_v$.

\subsection{The choice of local toric Whittaker functions}{\ell}abel{SS:Whittaker.W}
If $v$ is finite, we let $W^0_v$ denote the new Whittaker function in ${\mathcal W}(\pi,\psi)$. In other words, $W^0_v$ is invariant by $K^0_v(\mathfrak n)$ and $W^0_v(1)=1$. The existence of $W^0_v$ is a consequence of the theory of local new vectors \cite{Casselman:Atkin-Lehner}. Now we introduce special local Whittaker functions.

\subsubsection{The archimedean case} Suppose that $v=\sigma\in{\varSigma}$ is an archimedean place and $\mathbb Fv=\mathbf R$. Then $\pi_\sigma=\pi(\mathbf Abs^\frac{k_\sigma-1}{2},\mathbf Abs^\frac{1-k_\sigma}{2}\sigman^{k_\sigma})$ is the discrete series of minimal $\SO(2,\mathbf R)$-type $k_\sigma$. Let $W_{k_\sigma}\in{\mathcal W}(\pi_v,\psi_v)$ be the Whittaker function given by
\begin{equation}{\ell}abel{E:archiWhittaker.W}W_{k_\sigma}(z\DII{a}{1}\kappappapa_\theta)=a^\frac{k_\sigma}{2}e^{-2\pi a}\mathbb I_{\mathbf R_+}(a)\cdot e^{ik_\sigma\theta}\sigman(z)^{k_\sigma}\quad (z\in\mathbf R^{\times},\,\kappappapa_\theta=\MX{\cos\theta}{\sin\theta}{-\sin\theta}{\cos\theta}).\varepsilonsilonnd{equation}
Let $V_+$ and $V_-$ be the weight raising and lowering differential operators in \cite[p.165]{Jacquet_Langlands:GLtwo} given by
\[V_\pm=\MX{1}{0}{0}{-1}\otimes 1\pm\MX{0}{1}{1}{0}\otimes i\in\Lie(\GL_2(\mathbf R))\otimes_\mathbf R\mathbf C.\]
Define the normalized weight raising differential operator $\widetilde V_+$ by
\begin{equation}{\ell}abel{E:Shi_Maass}\widetilde V_+=\frac{1}{(-8\pi)}\cdot V_+.\varepsilonsilonnd{equation}
Then we have
\begin{equation}{\ell}abel{E:15.W}\widetilde V_+^{m_\sigma} W_{k_\sigma}(g\kappappapa_\theta)=\widetilde V_+^{m_\sigma} W_{k_\sigma}(g)e^{i(k_\sigma+2m_\sigma)\theta}.\varepsilonsilonnd{equation}

\subsubsection{The split case} Suppose that $v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}$ is split with $w|{\varSigma}bar_p\mathbf Csplit$ if $v|p\mathfrak N^+$. We introduce some smooth functions ${\ell}ocalK{{\mathbb I}i,v}$ on $F^{\times}$ in the Kirillov model ${\mathcal a}lK(\pi,\psi)$. Write ${\mathbb I}i=({\mathbb I}i_w,{\mathbb I}i_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}):F^{\times}\oplus F^{\times}\to\mathbf C^{\times}$. If the local $L$-function $L(s,\pi\otimes{\mathbb I}i_w)=1$, we simply put \[{\ell}ocalK{{\mathbb I}i,v}(a)=\mathbb I_{\mathcal O^{\times}}(a){\mathbb I}i_w(a^{-1}).\]
Suppose that $L(s,\pi\otimes{\mathbb I}i_w)\not =1$. Then $\pi=\pi(\mu,\nu)$ is a principal series or $\pi=\pi(\mu,\nu)$ is special with $\mu\nu^{-1}=\mathbf Abs$ and $\mu{\mathbb I}i_w$ is unramified. If $\pi\otimes{\mathbb I}i_w$ is unramified, we set
\[{\ell}ocalK{{\mathbb I}i,v}(a)=\mathbb I_{\mathcal O}(a)\cdot {\mathbb I}i_w^{-1}\mathbf Abs^\mathbf 1half(a)\sum_{i+j=v(a),\,i,j\geq 0}\mu{\mathbb I}i_w(\varpi^i)\nu{\mathbb I}i_w(\varpi^j).\]
If $\mu_i{\mathbb I}i_w$ is unramified and $\mu_j{\mathbb I}i_w$ is ramified for $\stt{\mu_1,\mu_2}=\stt{\mu,\nu}$, we set
\[{\ell}ocalK{{\mathbb I}i,v}(a)=\mu_i\mathbf Abs^\mathbf 1half(a)\mathbb I_{\mathcal O}(a).\]
If $\pi$ is special, we set
\[{\ell}ocalK{{\mathbb I}i,v}(a)=\mu\mathbf Abs^\mathbf 1half(a)\mathbb I_{\mathcal O}(a).\]

These functions ${\ell}ocalK{{\mathbb I}i,v}$ indeed belong to the Kirillov model ${\mathcal a}lK(\pi,\psi)$ in virtue of the description of the Kirillov models \cite[Lemma 14.3]{Jacquet:GLtwoPartII}.
For each ${\times}i\in {\mathcal a}lK(\pi,\psi)$, by the isomorphism \varepsilonsilonqref{E:KisoW} we denote by $W_{\times}i\in {\mathcal W}(\pi,\psi)$ the unique Whittaker function such that $W_{\times}i(\DII{a}{1})={\times}i(a)$.
We put
\[{\ell}ocalW{{\mathbb I}i,v}:=W_{{\ell}ocalK{{\mathbb I}i,v}}.\]
It follows from the choice of ${\ell}ocalW{{\mathbb I}i,v}$ that
\[{\ell}ocalW{{\mathbb I}i\phi,v}={\ell}ocalW{{\mathbb I}i,v}\text{ if $\phi: E^{\times}\to\mathbf C^{\times}$ is unramified}.\]
Recall that the zeta integral $\Psi(s,W,{\mathbb I}i_w)$ for $W\in {\mathcal W}(\pi,\psi)$ is defined by
\[\Psi(s,W,{\mathbb I}i_w):=\int_{F^{\times}}W(\DII{a}{1}){\mathbb I}i_w(a)abelian variety s{a}^{s-\mathbf 1half}d^\x a.\]
Then the zeta integral for ${\ell}ocalW{{\mathbb I}i,v}$ satisfies the following equation:
\begin{equation}{\ell}abel{E:DeflocalWsplit.W}\Psi(s,{\ell}ocalW{{\mathbb I}i,v},{\mathbb I}i_w)=L(s,\pi\otimes{\mathbb I}i_w)abelian variety s{\mathcal D_F}^\mathbf 1half\quad(\vol(\mathcal O_F^{\times},d^\x a)=abelian variety s{\mathcal D_F}^\mathbf 1half).\varepsilonsilonnd{equation}

Suppose that $v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}$ with $w\in{\varSigma}bar_p$. We define some $p$-modified Whittaker functions as follows. For each $u\in\mathcal O_F^{\times}$, we put
\[\bfa_{u,v}(a):=\mathbb I_{u(1+\varpi\mathcal O)}(a){\mathbb I}i_w(a^{-1})\text{ and }{\ell}ocalW{{\mathbb I}i,u,v}=W_{{\ell}ocalK{u,v}}.\]
Let ${\ell}ocalK{{\mathbb I}i,v}^\flat(a):=\mathbb I_{\mathcal O^{\times}}(a){\mathbb I}i_w(a^{-1})$ and let ${\ell}ocalW{{\mathbb I}i,v}^\flat$ be the $p$-modified Whittaker function given by
\begin{equation}{\ell}abel{E:DeflocalWp.W}
{\ell}ocalW{{\mathbb I}i,v}^\flat:=W_{{\ell}ocalK{{\mathbb I}i,v}^\flat}=\sum_{u\in\mathcal U_v}{\ell}ocalW{{\mathbb I}i,u,v},\varepsilonsilonnd{equation}
where $\mathcal U_v$ is the torsion subgroup of $\mathcal O^{\times}$.
It is easy to verify that
\begin{equation}{\ell}abel{E:10.W}\begin{aligned}\Psi(s,{\ell}ocalW{{\mathbb I}i,v}^\flat,{\mathbb I}i_w)=1\,;\,\pi(\MX{a}{b}{0}{d}){\ell}ocalW{{\mathbb I}i,v}^\flat=&{\mathbb I}i_w^{-1}(a){\mathbb I}i_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}^{-1}(d){\ell}ocalW{{\mathbb I}i,v}^\flat\text{ for }a,d\in\mathcal O^{\times},\,b\in\mathcal D_F^{-1}.
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}

\subsubsection{The inert and ramified case} Suppose that $v$ is an inert or ramified finite place. Then $E$ is a non-archimedean local field. Define the operators $\cR_v$ and ${\mathcal P}_{{\mathbb I}i,\varsigma}$ on $W\in{\mathcal W}(\pi,\psi)$ by
\begin{align*}\cR_v W(g):=&W(g\DII{1}{\varpi}),\\
{\mathcal P}_{{\mathbb I}i,\varsigma} W(g):=& \bfv_E^{-1}\cdot \int_{E^{\times}/\mathbb Fv^{\times}}\pi(\iota_\varsigma(t))W(g){\mathbb I}i(t)dt.\\
=&\bfv_E^{-1}\int_{E^{\times}/\mathbb Fv^{\times}}W(g\varsigma^{-1}\iota(t)\varsigma){\mathbb I}i(t)dt.\varepsilonsilonnd{align*}
Note that
\[\bfv_E=\vol(E^{\times}/\mathbb Fv^{\times},dt)=e_v\cdot abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half},\quad e_v=\begin{cases}1&\text{ if $v$ is inert},\\
2&\text{ if $v$ is ramified}.\varepsilonsilonnd{cases}
\]
We define the Whittaker function ${\ell}ocalW{{\mathbb I}i,v}$ by
\begin{equation}{\ell}abel{E:DeflocalWinert.W}{\ell}ocalW{{\mathbb I}i,v}:={\mathcal P}_{{\mathbb I}i,\varsigma} \cR_v^{m_v({\mathbb I}i,\pi)} W^0_v.
\varepsilonsilonnd{equation}
\subsubsection{}
Define the subgroup $\mathcal T_v$ of $E^{\times}$ by
\[\mathcal T_v=\begin{cases}\mathcal O^{\times}_{E}F^{\times}&\text{ if $v$ is split,}\\
E^{\times}&\text{ if $v$ is non-split.}\varepsilonsilonnd{cases}\]
Then $\mathcal T_v=\stt{x\in E\mid x/\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{x}\in\mathcal O_E^{\times}}$ if $v$ is finite.
\begin{defn}[Toric Whittaker functions]{\ell}abel{D:toric} We say that $W\in {\mathcal W}(\pi,\psi)$ is a toric Whittaker function of character ${\mathbb I}i$ if
\[\pi(\iota_\varsigma(t))W={\mathbb I}i^{-1}(t)\cdot W\text{ for all }t\in\mathcal T_v.\]
\varepsilonsilonnd{defn}
\begin{lm}{\ell}abel{L:toric.W}The Whittaker functions ${\ell}ocalW{{\mathbb I}i,v}$ chosen as above are toric. To be precise, we have \begin{mylist}
\item $\widetilde V_+^{m_\sigma} W_{k_\sigma}$ is a toric Whittaker function of the character ${\mathbb I}i_{\sigma}:\mathbf C^{\times}\to\mathbf C^{\times},\,z\mapsto z^{k_\sigma+m_\sigma}\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{z}^{-m_\sigma}abelian variety s{z\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{z}}^{-k_\sigma/2}$.
\item If $v$ is finite, then ${\ell}ocalW{{\mathbb I}i,v}$ are toric Whittaker functions of character ${\mathbb I}i_v$.
\item If $v|p$, then  ${\ell}ocalW{{\mathbb I}i,v}^\flat$ is toric, and for $u\in\mathcal O_F^{\times}$
\[\pi(\iota_\varsigma(t)){\ell}ocalW{{\mathbb I}i,u,v}={\mathbb I}i^{-1}(t){\ell}ocalW{{\mathbb I}i,u.t^{1-c},v},\]
where $u.t^{1-c}:=ut_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}} t_{w}^{-1}$, $t=t_we_w+t_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}} e_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}\in\mathcal O_E^{\times}$ with $w\in{\varSigma}bar_p$,
\varepsilonsilonnd{mylist}
\varepsilonsilonnd{lm}
\begin{proof} It follows immediately from the definitions of these Whittaker functions together with \varepsilonsilonqref{E:15.W}, \varepsilonsilonqref{E:cm1.W} and \varepsilonsilonqref{E:cm2.W}.\varepsilonsilonnd{proof}
\subsection{Local toric period integrals (I)}{\ell}abel{SS:toricintegralI.W}
\subsubsection{}
Define the local toric period integral for $W\in {\mathcal W}(\pi,\psi)$ by
\begin{align*}{\ell}ocalP{W}{{\mathbb I}i}:=&P(W,W,{\mathbb I}i)\\
=&\int_{E^{\times}/F^{\times}}\bfb_v(\pi(\iota(t))W,\pi(\boldsymbol{J})W){\mathbb I}i(t)dt\cdot\frac{L(1,\tau_{E/F})}{\zeta_F(1)}.\varepsilonsilonnd{align*}
The main task of this section is to evaluate ${\ell}ocalP{\pi(\varsigma){\ell}ocalW{{\mathbb I}i,v}}{{\mathbb I}i}$. Put
\[\bfd(a)=\DII{a}{1}\quad(a\in F^{\times}).\]
We first treat the archimedean and split cases.
\subsubsection{The archimedean case}{\ell}abel{SS:local_archi}
Suppose $v=\sigma\in{\varSigma}\stackrel{\sim}{\to}\Hom(F,\mathbf R)$ is an archimedean place.
\begin{prop}{\ell}abel{P:Architoric.W}We have
\[{\ell}ocalP{\pi(\varsigma) \widetilde V_+^{m_\sigma}W_{k_\sigma}}{{\mathbb I}i}=2^3\cdot \frac{\Gamma(m_\sigma+1)\Gamma(k_\sigma+m_\sigma)}{(4\pi)^{k_\sigma+1+2m_\sigma}}.\]
\varepsilonsilonnd{prop}
\begin{proof}Introduce the Hermitian inner product on ${\mathcal W}(\pi,\psi)$ defined by
\[\pair{W_1}{W_2}:=\bfb_v(W_1,c(W_2)),\text{ where }c(W_2)(g):=\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{W(\DII{-1}{1}g)}\omega(\det g).\]
Write $k=k_\sigma$ and $m=m_\sigma$. It is clear that
\[\pair{W_k}{W_k}=(4\pi)^{-k}\Gamma(k).\]
Since $c(V_+^mW_k)$ and $\pi(\DII{-1}{1})V_+^mW_k$ are both nonzero Whittaker functions of weight $-k-2m$, there exists some constant $\gamma$ such that
\[\pi(\DII{-1}{1})V_+^mW_k=\gamma\cdot c(V_+^mW_k)\iff V_+^mW_k(\bfd(a))=\gamma\cdot \overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{V_+^mW_k(\bfd(a))}\text{ for all }a\in\mathbf R_+.\]
Let $h_m(x):=V_+^mW_k(\bfd(x))$. Then $h_0(x)=W_k(\bfd(x))$ is a real-valued function in view of the definition \varepsilonsilonqref{E:archiWhittaker.W}. A simple calculation shows that
\[h_{m+1}=2x\frac{dh_m}{dx}+(2\pi x-k-2m)h_m,\]
so by induction $h_m(x)$ takes value in $\mathbf R$ (\mbox{{\it cf.} }\cite[p.189]{Jacquet_Langlands:GLtwo}). This implies that $\gamma=1$. We thus have
\[\bfb_v(\pi(\varsigma)V_+^mW_k,\pi(\boldsymbol{J}\varsigma)V_+^mW_k)=\pair{V_+^mW_k}{V_+^mW_k}\quad(\varsigma^{-1}\boldsymbol{J}\varsigma=\DII{-1}{1}).\]

To evaluate $\pair{V_+^mW_k}{V_+^mW_k}$, note that by \cite[p.166]{Jacquet_Langlands:GLtwo} we have
\begin{equation}{\ell}abel{E:6.N}\begin{aligned}V_-^mV_{+}^{m}W_{k}&=(-4)^m\frac{\Gamma(k+m)\Gamma(m+1)}{\Gamma(k)}\cdot W_k,
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
and hence we find that
\begin{align*}\pair{V_+^mW_k}{V_+^mW_k}=&(-1)^m\pair{W_k}{V_-^mV_+^mW_k}\\
=&4^m\frac{\Gamma(k+m)}{\Gamma(k)}\Gamma(m+1)\pair{W_k}{W_k}\\
=&4^m(4\pi)^{-k}\cdot \Gamma(k+m)\Gamma(m+1).\varepsilonsilonnd{align*}
Recall that $dt=2\pi^{-1}d\theta$ with$ t=e^{i\theta}$, so $\vol(\mathbf C^{\times}/\mathbf R^{\times},dt)=2\pi^{-1}\cdot \pi=2$. Combining these with {\ell}mref{L:toric.W} (1), we find that
\begin{align*}
{\ell}ocalP{\pi(\varsigma)\widetilde V_+^mW_k}{{\mathbb I}i}&=2\cdot (-8\pi)^{-2m}\cdot\bfb_v(\pi(\varsigma)V_+^mW_k,\pi(\boldsymbol{J}\varsigma)V_+^mW_k)\cdot\frac{\zeta_\mathbf R(2)}{\zeta_\mathbf R(1)}\\
&=2^3(4\pi)^{-2m-1}4^{-m}\cdot\pair{V_+^mW_k}{V_+^mW_k}\\
&=2^3\cdot (4\pi)^{-k-2m-1}\Gamma(k+m)\Gamma(m+1).\qedhere
\varepsilonsilonnd{align*}
\varepsilonsilonnd{proof}
\subsubsection{The split case}{\ell}abel{SS:toricI.W}
 Suppose that $v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}$ is a finite place split in $E$. Recall that we have assumed $w|{\varSigma}bar_p\mathbf Csplit$ if $v|p\mathfrak N^+ $.
\begin{lm}{\ell}abel{L:key.W} We have
\[{\ell}ocalP{\pi(\varsigma)W}{{\mathbb I}i}=\Psi(\mathbf 1half,W,{\mathbb I}i_w)^2\cdot \frac{L(\mathbf 1half,\pi\otimes{\mathbb I}i_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}})}{L(\mathbf 1half,\pi\otimes{\mathbb I}i_w)}\cdot\varepsilonsilonp(\mathbf 1half,\pi\otimes{\mathbb I}i_w,\psi)\cdot\omega^{-1}{\mathbb I}i_w^{-2}(-d_F)\omega(\det\varsigma).\]
\varepsilonsilonnd{lm}
\begin{proof}
Let $\widehat W(g):=W(g\MX{0}{1}{-1}{0})\omega^{-1}(\det g)$. By \cite[Thm.\,2.18 (iv)]{Jacquet_Langlands:GLtwo}, we have
the local functional equation:
\[\frac{\Psi(1-s,\widehat W,{\mathbb I}i_w^{-1})}{L(1-s,\pi^\vee\otimes{\mathbb I}i_w^{-1})}=\varepsilonsilonp(s,\pi\otimes{\mathbb I}i_w,\psi)\cdot\frac{\Psi(s,W,{\mathbb I}i_w)}{L(s,\pi\otimes{\mathbb I}i_w)}.\]
We note that
\[\varsigmav^{-1}\boldsymbol{J}\varsigmav=\MX{0}{d_F^{-1}}{d_F}{0}.\]
A straightforward computation shows that \begin{align*}
{\ell}ocalP{\pi(\varsigma)W}{{\mathbb I}i}&=\omega(\det\varsigma)\int_{F^{\times}}\int_{F^{\times}}W(\bfd(at_1))W(\bfd(-a)\MX{0}{d_F^{-1}}{d_F}{0}){\mathbb I}i_w(t_1)\omega^{-1}(a)d^\x a dt_1\\
&=\omega(-\det\varsigma\cdot d_F)\int_{F^{\times}}\int_{F^{\times}}W(\bfd(t_1))W(\bfd(a d_F^{-2})\MX{0}{1}{-1}{0}){\mathbb I}i_w(t_1)\omega^{-1}{\mathbb I}i_w^{-1}(a)d^\x a dt_1\\
&=\omega(-\det\varsigma)\omega^{-1}{\mathbb I}i_w^{-2}(d_F)\cdot \Psi(\mathbf 1half,W,{\mathbb I}i_w)\Psi(\mathbf 1half,\widehat W,{\mathbb I}i_w^{-1})\\
&=\omega(\det\varsigma)\omega^{-1}{\mathbb I}i_w^{-2}(-d_F)\Psi(\mathbf 1half,W,{\mathbb I}i_w)^2\cdot \varepsilonsilonp(\mathbf 1half,\pi\otimes{\mathbb I}i_w,\psi)\cdot \frac{L(\mathbf 1half,\pi^\vee\otimes{\mathbb I}i_w^{-1})}{L(\mathbf 1half,\pi\otimes{\mathbb I}i_w)}.
\varepsilonsilonnd{align*}
The lemma thus follows.
\varepsilonsilonnd{proof}
\begin{prop}{\ell}abel{P:toric_split.W}We have
\begin{align*}\frac{1}{L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)}\cdot{\ell}ocalP{\pi(\varsigma){\ell}ocalW{{\mathbb I}i,v}}{{\mathbb I}i}=&abelian variety s{\mathcal D_F}\cdot\begin{cases}\varepsilonsilonp(\mathbf 1half,\pi\otimes{\mathbb I}i_w,\psi)\omega^{-1}{\mathbb I}i_w^{-2}(-2\delta)
&\text{ if } v\mid\mathfrak N^+,\\
\omega(\det\varsigma)&\text{ if } v\nmid \mathfrak N^+.
\varepsilonsilonnd{cases}
\intertext{If $v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}$ with $w\in{\varSigma}bar_p$, then }
\frac{1}{L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)}\cdot{\ell}ocalP{\pi(\varsigma){\ell}ocalW{{\mathbb I}i,v}^\flat}{{\mathbb I}i}=&\frac{\varepsilonsilonp(\mathbf 1half,\pi\otimes{\mathbb I}i_w,\psi)}{L(\mathbf 1half,\pi\otimes{\mathbb I}i_w)^2}\cdot \omega^{-1}{\mathbb I}i_w^{-2}(-2\delta)abelian variety s{\mathcal D_F}.\varepsilonsilonnd{align*}
\varepsilonsilonnd{prop}
\begin{proof}
The proposition follows immediately from {\ell}mref{L:key.W}. \varepsilonsilonqref{E:DeflocalWsplit.W} and \varepsilonsilonqref{E:10.W} combined with our choices of $d_F$ for $v|p\mathfrak N^+$ and the fact that
\[\varepsilonsilonp(\mathbf 1half,\pi\otimes{\mathbb I}i_w,\psi)\cdot\omega^{-1}{\mathbb I}i_w^{-2}(-d_F)=1 \text{ if }v\nmid \mathfrak N^+.\qedhere\]
\varepsilonsilonnd{proof}
\subsection{Local toric period integrals (II)}{\ell}abel{SS:toricII.W}
In this subsection, we treat the case $v$ is inert or ramified. A large part of the computation in this subsection is inspired by \cite{Murase:WaldII}. Let
\begin{align*}\mathbf w=&\MX{0}{-d_F^{-1}}{d_F}{0}\text{ and put}\\
K^0(\varpi):=&\stt{g=\MX{a}{b}{c}{d}\in K_v^0\mid a-1\in\varpi\mathcal O,\,c\in\varpi\mathcal D_F}.\varepsilonsilonnd{align*}
Let $\cO_\cKbasis=\cO_\cKbasis_v\in \mathcal O_E$ be the element chosen in \subsecref{SS:choiceofcmpt} and write $W^0$ for the new local Whittaker function $W^0_v$ at $v$. Recall that $\stt{1,\cO_\cKbasis}$ is an $\mathcal O$-basis of $\mathcal O_E$ and $\cO_\cKbasis$ is a uniformizer if $E/F$ is ramified. 
\subsubsection{}
 We prepare some elementary lemmas.
\begin{lm}{\ell}abel{L:coset.W}Suppose that $v|\frakr$. Let $m$ be a non-negative integer and let
\begin{align*}B^1(\mathcal O)=&\stt{\MX{1}{x}{0}{d}\mid x\in\mathcal D_F^{-1},\,d\in\mathcal O^{\times}},\\
N(\mathcal D_F^{-1})=&\stt{\MX{1}{x}{0}{1}\mid x\in\mathcal D_F^{-1}}.
\varepsilonsilonnd{align*}
If $y\in\varpi^{m+1}\mathcal O$, then we have \[
\bfd(\varpi^{m})\iota_{\varsigma}(1+y\cO_\cKbasis)\bfd(\varpi^{-m})\in K^0(\varpi).\]
If $y\in\varpi^r\mathcal O^{\times}$ and $0{\ell}eq r{\ell}eq m$, then \begin{align*}
N(\mathcal D_F^{-1})\bfd(\varpi^{m})\iota_{\varsigma}(1+y\cO_\cKbasis)\bfd(\varpi^{-m}) B^1(\mathcal O)=&
N(\mathcal D_F^{-1})\DII{\varpi^{m-r}}{y\varpi^{-m}}\mathbf w B^1(\mathcal O).

\intertext{If $y\in \varpi\mathcal O$, then  }
N(\mathcal D_F^{-1})\bfd(\varpi^{m})\iota_\varsigma(y+\cO_\cKbasis)\bfd(\varpi^{-m}) B^1(\mathcal O)=&

N(\mathcal D_F^{-1})\DII{\varpi^{m+e_v-1}}{\varpi^{-m}}\mathbf w B^1(\mathcal O).

\varepsilonsilonnd{align*}
\varepsilonsilonnd{lm}
\begin{proof} Recall that if $v|\frakr$, then $\cO_\cKbasis=d_{F}^{-1}\mathbf CMP$, $\varsigma=\DII{d_F}{d_F^{-1}}$, and hence 
\[\iota_{\varsigma}(x+y\cO_\cKbasis)=\MX{x+y{\mathrm T}(\cO_\cKbasis)}{yd_F^{-1}\mathrm N(\cO_\cKbasis)}{yd_F}{x}\quad(x,y\in F).\]
Then the proof is a straightforward calculation, so we omit the details.

\varepsilonsilonnd{proof}

\begin{lm}{\ell}abel{L:halfsum.W}Suppose that ${\mathbb I}i|_{F^{\times}}$ is trivial on $1+\varpi\mathcal O$. For each non-negative integer $r$, we set
\[X_r:=\int_{\varpi^r\mathcal O}{\mathbb I}i(1+y\cO_\cKbasis)d'y,\]
where $d'y$ is the Haar measure on $\mathcal O$ such that $\vol(\mathcal O,d'y)=L(1,\tau_{E/F})abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}$. Then $X_r=0$ if $c_v({\mathbb I}i)>1$ and $0<r<c_v({\mathbb I}i)$ and $X_{r}=abelian variety s{\varpi^r}\cdot L(1,\tau_{E/F})abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}$ if $r\geq c_v({\mathbb I}i)$.
\varepsilonsilonnd{lm}
\begin{proof}Let $Q_r:=1+\varpi^r\mathcal O_E/1+\varpi^r\mathcal O$. If $0<r<c_v({\mathbb I}i)$, then ${\mathbb I}i$ is a non-trivial character on the group $Q_r$. Note that we have a bijection $\varpi^r\mathcal O\stackrel{\sim}{\to} Q_r,\,y\mapsto 1+y\cO_\cKbasis$ and the pull-back of the quotient measure $dt$ on $Q_r$ is $d'y$. Therefore, we have
\[\int_{\varpi^r\mathcal O}d'y=\int_{Q_r}{\mathbb I}i(t)dt=\begin{cases} 0&\text{ if }0<r<c_v({\mathbb I}i)\\
\vol(\varpi^r\mathcal O,d'y)&\text{ if }r\geq c_v({\mathbb I}i).\varepsilonsilonnd{cases}\]
This finishes the proof.
\varepsilonsilonnd{proof}
Define the matrix coefficient $\bfm^0:\GL_2(F)\to\mathbf C$ by
\begin{align*}\bfm^0(g):=&\bfb_v(\pi(g)W^0,\pi(\DII{-1}{1})W^0)\\
=&\int_{F^{\times}}W^0(\bfd(a)g)W^0(\bfd(a))\omega^{-1}(a)d^\x a.
\varepsilonsilonnd{align*}
Since $W^0$ is invariant by $K^0(\varpi)$, $\bfm^0(g)$ only depends on the double coset $K^0(\varpi)gK^0(\varpi)$ by \varepsilonsilonqref{E:bilinear.W}.  Put \[m=m_v({\mathbb I}i,\pi)=c_v({\mathbb I}i)-v(\mathfrak n^-)\geq -1.\] We set
\begin{equation}{\ell}abel{E:def.W}\begin{aligned}\boldsymbol{P}^*(\pi(\varsigma)\cR_v^{m}W^0,{\mathbb I}i)&:={\ell}ocalP{\pi(\varsigma)\cR_v^{m}W^0}{{\mathbb I}i}\cdot \frac{\zeta_F(1)}{L(1,\tau_{E/F})}\omega(\varpi^{-m}\det\varsigma^{-1})\\
&=\int_{E^{\times}/F^{\times}}\bfb_v(\cR_v^{-m}\pi(\iota_\varsigma(t))\cR_v^{m}W^0,\pi(\DII{-1}{1})W^0){\mathbb I}i(t)dt\\
&=\int_{E^{\times}/F^{\times}}\bfm^0(\bfd(\varpi^{m}) \iota_\varsigma(t) \bfd(\varpi^{-m}) ){\mathbb I}i(t)dt.
\varepsilonsilonnd{aligned}
\varepsilonsilonnd{equation}
Here we have used the fact that $m+v({\mathrm T}(\cO_\cKbasis))\geq 0$ in the second equality. It follows immediately from the definition of the projector ${\mathcal P}_{{\mathbb I}i,\varsigma}$ that
\begin{equation}{\ell}abel{E:7.W}\begin{aligned}{\ell}ocalP{\pi(\varsigma){\ell}ocalW{{\mathbb I}i,v}}{{\mathbb I}i}=& {\ell}ocalP{\pi(\varsigma){\mathcal P}_{{\mathbb I}i,\varsigma}\cR^m_vW^0}{{\mathbb I}i}\\
=&\boldsymbol{P}^*(\pi(\varsigma)\cR_v^{m} W^0,{\mathbb I}i)\cdot\frac{\omega(\varpi^{m}\det\varsigma)L(1,\tau_{E/F})}{\zeta_F(1)}.\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}

Using the decomposition \[E^{\times}=F^{\times}(1+\mathcal O\cO_\cKbasis)\bigsqcup F^{\times}(\varpi\mathcal O+\cO_\cKbasis)\]
and {\ell}mref{L:coset.W}, we find that
\begin{equation}{\ell}abel{E:long.W}
\begin{aligned}
\boldsymbol{P}^*(\pi(\varsigma)\cR_v^{m}W^0,{\mathbb I}i)=&\int_{\mathcal O}{\mathbb I}i(1+y\cO_\cKbasis)\bfm^0(\bfd(\varpi^m)\iota_\varsigma(1+y\cO_\cKbasis)\bfd(\varpi^{-m}) )d'y\\
&+\int_{\varpi\mathcal O}{\mathbb I}i(x+\cO_\cKbasis)\bfm^0(\bfd(\varpi^m)\iota_\varsigma (y+\cO_\cKbasis)\bfd(\varpi^{-m}) )abelian variety s{y+\cO_\cKbasis}_E^{-1}d'y\\
=&X_{m+1}\cdot\bfm^0(1)+\sum_{r=0}^{m}\int_{\varpi^r\mathcal O^{\times}}{\mathbb I}i(1+y\cO_\cKbasis)\omega(\varpi^{-m}y)d'y\cdot \bfm^0(\bfd(\varpi^{2(m-r)})\mathbf w)\\
&+Y_0\cdot \omega(\varpi^{-m})\bfm^0(\bfd(\varpi^{2m+e_v-1})\mathbf w),
\varepsilonsilonnd{aligned}
\varepsilonsilonnd{equation}
where
\[Y_0:=\int_{\varpi\mathcal O}{\mathbb I}i(y+\cO_\cKbasis)d'y\cdotabelian variety s{\varpi}^{1-e_v}.\]
In what follows, we use {\ell}mref{L:halfsum.W} and \varepsilonsilonqref{E:long.W} to calculate ${\ell}ocalP{\pi(\varsigma){\ell}ocalW{{\mathbb I}i,v}}{{\mathbb I}i}$.
\subsubsection{The case $v\nmid \mathfrak n^-_r$}
Suppose that $v\nmid \mathfrak n^-_r$, i.e.  the central character $\omega$ is unramified. Then \varepsilonsilonqref{sf} implies that $\pi$ is either an unramified principal series or an unramified special representation.
\begin{prop}{\ell}abel{P:toric_inert_un.W}Suppose that $\pi$ is an unramified principal series. Then
\begin{align*}\frac{1}{L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)}\cdot{\ell}ocalP{\pi(\varsigma){\ell}ocalW{{\mathbb I}i,v}}{{\mathbb I}i}=\omega(\varpi^{m})abelian variety s{\varpi^{c_v({\mathbb I}i)}}abelian variety s{\mathcal D_E}_E^\mathbf 1half\cdot \begin{cases}\omega(\det\varsigma)&\text{ if } c_v({\mathbb I}i)=0,\\
L(1,\tau_{E/F})^2&\text{ if } c_v({\mathbb I}i)>0.\varepsilonsilonnd{cases}\varepsilonsilonnd{align*}
\varepsilonsilonnd{prop}
\begin{proof} Since $\pi$ is unramified, $\omega$ is unramified and $m=c_v({\mathbb I}i)$. Write $\pi=\pi(\mu,\nu)$ and let $\alpha=\mu(\varpi)$ and $\beta=\nu(\varpi)$.
The matrix coefficient $\bfm^0$ is a spherical function on $\GL_2(F)$ in the sense of \cite[Definition 4.1, p.150]{Cartier:corvallis}, and $\bfm^0(g)$ only depends on the double coset $K^0_vgK^0_v$. By Macdonald formula,
\begin{align}{\ell}abel{E:MCF1.W}\bfm^0(1)=&\frac{\zeta_F(1)L(1,\mathbf Ad\pi)}{\zeta_F(2)}\cdotabelian variety s{\mathcal D_F}^\mathbf 1half
=\frac{(1+abelian variety s{\varpi})\zeta_F(1)}{(1-\alpha\beta^{-1}abelian variety s{\varpi})(1-\alpha^{-1}\betaabelian variety s{\varpi})}\cdotabelian variety s{\mathcal D_{F}}^\mathbf 1half;\\
{\ell}abel{E:2W}
\bfm^0(\bfd(\varpi))=&\frac{abelian variety s{\varpi}^\mathbf 1half}{1+abelian variety s{\varpi}}\cdot (\alpha+\beta)\cdot \bfm^0(1);\\
{\ell}abel{E:3W}\bfm^0(\bfd(\varpi^2))=&\frac{abelian variety s{\varpi}}{1+abelian variety s{\varpi}}\cdot (\alpha^2+\beta^2+(1-abelian variety s{\varpi})\alpha\beta)\cdot \bfm^0(1).\varepsilonsilonnd{align}
If $v$ is inert and $m=0$, then
\begin{align*}\omega(\det\varsigma^{-1}){\ell}ocalP{\pi(\varsigma)W^0}{{\mathbb I}i}=&\bfm^0(1)\cdot \frac{L(1,\tau_{E/F})}{\zeta_F(1)}\cdotabelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}\\
=&\frac{1}{(1-\alpha\beta^{-1}abelian variety s{\varpi})(1-\alpha^{-1}\betaabelian variety s{\varpi})}\cdotabelian variety s{\mathcal D_E}_E^{\mathbf 1half}\\
=&L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)\cdot abelian variety s{\mathcal D_E}_E^\mathbf 1half.\varepsilonsilonnd{align*}

Suppose that either $v$ is ramified or $m>0$ (so $v|\frakr$ and $\det\varsigma=1$). Then we deduce from \varepsilonsilonqref{E:long.W} that
\begin{equation}{\ell}abel{E:16.W}\begin{aligned}\boldsymbol{P}^*(\pi(\varsigma)\cR_v^{m}W^0,{\mathbb I}i)=&X_m\cdot\bfm^0(1)+\sum_{r=0}^{m-1}(X_r-X_{r+1})\omega(\varpi^{r-m})\cdot\bfm^0(\bfd(\varpi^{2(m-r)}))\\
&+Y_0\cdot \omega(\varpi^{-m})\bfm^0(\bfd(\varpi^{2m+e_v-1})).
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
If $v$ is ramified and $m=0$, then $X_0=abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}$ and $Y_0={\mathbb I}i(\varpi_E)abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}$. By \varepsilonsilonqref{E:16.W}, we find that
\begin{align*}
{\ell}ocalP{\pi(\varsigma)W^0}{{\mathbb I}i}
=&(\bfm^0(1)+{\mathbb I}i(\varpi_E)\bfm^0(\bfd(\varpi)))\frac{L(1,\tau_{E/F})}{\zeta_F(1)}abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}\\
=&(1+\frac{\alpha+\beta}{1+abelian variety s{\varpi}}\cdotabelian variety s{\varpi}^\mathbf 1half{\mathbb I}i(\varpi_E))\cdot\bfm^0(1)\cdot\frac{abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}}{\zeta_F(1)}\quad(\text{by \varepsilonsilonqref{E:2W}})\\
=&\frac{(1+{\mathbb I}i(\varpi_E)\alphaabelian variety s{\varpi}^\mathbf 1half)(1+{\mathbb I}i(\varpi_E)\betaabelian variety s{\varpi}^\mathbf 1half)}{1+abelian variety s{\varpi}}\cdot\bfm^0(1)\cdot\frac{abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}}{\zeta_F(1)}\\
=&abelian variety s{\mathcal D_E}_E^\mathbf 1half\cdot L(\mathbf 1half,\pi_E\otimes{\mathbb I}i).
\varepsilonsilonnd{align*}
Suppose that $m>0$. Note that since ${\mathbb I}i|_{\mathcal O^{\times}}=1$, $Y_0=-X_0$ if $v$ is inert and $Y_0=X_0=0$ if $v$ is ramified. Combining with {\ell}mref{L:halfsum.W}, \varepsilonsilonqref{E:3W} and \varepsilonsilonqref{E:16.W}, we find that
\begin{align*}
{\ell}ocalP{\pi(\varsigma)\cR_v^mW^0}{{\mathbb I}i}=&X_m\cdot (\bfm^0(1)-\omega(\varpi^{-1})\bfm^0(\bfd(\varpi^2))\cdot\omega(\varpi^{-m}) \frac{L(1,\tau_{E/F})}{\zeta_F(1)}\\
=&\omega(\varpi^{m})abelian variety s{\varpi^m}\cdot\frac{(1-\alpha\beta^{-1}abelian variety s{\varpi})(1-\alpha^{-1}\betaabelian variety s{\varpi})}{1+abelian variety s{\varpi}}\cdot \bfm^0(1)\\
&\times \frac{L(1,\tau_{E/F})^2}{\zeta_F(1)}abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}\\
=&\omega(\varpi^{m})abelian variety s{\varpi^m}abelian variety s{\mathcal D_E}_E^\mathbf 1half\cdot L(1,\tau_{E/F})^2.
\varepsilonsilonnd{align*}
The proposition follows immediately.
\varepsilonsilonnd{proof}

\begin{prop}{\ell}abel{P:toric_inert_sp.W}Suppose that $\pi$ is an unramified special representation. Then
\begin{align*}\frac{1}{L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)}\cdot {\ell}ocalP{\pi(\varsigma){\ell}ocalW{{\mathbb I}i,v}}{{\mathbb I}i}=&\omega(\varpi^{m})abelian variety s{\varpi^{c_v({\mathbb I}i)}}abelian variety s{\mathcal D_E}_E^\mathbf 1half\\
&{\times} \begin{cases}L(1,\tau_{E/F})^2&\text{ if $c_v({\mathbb I}i)>0$},\\
2&\text{ if $v$ is ramified and $c_v({\mathbb I}i)=0$.}\varepsilonsilonnd{cases}\varepsilonsilonnd{align*}
\varepsilonsilonnd{prop}
\begin{proof} Suppose that $v|\mathfrak n^-_s$. Then $m=m_v({\mathbb I}i,\pi)=c_v({\mathbb I}i)-1$. Recall that $\pi=\pi(\mu,\nu)$ is a special representation with a unramified character $\mu$ and $\mu\nu^{-1}=\mathbf Abs$. We have
\begin{align*} W^0(\bfd(a))=&\mu(a)abelian variety s{a}^\mathbf 1half\mathbb I_{\mathcal O}(a),\\
W^0(\bfd(a)\mathbf w)=&-\mu(a)abelian variety s{a}^\mathbf 1halfabelian variety s{\varpi}\mathbb I_{\varpi^{-1}\mathcal O}(a).

\varepsilonsilonnd{align*}
A direct computation shows that
\[\bfm^0(1)=\frac{abelian variety s{\mathcal D_{F}}^\mathbf 1half}{1-abelian variety s{\varpi}^2}\,;\,\bfm^0(\mathbf w)=(-abelian variety s{\varpi})\cdot \bfm^0(1)\,;\,\bfm^0(\DII{1}{\varpi}\mathbf w)=(-\mu(\varpi)abelian variety s{\varpi}^{-\mathbf 1half})\cdot\bfm^0(1).\]
If $c_v({\mathbb I}i)>0$, then it follows from \varepsilonsilonqref{E:long.W} and {\ell}mref{L:halfsum.W} that
\begin{align*}{\ell}ocalP{\pi(\varsigma)\cR_v^mW^0}{{\mathbb I}i}=&X_{m+1}\cdot(\bfm^0(1)-\bfm^0(\mathbf w))\cdot\omega(\varpi^{m})\frac{L(1,\tau_{E/F})}{\zeta_F(1)}\\
=&\omega(\varpi^{m})abelian variety s{\varpi^{m+1}}\cdotabelian variety s{\mathcal D_E}_E^\mathbf 1half L(1,\tau_{E/F})^2.
\varepsilonsilonnd{align*}
If $c_v({\mathbb I}i)=0$ ($m=-1$), then $v$ is ramified, $X_0=abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half},$, $Y_0={\mathbb I}i(\varpi_E)abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half},$ and
\begin{align*}
{\ell}ocalP{\pi(\varsigma)\cR_v^mW^0}{{\mathbb I}i}&={\ell}eft(X_0\cdot \bfm^0(1)+Y_0\cdot \bfm^0(\DII{1}{\varpi}\mathbf w)\right)\omega(\varpi^{-1})\cdot\frac{L(1,\tau_{E/F})}{\zeta_F(1)}\\
&=(1-\mu(\varpi){\mathbb I}i(\varpi_E)abelian variety s{\varpi}^{-\mathbf 1half})abelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}\cdot \bfm^0(1)(1-abelian variety s{\varpi})\omega(\varpi^{-1})\\
&=\frac{2abelian variety s{\mathcal D_E}_E^\mathbf 1half\omega(\varpi^{-1})}{1+abelian variety s{\varpi}}\quad(\text{by (R1)})\\
&=2abelian variety s{\mathcal D_E}_E^\mathbf 1half\omega(\varpi^{-1})\cdot L(\mathbf 1half,\pi_E\otimes{\mathbb I}i).\qedhere
\varepsilonsilonnd{align*}
\varepsilonsilonnd{proof}

\subsubsection{The case $v|\mathfrak n^-_r$}
We consider the case $\pi$ is a ramified principal series. Recall that \varepsilonsilonqref{sf} suggests that $\pi=\pi(\mu,\nu)$, where $\mu$ is unramified and $\nu$ is ramified, and the conductor $a(\nu)=a(\omega)=1$. Since ${\mathbb I}i|_{F^{\times}}=\omega^{-1}$, we must have $m=c_v({\mathbb I}i)-1\geq 0$. Let $\boldsymbol{\delta}_v:=\cO_\cKbasis-\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\cO_\cKbasis}$. Let $D_{E/F}$ be the discriminant of $E/F$. We begin with a lemma.
\begin{lm}{\ell}abel{L:1.W}Suppose that ${\mathbb I}i|_{\mathcal O^{\times}}\not =1$ and ${\mathbb I}i|_{1+\varpi\mathcal O}=1$. Then
\[\int_{\varpi^{-m}\mathcal O}{\mathbb I}i(y+\cO_\cKbasis)d'y={\mathbb I}i(\boldsymbol{\delta}_v)abelian variety s{\boldsymbol{\delta}_v}_E^{\mathbf 1half}\cdot\frac{\varepsilonsilonp(0,{\mathbb I}i^{-1},\psi_E)}{\varepsilonsilonp(-1,\omega,\psi)}\cdot L(1,\tau_{E/F})abelian variety s{D_{E/F}}^\mathbf 1half.\]
\varepsilonsilonnd{lm}
\begin{proof}
 By \cite[Prop.\,8.2]{Harris:Theta_dichotomy}, we have
\begin{align*}
\int_F{\mathbb I}i(y+2^{-1}\boldsymbol{\delta}_v)dy:=&\int_F{\mathbb I}i(y+2^{-1}\boldsymbol{\delta}_v)abelian variety s{y+2^{-1}\boldsymbol{\delta}_v}_E^{-s-\mathbf 1half}dy|_{s=-\mathbf 1half}\\
=&{\mathbb I}i(\boldsymbol{\delta}_v)abelian variety s{\boldsymbol{\delta}_v}_E^{\mathbf 1half}\cdot\frac{\varepsilonsilonp(0,{\mathbb I}i^{-1},\psi_E)}{\varepsilonsilonp(-1,\omega,\psi)}.
\varepsilonsilonnd{align*}
By the assumption, for all $r\geq m+1$ we have
\[\int_{\varpi^{-r}\mathcal O^{\times}}{\mathbb I}i(y+\cO_\cKbasis)dy={\mathbb I}i(\varpi^{-r})\cdot \int_{\mathcal O^{\times}}{\mathbb I}i(y)dy=0.\]
Thus \[\int_{\varpi^{-m}\mathcal O}{\mathbb I}i(y+\cO_\cKbasis)dy=\varinjlim_r\int_{\varpi^{-r}\mathcal O}{\mathbb I}i(y+\cO_\cKbasis)dy=\int_{F}{\mathbb I}i(y+\cO_\cKbasis)dy=\int_F{\mathbb I}i(y+2^{-1}\boldsymbol{\delta}_v)dy.\]
The lemma follows from the fact that
\[d'y=L(1,\tau_{E/F})abelian variety s{D_{E/F}}^\mathbf 1half\cdot dy.\qedhere\]
\varepsilonsilonnd{proof}
\begin{prop}{\ell}abel{P:toric_ramified.W} Then we have
\[\frac{1}{L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)}\cdot {\ell}ocalP{\pi(\varsigma){\ell}ocalW{{\mathbb I}i,v}}{{\mathbb I}i}=abelian variety s{\varpi^{c_v({\mathbb I}i)}}abelian variety s{\mathcal D_E}_E^\mathbf 1half{\mathbb I}i(\boldsymbol{\delta}_v d_F^{-1})abelian variety s{\boldsymbol{\delta}_v}_E^\mathbf 1half\varepsilonsilonp(0,{\mathbb I}i,\psi_E)\cdot L(1,\tau_{E/F})^2\cdot n_v^2,\]
where $n_v$ is given by
\begin{equation}{\ell}abel{E:normalized.W}n_v:=\frac{\mu(\varpi)abelian variety s{\varpi}^{m/2}abelian variety s{\mathcal D_F}^{\frac{1}{4}}}{\varepsilonsilonp(0,\omega,\psi)}\in\mathbf Zbar_{(p)}^{\times}.\varepsilonsilonnd{equation}
\varepsilonsilonnd{prop}
\begin{proof} Note that $L(s,\pi_E\otimes{\mathbb I}i)=1$ and the conductor $a(\nu)=1$. A straightforward computation shows that
\begin{align*}W^0(\bfd(a))=&\nu\mathbf Abs^\mathbf 1half(a)\mathbb I_{\mathcal O}(a),\\
W^0(\bfd(a)\mathbf w)=&\mu\mathbf Abs^\mathbf 1half(a)\mathbb I_{\varpi^{-1}\mathcal O}(a)\cdot\frac{\omega(d_F)\mu(\varpi^2)}{\varepsilonsilonp(0,\omega,\psi)}.\varepsilonsilonnd{align*}
\[\bfm^0(1)=0,\,\bfm^0(\mathbf w)=\frac{\omega(d_F)\mu(\varpi^2)}{\varepsilonsilonp(0,\omega,\psi)\cdot (1-abelian variety s{\varpi})}\cdotabelian variety s{\mathcal D_{F}}^\mathbf 1half.\]

It is not difficult to show that if $m=c_v({\mathbb I}i)-1>0$, then
\begin{align*}\int_{\varpi^r\mathcal O^{\times}}{\mathbb I}i(y^{-1}+\cO_\cKbasis)d'y&=0 \text{ for }0<r<m\text{ and }\\
\int_{\mathcal O}{\mathbb I}i(y+\cO_\cKbasis)d'y&=0,\varepsilonsilonnd{align*}
and that if $v$ is ramified, then
\[Y_0=\int_{\varpi\mathcal O}{\mathbb I}i(y+\cO_\cKbasis)d'y=0.\]
From the above equations, we find that
\begin{align*}
\boldsymbol{P}^*(\pi(\varsigma)\cR_v^{m}W^0,{\mathbb I}i)
=&X_{m+1}\cdot\bfm^0(1)+\sum_{r=0}^{m}\int_{\varpi^r\mathcal O^{\times}}{\mathbb I}i(y^{-1}+\cO_\cKbasis)d'y\cdot\omega(\varpi^{-m})\bfm^0(\bfd(\varpi^{2m-2r})\mathbf w)\\
&+Y_0\cdot \omega(\varpi^{-m}) \bfm^0(\bfd(\varpi^{2m+e_v-1})\mathbf w)\\
=&\int_{\varpi^{-m}\mathcal O}{\mathbb I}i(y+\cO_\cKbasis)d'y\cdot \omega(\varpi^{-m})abelian variety s{\varpi^{2m}}\bfm^0(\mathbf w).
\varepsilonsilonnd{align*}
By {\ell}mref{L:1.W}, we obtain
\begin{align*}
{\ell}ocalP{\pi(\varsigma)\cR_v^{m}W^0}{{\mathbb I}i}=&\boldsymbol{P}^*(\pi(\varsigma)\cR_v^{m}W^0,{\mathbb I}i)\cdot\omega(\varpi^m)\frac{L(1,\tau_{E/F})}{\zeta_F(1)}\\
=&abelian variety s{\varpi^{2m}}\cdot {\mathbb I}i(\boldsymbol{\delta}_v)abelian variety s{\boldsymbol{\delta}_v}_E^\mathbf 1halfabelian variety s{\mathcal D_E}_E^\mathbf 1halfabelian variety s{\mathcal D_F}^{-\mathbf 1half}\cdot\frac{\varepsilonsilonp(0,{\mathbb I}i^{-1},\psi_E)}{\varepsilonsilonp(-1,\omega,\psi)}\cdot\frac{\mu(\varpi^2)\omega(d_F)}{\varepsilonsilonp(0,\omega,\psi)(1-abelian variety s{\varpi})}\cdot \frac{L(1,\tau_{E/F})^2}{\zeta_F(1)}\\
=&\frac{L(1,\tau_{E/F})^2\mu(\varpi^2)abelian variety s{\mathcal D_F}^\mathbf 1halfabelian variety s{\varpi^{2m+1}}}{\varepsilonsilonp(0,\omega,\psi)^2 }\cdot abelian variety s{\mathcal D_{E}}_E^\mathbf 1half{\mathbb I}i(\boldsymbol{\delta}_vd_F^{-1})abelian variety s{\boldsymbol{\delta}_v}_E^\mathbf 1half\varepsilonsilonp(0,{\mathbb I}i^{-1},\psi_E).
\varepsilonsilonnd{align*}
The last equality follows from
\[\varepsilonsilonp(-1,\omega,\psi)=abelian variety s{\varpi\mathcal D_F}^{-1}\varepsilonsilonp(0,\omega,\psi).\]
The proposition follows.
\varepsilonsilonnd{proof}
\def{\mathbb I}pi{\varPsi_{\pi,{\mathbb I}i,v}}
\subsection{Non-vanishing of the local Fourier coefficients}{\ell}abel{SS:localFourier}
The function ${\ell}ocalK{{\mathbb I}i,v}:F^{\times}\to\mathbf C$ defined by
\[{\ell}ocalK{{\mathbb I}i,v}(a)={\ell}ocalW{{\mathbb I}i,v}(\bfd(a))\] is called the \varepsilonsilonmph{local Fourier coefficient} associated to ${\ell}ocalW{{\mathbb I}i,v}$.
\begin{defn}[Normalized local Fourier coefficients]{\ell}abel{D:localFourier.W} Let \[B({\mathbb I}i)=\stt{v\in\mathbf h\mid\text{$v$ is non-split with $c_v({\mathbb I}i)>0$}}.\]
For $v\in B({\mathbb I}i)$, let $n_v$ be defined as in \varepsilonsilonqref{E:normalized.W} if $v|\mathfrak n^-_r$ and $n_v=1$ if $v\nmid \mathfrak n^-_r$. Define the normalized local Fourier coefficient ${\ell}ocalK{{\mathbb I}i,v}^*$ by
\[{\ell}ocalK{{\mathbb I}i,v}^*:= {\ell}ocalK{{\mathbb I}i,v}\cdot\begin{cases}n_v^{-1}L(1,\tau_{E_v/F_v})^{-1}&\text{ if } v\in B({\mathbb I}i),\\
1&\text{ if }v\in A({\mathbb I}i),\\
e_v&\text{ otherwise.}\varepsilonsilonnd{cases}\]
Recall that $e_v=1$ if $v$ is unramified and $e_v=2$ if $v$ is ramified.
\varepsilonsilonnd{defn}
Let $v\nmid p$ be a finite place. We are going to show the normalized local Fourier coefficients ${\ell}ocalK{{\mathbb I}i,v}^*$ indeed take value in a finite extension of $\mathbf Zp$ when regarded as a $\mathbf Cp$-valued function via $\iota_p:\mathbf C\simeq\mathbf Cp$ and is not identically zero modulo $\mathfrak m$. This is clear if $v$ is split in view of the definition of ${\ell}ocalK{{\mathbb I}i,v}^*={\ell}ocalK{{\mathbb I}i,v}$ in \subsecref{SS:Whittaker.W}. We give the formulae of ${\ell}ocalK{{\mathbb I}i,v}$.
\begin{lm}Suppose that $c_v({\mathbb I}i)=0$. Then
\[{\ell}ocalK{{\mathbb I}i,v}^*(a)=\begin{cases}W^0_v(\bfd(a))&\text{ if $v\nmid \mathfrak n$ is unramified,}\\
W^0_v(\bfd(a))+W^0_v(\bfd(a\varpi_v)){\mathbb I}i(\varpi_{E_v})&\text{ if $v\nmid \mathfrak n$ is ramified.}\varepsilonsilonnd{cases}\]
If $v\mid \mathfrak n$, then $v$ is ramified and
\[{\ell}ocalK{{\mathbb I}i,v}^*(a)= \mu\mathbf Abs^\mathbf 1half(a)\mathbb I_{\varpi^{-1}\mathcal O}(a).\]
\varepsilonsilonnd{lm}
\begin{proof} It is well-known that if $\pi=\pi(\mu,\nu)$ is a unramified principal series, then
\[W^0_v(\bfd(a))=\mathbb I_{\mathcal O}(a)abelian variety s{a}^\mathbf 1half\cdot \sum_{i+j=v(a),\,i,j\geq 0}\mu(\varpi^i)\nu(\varpi^j)\]
(\mbox{{\it cf.} } \cite[Thm.\,4.6.5]{Bump:AutoRep}). It follows from the definition of ${\ell}ocalW{{\mathbb I}i,v}$ that ${\ell}ocalW{{\mathbb I}i,v}=W^0_v$ if $v\nmid \mathfrak n$ is unramified and
\[{\ell}ocalW{{\mathbb I}i,v}(g)=\mathbf 1half \cdot W^0_v(g)+\mathbf 1half\cdot W^0_v(g\bfd(\varpi)){\mathbb I}i(\varpi_E)\text{ if $v\nmid \mathfrak n$ is ramified }.\]
If $v|\mathfrak n$, then $v\in A({\mathbb I}i)$. By \varepsilonsilonqref{R1} $v$ is ramified, and we find that
\[{\ell}ocalW{{\mathbb I}i,v}(g)=\mathbf 1half\cdot W^0_v(g\mathbf d(\varpi))+\mathbf 1half\cdot W^0_v(g\mathbf w)\omega(\varpi){\mathbb I}i(\varpi_E).\]
The assertion follows from the formulas of $W^0_v$ in \indent opref{P:toric_inert_sp.W}.
\varepsilonsilonnd{proof}
To treat the case $v$ is non-split with $c_v({\mathbb I}i)>0$, i.e.  $v\in B({\mathbb I}i)$, we need to introduce certain partial Gauss sums. For a non-split place $v$, write $\pi=\pi(\mu,\nu)$ with unramified $\mu$ and $\mu\nu^{-1}(\varpi)\not =abelian variety s{\varpi}^{-1}$ if $\pi$ is unramified or special. Define a character ${\mathbb I}pi:E^{\times}\to\mathbf C^{\times}$ by
\begin{equation}{\ell}abel{E:char.W}{\mathbb I}pi(t):=\mu(\mathrm N(t))\cdot{\mathbb I}i\mathbf Abs_E^\mathbf 1half(t).\varepsilonsilonnd{equation}
Recall that the partial Gauss sum $\widetilde A_\beta({\mathbb I}pi)$ in \cite[(4.17)]{Hsieh:Hecke_CM} is defined by
\[\widetilde A_\beta({\mathbb I}pi):={\ell}im_{n\to\infty}\int_{\varpi^{-n}\mathcal O}{\mathbb I}pi^{-1}(x+\cO_\cKbasis)\psi(-d_F^{-1}\beta x)dx \quad(\beta\in F^{\times}).\]
\begin{lm}{\ell}abel{L:FCformula.W}Let $v\in B({\mathbb I}i)$ be a non-split place with $c_v({\mathbb I}i)>0$. Then we have
\begin{align*}\frac{e_v}{L(1,\tau_{E/F})}\cdot {\ell}ocalK{{\mathbb I}i,v}(a)=&\widetilde A_a({\mathbb I}pi)\nu\mathbf Abs^\mathbf 1half(a)\cdot \nu\mathbf Abs^\mathbf 1half(\varpi^m)abelian variety s{D_{F}}^{-\mathbf 1half}\\
&\times \begin{cases}1&\text{ if } v\nmid \mathfrak n^-,\\
-1&\text{ if } v\mid \mathfrak n^-_s,\\
abelian variety s{\varpi^m}{\mathbb I}i(\boldsymbol{\delta}_v)abelian variety s{\boldsymbol{\delta}_v}_E^\mathbf 1half\frac{\varepsilonsilonp(0,{\mathbb I}i^{-1},\psi_E)}{\varepsilonsilonp(-1,\omega,\psi)}abelian variety s{\mathcal D_F}^{-\mathbf 1half}&\text{ if } v\mid \mathfrak n^-_r.
\varepsilonsilonnd{cases}\varepsilonsilonnd{align*}
\varepsilonsilonnd{lm}
\begin{proof}
We recall the Whittaker linear functional $\Lambda:I(\mu,\nu)\to\mathbf C$ (\cite[(6.9), p.498]{Bump:AutoRep}) is defined by
\[\Lambda(f)=\int_{F^{\times}}f(\MX{0}{-1}{1}{x})\psi(-x)dx:={\ell}im_{n\to\infty}\int_{\varpi^{-n}\mathcal O}f(\MX{0}{-1}{1}{x})\psi(-x)dx.\]
Let $\varsigma=\varsigmav=\DII{d_F}{d_F^{-1}}$ and $m=m_v({\mathbb I}i,\pi)$. Define ${\mathcal P}_{{\mathbb I}i,\varsigma} \cR_v^m\in\End_\mathbf C I(\mu,\nu)$ by
\[{\mathcal P}_{{\mathbb I}i,\varsigma} \cR_v^m f(g)=\vol(E^{\times}/F^{\times},dt)^{-1}\int_{E^{\times}/F^{\times}}f(g\DII{1}{\varpi^m}\iota_\varsigma(t))dt.\]
We are going to choose a section $f^0$ in $I(\mu,\nu)^{K^0(\varpi)}$ such that
\[W^0_v(g)=\Lambda(\pi(g)f^0).\]
Let $f^0_{\mathbb I}i={\mathcal P}_{{\mathbb I}i,\varsigma}\cR_v^m f^0$. Then it is not difficult to see that
\[{\ell}ocalK{{\mathbb I}i,v}(a)={\ell}ocalW{{\mathbb I}i,v}(\bfd(a))=\Lambda(\pi(\bfd(a))f^0_{\mathbb I}i).\]
Put \[f^0_{\mathbb I}i(\varsigma)^*:=\nu^{-1}\mathbf Abs^\mathbf 1half(\varpi^m)\bfv_E\cdot  f^0_{\mathbb I}i(\varsigma)\quad(\bfv_E=e_vabelian variety s{\mathcal D_E}_E^{\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}).\]
Therefore we have
\begin{align*}{\ell}ocalK{{\mathbb I}i,v}(a)=&\nu\mathbf Abs^\mathbf 1half(a)\int_{F}f^0_{\mathbb I}i(\MX{0}{-1}{1}{x})\psi(-ax)dx\\
=&f^0_{\mathbb I}i(\DII{d_F}{d_F^{-1}})abelian variety s{\mathcal D_F}^{-1}\cdot\nu\mathbf Abs^\mathbf 1half(a)\int_{F}{\mathbb I}pi^{-1}(x+\cO_\cKbasis)\psi(-d_F^{-1}ax)dx\\
=&f^0_{\mathbb I}i(\varsigma)^*\cdot \nu\mathbf Abs^\mathbf 1half(a)\widetilde A_a({\mathbb I}pi)\cdot e_v^{-1}abelian variety s{\mathcal D_E}_E^{-\mathbf 1half}abelian variety s{\mathcal D_F}^{-\mathbf 1half}\nu\mathbf Abs^{-\mathbf 1half}(\varpi^m).
\varepsilonsilonnd{align*}
It remains to determine $f^0$ and $f^0_{\mathbb I}i(\varsigma)$. Since $W^0$ is uniquely characterized by the property that $W^0\in{\mathcal W}(\pi,\psi)$ is right $K^0(\varpi)$-invariant with $W^0(1)=1$, it suffices to construct $f^0\in \pi(\mu,\nu)^{K^0(\varpi)}$ with $\Lambda(f^0)=1$. To calculate $f^0_{\mathbb I}i(\varsigma)$, following \varepsilonsilonqref{E:long.W} we have
\begin{align*}f^0_{\mathbb I}i(\varsigma)^*
=&\int_{\mathcal O}{\mathbb I}i(1+y\cO_\cKbasis)f^0(\varsigma\cdot \cR_v^{-m}\iota_\varsigma(1+y\cO_\cKbasis)\cR_v^m )d'y
+\int_{\varpi\mathcal O}{\mathbb I}i(y+\cO_\cKbasis)f^0(\varsigma\cdot \cR_v^{-m}\iota_\varsigma(y+\cO_\cKbasis)\cR_v^m)abelian variety s{\mathcal D_E}_E^{-1}d'y\\
=&X_{m+1}\cdot f^0(\varsigma)+\sum_{r=0}^m \int_{\varpi^r\mathcal O^{\times}}{\mathbb I}i(1+y\cO_\cKbasis)\omega(\varpi^{-m}y)d'y\cdot f^0(\varsigma\bfd(\varpi^{2(m-r)}))\\
&+Y_0\cdot \omega(\varpi^{-m})f^0(\varsigma\bfd(\varpi^{2m+e_v-1})).
\varepsilonsilonnd{align*}

Suppose that $\pi$ is a unramified principal series ($v\nmid \mathfrak n^-$) or special representation ($v\mid\mathfrak n^-_s$). Then \begin{align*}f^0_{\mathbb I}i(\varsigma)^*
=&X_m\cdot f^0(\varsigma)+\sum_{r=0}^{m-1}(X_r-X_{r+1})\cdot\omega(\varpi^{r-m})\cdot f^0(\varsigma\bfd(\varpi^{2(m-r)}))\\
&+Y_0\cdot \omega(\varpi^{-m})f^0(\varsigma\bfd(\varpi^{2m+e_v-1})).\varepsilonsilonnd{align*}

Let $f^{sph}$ be the unique $K^0_v$-invariant function in $I(\mu,\nu)$ with $f^{sph}(\varsigma)=L(1,\mu\nu^{-1})abelian variety s{\mathcal D_{F}}^\mathbf 1half$.
If $\pi$ is an unramified principal series, then we can take $f^0=f^{sph}$ (\cite[Prop.\,4.6.8]{Bump:AutoRep}), and following the computation of the case $c_v({\mathbb I}i)>0$ in \indent opref{P:toric_inert_un.W} we find that
\begin{align*}f^0_{\mathbb I}i(\varsigma)^*

=&X_m\cdot (f^0(\varsigma)-\omega(\varpi^{-1})f^0(\bfd(\varpi^2)\varsigma))=X_m \cdot(1-\mu\nu^{-1}\mathbf Abs(\varpi))\cdot f^0(\varsigma)\\
=&abelian variety s{\varpi^m}abelian variety s{\mathcal D_E}_E^\mathbf 1half L(1,\tau_{E/F}).
\varepsilonsilonnd{align*}
If $\pi$ is special, then
\[f^0=f^{sph}-\mu^{-1}\mathbf Abs^\mathbf 1half(\varpi)^{-1}\pi(\DII{1}{\varpi})f^{sph},\]
and following the computation of the case $c_v({\mathbb I}i)>0$ in \indent opref{P:toric_inert_sp.W} we find that
\begin{align*}f^0_{\mathbb I}i(\varsigma)^*=&X_{m+1}\cdot (f^0(\varsigma)-f^0(\varsigma\cdot \mathbf w))\\
=&X_{m+1}\cdot (-abelian variety s{\varpi}^{-1}+abelian variety s{\varpi})f^{sph}(\varsigma)\\
=&(-1)\cdotabelian variety s{\varpi^{m}}abelian variety s{\mathcal D_E}_E^\mathbf 1half\cdot L(1,\tau_{E/F}).\varepsilonsilonnd{align*}

Finally, suppose that $\pi$ is a ramified principal series with ramified $\nu$ ($v\mid\mathfrak n^-_r$). Let $B(F)$ be the group of upper triangular matrices in $\GL_2(F)$. Let $f^0\in I(\mu,\nu)$ be the function supported in $B(F)\mathbf w N(\mathcal D_F^{-1})$ such that
\[f^0(\varsigma\mathbf w n)=abelian variety s{\mathcal D_F}^\mathbf 1half\text{ for every }n\in N(\mathcal D_F^{-1}).\]

Then one checks easily that $f^0$ does the job. Following the computation in \indent opref{P:toric_ramified.W}, we find that
\begin{align*}
f^0_{\mathbb I}i(\varsigma)^*&=\omega(\varpi^{-m})abelian variety s{\varpi^{2m}}\cdot{\mathbb I}i(\boldsymbol{\delta}_v)abelian variety s{\boldsymbol{\delta}_v}_E^\mathbf 1half\frac{\varepsilonsilonp(0,{\mathbb I}i^{-1},\psi_E)}{\varepsilonsilonp(-1,\omega,\psi)}L(1,\tau_{E/F})abelian variety s{\mathcal D_{E}}_E^\mathbf 1halfabelian variety s{\mathcal D_{F}}^{-1}\cdot f^0(\varsigma\cdot\mathbf w)\\
&=abelian variety s{\varpi^m}\cdot L(1,\tau_{E/F})\cdot \omega^{-1}\mathbf Abs(\varpi^m)\cdot {\mathbb I}i(\boldsymbol{\delta}_v)\frac{\varepsilonsilonp(0,{\mathbb I}i^{-1},\psi_E)}{\varepsilonsilonp(-1,\omega,\psi)}abelian variety s{\mathcal D_E}_E^\mathbf 1halfabelian variety s{\mathcal D_F}^{-\mathbf 1half}.
\varepsilonsilonnd{align*}
This completes the proof in all cases.
\varepsilonsilonnd{proof}

To investigate the $p$-integrality of ${\ell}ocalK{{\mathbb I}i,v}^*$, we define the local invariant $\mu_p({\mathbb I}pi)$ by
\begin{equation}{\ell}abel{E:9.W}\mu_p({\mathbb I}pi):=\inf_{x\inE_v^{\times}}v_p({\mathbb I}pi(x)-1).\varepsilonsilonnd{equation}
By \cite[(4.17)]{Hsieh:Hecke_CM}, $\widetilde A_\beta({\mathbb I}pi)$ is indeed an algebraic integer. 
Moreover, it is proved in \cite[Lemma 6.4]{Hsieh:Hecke_CM} that

\[\mu_p({\mathbb I}pi)>0\iff \widetilde A_\beta({\mathbb I}pi)\equiv 0\pmod{\mathfrak m}\text{ for all $\beta\in F^{\times}$.}\]

Therefore, it follows from {\ell}mref{L:FCformula.W} that if $v\in B({\mathbb I}i)$, then ${\ell}ocalK{{\mathbb I}i,v}^*$ takes values in $\mathbf Zbarp$ and \[{\ell}ocalK{{\mathbb I}i,v}^*\equiv 0\pmod{\mathfrak m}\iff \mu_p({\mathbb I}pi)>0.\] We summarize our discussion in the following proposition.
\begin{prop}{\ell}abel{P:4.W} Let ${\EuScript O}$ be the finite extension of $\mathcal O_{L_\pi}$ generated by $\stt{{\ell}ocalK{{\mathbb I}i,v}^*(1)}_{v\in B({\mathbb I}i)}$ and the values of $\widehat{\mathbb I}i$. Then we have
\begin{mylist}\item
the normalized local Fourier coefficient ${\ell}ocalK{{\mathbb I}i,v}^*$ takes values in ${\EuScript O}$ for every finite place $v\nmid p$,
 \item if either $v\not\in B({\mathbb I}i)$ is unramified or $v\in A({\mathbb I}i)$, then ${\ell}ocalK{{\mathbb I}i,v}^*(1)=1$,
 \item if $v\nmid \mathfrak n$ is ramified with $c_v({\mathbb I}i)=0$, then ${\ell}ocalK{{\mathbb I}i,v}^*(\varpi^{-1})=1$,
 \item if $v\in B({\mathbb I}i)$, then $\mu_p({\mathbb I}pi)=0$ if and only if there exists ${algebraically closed ute{e}t}a_v\in F^{\times}$ such that
\[{\ell}ocalK{{\mathbb I}i,v}^*({algebraically closed ute{e}t}a_v)\not\equiv 0\pmod{\mathfrak m}.\]
\varepsilonsilonnd{mylist}
\varepsilonsilonnd{prop}

\subsection{The global toric period integral}We return to the global situation. Let ${\ell}ocalW{{\mathbb I}i,f}^{(p)}$ be the prime-to-$p$ Whittaker function given by
\[{\ell}ocalW{{\mathbb I}i,f}^{(p)}=\indent od_{v\in\mathbf h,\,v\nmid p}{\ell}ocalW{{\mathbb I}i,v}\in\bigotimes_{v\in\mathbf h,v\nmid p}{\mathcal W}(\pi_v,\psi_v).\]

\begin{defn}{\ell}abel{D:toricWhittaker.W}Let ${\ell}ocalW{{\mathbb I}i,\infty}:=\indent od_{\sigma\in{\varSigma}}W_{k_\sigma}$. Define the $p$-modified toric Whittaker function ${\ell}ocalW{{\mathbb I}i}$ by
\begin{equation}{\ell}abel{E:globalWhitaker.W}\begin{aligned}
{\ell}ocalW{{\mathbb I}i}=&W_{{\mathbb I}i,\infty}\cdot{\ell}ocalW{{\mathbb I}i,f}^{(p)}\cdot\indent od_{v|p}{\ell}ocalW{{\mathbb I}i,v}^\flat\in{\mathcal W}(\pi,\psi).\\
\varepsilonsilonnd{aligned} \varepsilonsilonnd{equation}
Let $u=(u_v)\in (\mathcal O_{F}\otimes_\mathbf Z\mathbf Zp)^{\times}=\indent od_v\mathcal O_{F_v}^{\times}$. The \varepsilonsilonmph{$u$-component} ${\ell}ocalW{{\mathbb I}i,u}$ of ${\ell}ocalW{{\mathbb I}i}$ is defined by
\begin{equation}{\ell}abel{E:Wu.W}{\ell}ocalW{{\mathbb I}i,u}={\ell}ocalW{{\mathbb I}i,\infty}\cdot{\ell}ocalW{{\mathbb I}i,f}^{(p)}\cdot\indent od_{v|p}{\ell}ocalW{{\mathbb I}i,u_v,v}.\varepsilonsilonnd{equation}
Recall that the automorphic form $\varphi_{W}\in{\mathcal A}(\pi)$ associated to $W\in{\mathcal W}(\pi,\psi)$ is defined by
\begin{equation}{\ell}abel{E:WhittakerAuto.W}\varphi_{W}(g):=\sum_{\beta\inF}W(\DII{\beta}{1}g).\varepsilonsilonnd{equation}
Let $\varphi_{\mathbb I}i$ (resp. $\varphi_{{\mathbb I}i,u}$) be the automorphic form associated to ${\ell}ocalW{{\mathbb I}i}$ (resp. ${\ell}ocalW{{\mathbb I}i,u}$). Let $\mathcal U_p=\indent od_{v|p}\mathcal U_v$ be the torsion subgroup of $(\mathcal O_{F}\otimes_\mathbf Z\mathbf Zp)^{\times}$. It follows immediately from the definition \varepsilonsilonqref{E:DeflocalWp.W} that
\begin{equation}{\ell}abel{E:ucompnent.W}\varphi_{\mathbb I}i=\sum_{u\in\mathcal U_p}\varphi_{{\mathbb I}i,u}.\varepsilonsilonnd{equation}
\varepsilonsilonnd{defn}

Choose a sufficiently small prime-to-$p$ integral ideal $\mathfrak n_1$ such that ${\ell}ocalW{{\mathbb I}i,v}$ is invariant by $U_v(\mathfrak n_1)$ for all $v\nmid p$. Let $K=\indent od_v K_v\subset \GL_2(\mathbf AFf)$ be an open-compact subgroup such that
\begin{equation}{\ell}abel{E:opncpt.W}
K_v=K^0_v\text{ if }v\mid p\,;\,
K_v\subset U_v(\mathfrak n_1)\text{ if }v\nmid p.\varepsilonsilonnd{equation}
For each positive integer $n$, put
\[{\ell}sgN:=\stt{g\in K\mid g_v\equiv\MX{1}{*}{0}{1}\pmod{p^n}\text{ for all }v|p}.\]
It is easy to verify that ${\ell}ocalW{{\mathbb I}i}$ and ${\ell}ocalW{{\mathbb I}i,u}$ (and hence $\varphi_{\mathbb I}i$ and $\varphi_{{\mathbb I}i,u}$) are invariant by ${\ell}sgN$ for sufficiently large $n$. The following lemma immediately follows from {\ell}mref{L:toric.W},
\begin{lm}{\ell}abel{L:4.W} Let $\mathcal T=\indent od_v'\mathcal T_v\subset \mathbf AK^{\times}$. Then $\varphi_{\mathbb I}i$ is a toric automorphic form in the sense that for all $t\in\mathcal T$, we have
\begin{align*}\pi(\iota_\varsigma(t))\widetilde V_+^m\varphi_{\mathbb I}i
=&{\mathbb I}i^{-1}(t)\widetilde V_+^m\varphi_{\mathbb I}i.
\intertext{In addition, for all $t\in\mathcal T_f=\indent od_{v\in\mathbf h}'\mathcal T_v$, we have}
\pi(\iota_\varsigma(t))\varphi_{{\mathbb I}i,u}=&{\mathbb I}i^{-1}(t)\varphi_{{\mathbb I}i,u\cdot t^{1-c}},\varepsilonsilonnd{align*}
where $u\cdot t^{1-c}:=ut_{{\varSigma}_p}t_{{\varSigma}bar_p}^{-1}\in(\mathcal O_F\otimes_\mathbf Z\mathbf Zp)^{\times}$.
\varepsilonsilonnd{lm}
Decompose ${\mathfrak c}^-_{\mathbb I}i={\mathfrak c}^-_{{\mathbb I}i,1}{\mathfrak c}^-_{{\mathbb I}i,2}$ such that $({\mathfrak c}^-_{{\mathbb I}i,1},\mathfrak n^-_r)=1$ and ${\mathfrak c}^-_{{\mathbb I}i,2}$ has the same support with $\mathfrak n^-_r$. Define a constant $C'(\pi,{\mathbb I}i)$ by
\begin{equation}{\ell}abel{E:constantC.W}\begin{aligned}
C'(\pi,{\mathbb I}i):=&2^{\sharp(A({\mathbb I}i))+3[F:\mathbf Q]}\cdot\mathrm N_{F/\mathbf Q}({\mathfrak c}^-_{\mathbb I}i)^{-1}\omega({\mathfrak c}^-_{{\mathbb I}i,1})\omega(\mathfrak n_s^-)^{-1}\indent od_{v\nmid p\frakr}\omega(\det \varsigmav)\\ &{\times}\indent od_{w|\mathbf Csplit,\,v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}\varepsilonsilonp(\mathbf 1half,\pi_v\otimes{\mathbb I}i_w,\psi_v)\omega^{-1}{\mathbb I}i_w^{-2}(-2\delta)\cdot\indent od_{v|\mathfrak n^-_r}{\mathbb I}i_v(-\boldsymbol{\delta}_vd_{F_v}^{-1})abelian variety s{\boldsymbol{\delta}_v}_{E_v}^\mathbf 1half\varepsilonsilonp(0,{\mathbb I}i_v^{-1},\psi_{E_v})
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
Note that $C'(\pi,{\mathbb I}i)$ is actually a \text{$p$-adic } unit as $p>2$ and $(p,\mathbf Csplit\mathfrak n^-)=1$. We introduce the normalization factor $N(\pi,{\mathbb I}i)$ given by
\begin{equation}{\ell}abel{E:constantN.W}N(\pi,{\mathbb I}i):=\indent od_{v\in B({\mathbb I}i)}L(1,\tau_{E_v/F_v})n_v.
\varepsilonsilonnd{equation}
 We have the following central value formula of the toric integral $P_{\mathbb I}i(\pi(\varsigma)\widetilde V_+^m\varphi_{\mathbb I}i)$.
\begin{thm}{\ell}abel{T:main1.W}We have
\begin{align*}
P_{\mathbb I}i(\pi(\varsigma)\widetilde V_+^m\varphi_{\mathbb I}i)^2=&abelian variety s{D_{E}}_\mathbf R^{-\mathbf 1half}\frac{\Gamma_{\varSigma}(k+m)\Gamma_{\varSigma}(m+1)}{(4\pi)^{k+2m+1\cdot {\varSigma}}}\cdot E_{{\varSigma}_p}(\pi,{\mathbb I}i)\cdot L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)\cdot C'(\pi,{\mathbb I}i)N(\pi,{\mathbb I}i)^2,
\varepsilonsilonnd{align*}
where $E_{{\varSigma}_p}(\pi,{\mathbb I}i)$ is the Coates' \text{$p$-adic } multiplier given by
\[E_{{\varSigma}_p}(\pi,{\mathbb I}i)=
\indent od_{w\in{\varSigma}_p,v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}}\varepsilonsilonp(\mathbf 1half,\pi_v\otimes{\mathbb I}i_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}},\psi_v)L(\mathbf 1half,\pi_v\otimes{\mathbb I}i_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w}})^{-2}\omega^{-1}{\mathbb I}i_w^{-2}(-2\delta).\]
\varepsilonsilonnd{thm}
\begin{proof}Note that $\widetilde V_+^m\varphi_{\mathbb I}i$ is the automorphic form associated to the Whittaker function
\[\widetilde V_+^m{\ell}ocalW{{\mathbb I}i}=\widetilde V_+^m {\ell}ocalW{{\mathbb I}i,\infty}\cdot {\ell}ocalW{{\mathbb I}i,f}^{(p)}\cdot \indent od_{v|p}{\ell}ocalW{{\mathbb I}i,v}^\flat.\]
Hence, by \indent opref{P:Waldformula.W} we find that
\begin{align*}&P_{\mathbb I}i(\pi(\varsigma)\widetilde V_+^m\varphi_{\mathbb I}i)^2\\
=&\indent od_{\sigma\in{\varSigma}}{\ell}ocalP{\pi(\varsigma_\sigma)\widetilde V^{m_\sigma}_+W_{k_\sigma}}{{\mathbb I}i_\sigma}\indent od_{v|p}\frac{1}{L(\mathbf 1half,\pi_{E_v}\otimes{\mathbb I}i_v)}\cdot{\ell}ocalP{\pi(\varsigmav){\ell}ocalW{{\mathbb I}i,v}^\flat}{{\mathbb I}i_v}\\
&\times \indent od_{v\in\mathbf h,v\nmid p}\frac{1}{L(\mathbf 1half,\pi_{E_v}\otimes{\mathbb I}i_v)}\cdot{\ell}ocalP{\pi(\varsigmav){\ell}ocalW{{\mathbb I}i,v}}{{\mathbb I}i_v}\cdot L(\mathbf 1half,\pi_E\otimes{\mathbb I}i).
\varepsilonsilonnd{align*}
Combining the local calculations of toric integrals of our Whittaker functions (\indent opref{P:Architoric.W}, \indent opref{P:toric_split.W}, \indent opref{P:toric_inert_un.W}, \indent opref{P:toric_inert_sp.W} and \indent opref{P:toric_ramified.W}) yields the central value formula.
\varepsilonsilonnd{proof}
\begin{Remark}Let $\varphi_{\mathbb I}i^Sharp$ be the automorphic form associated to the toric Whittaker function ${\ell}ocalW{{\mathbb I}i}^Sharp:={\ell}ocalW{{\mathbb I}i,\infty}\cdot \indent od_{v\in\mathbf h}{\ell}ocalW{{\mathbb I}i,v}$. Then we obtain the following central value formula:
\[P_{\mathbb I}i(\pi(\varsigma)\widetilde V_+^m\varphi_{\mathbb I}i^Sharp)^2=abelian variety s{D_E}_\mathbf R^{-\mathbf 1half}\frac{\Gamma_{\varSigma}(k+m)\Gamma_{\varSigma}(m+1)}{(4\pi)^{k+2m+1\cdot{\varSigma}}}\cdot L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)\cdot C'(\pi,{\mathbb I}i)N(\pi,{\mathbb I}i)^2.\]
\varepsilonsilonnd{Remark}

\defI{I}
\def{(p)}N{{(pN)}}
\def\cO_{\cF}{\mathcal O_{F}}
\defSCH{SCH}
\defSETS{SETS}
\defU{U}
\def\cW{{\mathcal W}}
\defN{N}
\section{Review of Hilbert modular forms}{\ell}abel{S:Hilbert.W}
In this section, we review some standard facts about Hilbert modular Shimura varieties and Hilbert modular forms. The main purpose of this section is to recall the notation in
\subsection{}Let $V=F e_1\oplusF e_2$ be a two dimensional
$F$-vector space and $\pairing:V{\times} V\to F$ be the $F$-bilinear
alternating pairing defined by $\pair{e_1}{e_2}=1$.  Let $\mathscr L=\cO_{\cF} e_1\oplus \mathcal D_F^{-1} e_2$ be the standard $\cO_{\cF}$-lattice in $V$, which is self-dual with respect to $\psi(\pairing)$. For $g=\MX{a}{b}{c}{d}\in M_2(F)$, we define an involution $g\mapsto g':=\MX{d}{-b}{-c}{a}$. We
identify vectors in $V$ with row vectors according to the basis $e_1,e_2$, so $G(F)=\GL_2(F)$ has a natural right
action on $V$. If $g\in G(F)$, then $g'=g^{-1}\det g$. Define a left action of $G$ on $V$ by $g*x:=x\cdot g',\,x\in V$.

Hereafter, we let $K$ be an open-compact subgroup of $G(\mathbf AFf)$ satisfying \varepsilonsilonqref{E:opncpt.W} and the following conditions:
\begin{equation}cd{neat}K\text{ is neat and }\det (K){\mathcal a}p
\mathcal O_{F,+}^{\times}\subset (K{\mathcal a}p \mathcal O_F^{\times})^2.\varepsilonsilonnd{equation}cd
We also fix a prime-to-$p$ positive integer $N$ such that $U(N)\subset K$.

\subsection{Kottwitz models}
We recall Kottwitz models of Hilbert modular Shimura varieties following the exposition in \cite{Hida:p-adic-automorphic-forms}.
\begin{defn}[$S$-quadruples]{\ell}abel{D:6.H} Let $\Box$ be a finite set of
rational primes not dividing $N$ and let $U$ be an open-compact subgroup of $K^0$ such that $U(N)\subset U$.
Let $\cW_U=\mathbf Z_{(\Box)}[\zeta_{N}]$ with $\zeta_N=\varepsilonsilonxp(\frac{2\pi i}{N})$. Define the fibered category ${\mathcal A}^{(\Box)}_{U}$ over the category $SCH_{/\cW_U}$ of schemes over $\cW_U$ as follows. Let $S$ be a locally noethoerian connected $\cW_U$-scheme and let $\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{s}$ be a geometric point of $S$. Objects are abelian varieties with real
multiplication (AVRM) over $S$ of level $U$, i.e.  a
$S$-\varepsilonsilonmph{quadruple} $(A,\bar{{\ell}am},\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(\Box)})_S$ consisting of the
following data:
\begin{enumerate}
\item $A$ is an abelian scheme of dimension $d$ over $S$.
\item $\iota :\cO_{\cF}\hookrightarrow \End_S A\otimes_\mathbf Z\mathbf ZZbox$.
\item ${\ell}am$ is a prime-to-$\Box$ polarization of $A$ over $S$ and
$\bar{{\ell}am}$ is the $\mathcal O_{F,{(\Box)},+}$-orbit of ${\ell}am$. Namely
\[\bar{{\ell}am}=\mathcal O_{F,{(\Box)},+}{\ell}am:=\stt{{\ell}am'\in\Hom(A,A^t)\otimes_\mathbf Z\mathbf ZZbox\mid {\ell}am'={\ell}am\circ a,\,a\in \mathcal O_{F,{(\Box)},+}}.\]
\item $\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(\Box)}={algebraically closed ute{e}t}a^{(\Box)}U^{(\Box)}$ is a $\pi_1(S,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{s})$-invariant $U^{(\Box)}$-orbit of isomorphisms of $\cO_{\cF}$-modules ${algebraically closed ute{e}t}a^{(\Box)}: \mathscr L\otimes_\mathbf Z\mathbf A_f^{(\Box)}\stackrel{\sim}{\to} V^{(\Box)}(A_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{s}}):=H_1(A_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{s}},\mathbf Zhat^{(\Box)})\otimes_\mathbf Z\mathbf A_f^{(\Box)}$. Here we define ${algebraically closed ute{e}t}a^{(\Box)} g$ for $g\in G(\mathbf AFf^{(\Box)})$ by ${algebraically closed ute{e}t}a^{(\Box)} g(x)={algebraically closed ute{e}t}a^{(\Box)}(g*x)$.
\varepsilonsilonnd{enumerate}
Furthermore, $(A,\bar{{\ell}am},\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(\Box)} )_S$ satisfies the
following conditions:
\begin{itemize}
\item Let ${}^t$ denote the Rosati involution induced by ${\ell}am$ on
$\End_SA\otimes\mathbf ZZbox$. Then $\iota(b)^t=\iota(b),\, \forall\, b\in
\cO_{\cF}.$
\item Let $e^{\ell}am$ be the Weil pairing induced by ${\ell}am$. Lifting the isomorphism $\mathbf Z/N\mathbf Z\simeq \mathbf Z/N\mathbf Z(1)$ induced by $\zeta_{N}$ to an isomorphism $\zeta:\mathbf Zhat\simeq\mathbf Zhat(1)$, we can regard $e^{\ell}am$ as an $F$-alternating form
$e^{\ell}am:V^{(\Box)}(A)\times V^{(\Box)}(A)\to \mathcal D_F^{-1}\otimes_\mathbf Z\mathbf A_f^{(\Box)}$. Let $e^{algebraically closed ute{e}t}a$ denote the
$F$-alternating form on $V^{(\Box)}(A)$ induced by
$e^{algebraically closed ute{e}t}a(x,x')=\pair{x{algebraically closed ute{e}t}a}{x'{algebraically closed ute{e}t}a}$. Then
\[e^{\ell}am=u\cdot e^{algebraically closed ute{e}t}a\text{ for some }u\in\mathbf AFf^{(\Box)}.\]
\item As
$\cO_{\cF}\otimes_\mathbf Z\mathcal O_S$-modules, we have an isomorphism $\Lie A\simeq \cO_{\cF}\otimes_\mathbf Z\mathcal O_S$ locally under Zariski topology of $S$.

\varepsilonsilonnd{itemize}
For two $S$-quadruples $\ul{A}=(A,\bar{{\ell}am},\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(\Box)} )_S$ and $\ul{A'}=(A',\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{\ell}am'},\iota',\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a'}{}^{(\Box)})_S$,
we define morphisms by
\[\Hom_{{\mathcal A}^{(\Box)}_{K}}(\ul{A},\ul{A'})=\stt{\phi\in \Hom_{\cO_{\cF}}(A,A')\mid
\phi^*\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{\ell}am'}=\bar{{\ell}am},\,\phi\circ \overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a'}{}^{(\Box)}=\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(\Box)} }.\] We
say $\ul{A}\sim \ul{A'}$ (resp. $\ul{A}\simeq \ul{A'}$) if there exists a
prime-to-$\Box$ isogeny (resp. isomorphism) in
$\Hom_{{\mathcal A}_K^{(\Box)}}(\ul{A},\ul{A'})$.
\varepsilonsilonnd{defn}
We consider the cases when $\Box=\varepsilonsilonmptyset$ and $\stt{p}$. When
$\Box=\varepsilonsilonmptyset$ is the empty set and $U$ is an open-compact
subgroup in $G(\mathbf AFf^{(\Box)})=G(\mathbf AFf)$, we define the functor
${\mathcal E}_{U}:SCH_{/\cW_U}\toSETS$ by
\[{\mathcal E}_U(S)=\stt{(A,\bar{{\ell}am},\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a} )_S\in{\mathcal A}_K(S)}/\sim.\] By the theory of Shimura-Deligne, ${\mathcal E}_{U}$ is represented by
$Sh_U$ which is a quasi-projective scheme over $\cW_{U}$. We define
the functor $\mathfrak E_U:SCH_{/\cW_{U}}\toSETS$ by
\[\mathfrak E_U(S)=\stt{(A,\bar{{\ell}am},\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a})\in{\mathcal A}^{(\Box)}_U(S)\mid {algebraically closed ute{e}t}a^{(\Box)}(\mathscr L\otimes_\mathbf Z\mathbf Zhat)=H_1(A_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{s}},\mathbf Zhat)}/\simeq.\]
By the discussion in \cite[p.136]{Hida:p-adic-automorphic-forms}, we have $\mathfrak E_K\stackrel{\sim}{\to}{\mathcal E}_K$ under the hypothesis \varepsilonsilonqref{neat}.

When $\Box=\stt{p}$ and $U=K$, we let $\cW=\cW_K=\mathbf Z_{(p)}[\zeta_N]$ and define functor
${\mathcal E}^{(p)}_{K}:SCH_{/\cW}\toSETS$ by
\[{\mathcal E}^{(p)}_{K}(S)=\stt{(A,\bar{{\ell}am},\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(p)})_S\in{\mathcal A}_{K^{(p)}}^{(p)}(S)}/\sim.\]
In \cite{Kottwitz:Points-On-Shimura-Varieties}, Kottwitz shows
${\mathcal E}^{(p)}_{K}$ is representable by a quasi-projective
scheme $Sh^{(p)}_{K}$ over $\cW$ if $K$ is neat. Similarly we
define the functor $\mathfrak E_K^{(p)}:SCH_{/\cW}\toSETS$ by
\[\mathfrak E_K^{(p)}(S)=\stt{(A,\bar{{\ell}am},\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(p)})\in{\mathcal A}^{(p)}_K(S)\mid {algebraically closed ute{e}t}a^{(p)}(\mathscr L\otimes_\mathbf Z\mathbf Zhat^{(p)})=H_1(A_{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{s}},\mathbf Zhat^{(p)})}/\simeq.\]
It is shown in \cite[\S 4.2.1]{Hida:p-adic-automorphic-forms} that
$\mathfrak E^{(p)}_K\stackrel{\sim}{\to}{\mathcal E}^{(p)}_K$.

Let $\frakc$ be a prime-to-$pN$ ideal of $\cO_{\cF}$ and let $\mathbf c\in (\mathbf AFf^{(p)}N)^{\times}$ such that $\frakc=\mathfrak i\frakl_{F}(\mathbf c)$. We say $(A,{\ell}am,\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(p)})$ is $\frakc$-polarized if ${\ell}am\in\bar{{\ell}am}$ such that $e^{\ell}am=ue^{{algebraically closed ute{e}t}a},\,u\in \mathbf c\det(K)$.
The isomorphism class $[(A,{\ell}am,\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(p)})]$ is independent of a choice of ${\ell}am$ in $\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{\ell}am}$ under the assumption \varepsilonsilonqref{neat} (\mbox{{\it cf.} }\cite[p.136]{Hida:p-adic-automorphic-forms}).
We consider the functor
\[\mathfrak E^{(p)}_{\frakc,K}(S)=\stt{\text{$\frakc$-polarized $S$-quadruple }[(A,{\ell}am,\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(p)})_S]\in\mathfrak E^{(p)}_{K}(S)}.\]
Then $\mathfrak E^{(p)}_{\frakc,K}$ is represented by a geometrically irreducible scheme $Sh^{(p)}_K(\frakc)_{/\cW}$, and we have
\begin{equation}{\ell}abel{E:decomposition}Sh^{(p)}_K{}_{/\cW}=\bigsqcup_{[\frakc]\in\mathbf Cl^+_F(K)}Sh^{(p)}_K(\frakc)_{/\cW},\varepsilonsilonnd{equation}
where $\mathbf Cl^+_F(K)$ is the narrow ray class group of $F$ with level $\det(K)$.
\defIusa{I_{K,n}}
\subsection{Igusa schemes}{\ell}abel{17.H}
Let $n$ be a positive integer. Define the functor $\mathcal I^{(p)}_{K,n}:SCH_{/\cW}\toSETS$ by
\[S\mapsto \mathcal I^{(p)}_{K,n}(S)=\stt{(A,\bar{{\ell}am},\iota,{\ell}pp,{\ell}p)_S}/\sim,
\]
where $(A,\bar{{\ell}am},\iota,{\ell}pp)_S$ is a $S$-quadruple, ${\ell}p$ is a level $p^n$-structure, i.e.  an $\cO_{\cF}$-group scheme morphism:
\[{\ell}p:\mathcal D_F^{-1}\otimes_\mathbf Z\boldsymbol{\mu}_{p^n}\hookrightarrow A[p^n],\]
and $\sim$ means modulo prime-to-$p$ isogeny. It is known that $\mathcal I^{(p)}_{K,n}$ is relatively representable over ${\mathcal E}^{(p)}_{K}$ (\mbox{{\it cf.} }\cite[Lemma (2.1.6.4)]{HLS}) and thus is represented by a scheme $Iusa$.

Now we consider $S$-quintuples $(A,{\ell}am,\iota,{\ell}pp,{\ell}p)_S$ such that $[(A,{\ell}am,\iota,{\ell}pp)]\in\mathfrak E^{(p)}_{\frakc,K}(S)$.

Define the functor $\mathcal I^{(p)}_{K,n}(\frakc):SCH_{/\cW}\toSETS$ by
\begin{align*}S\mapsto\mathcal I^{(p)}_{K,n}(\frakc)(S)&=\stt{(A,{\ell}am,\iota,{\ell}pp,{\ell}p)_S\text{ as above}}/\simeq.\\
\varepsilonsilonnd{align*}
Then $\mathcal I^{(p)}_{K,n}(\frakc)$ is represented by a scheme $Iusa(\frakc)$ over $Sh^{(p)}_K(\frakc)$, and $Iusa(\frakc)$ can be identified with a geometrically irreducible subscheme of $Iusa$ (\cite[Thm.\,(4.5)]{DR_padic_L}).
For $n\geq n'>0$, the natural morphism
$\pi_{n,n'}:Iusa(\frakc)\toI_{K,n'}(\frakc)$ induced by the
inclusion $\mathcal D_F^{-1}\otimes\boldsymbol{\mu}_{p^{n'}}\hookrightarrow \mathcal D_F^{-1}\otimes\boldsymbol{\mu}_{p^n}$ is finite
{algebraically closed ute{e}t}ale.

The forgetful
morphism $\pi:Iusa(\frakc)\to Sh^{(p)}_{K}(\frakc)$ defined by
$\pi:(\ul{A},{\ell}p)\mapsto \ul{A}$ is {algebraically closed ute{e}t}ale for all $n>0$. Hence
$Iusa(\frakc)$ is smooth over $\mathrm{Spec}\,\cW$. We write $I_K(\frakc)$ for $\indent olim_nIusa(\frakc)$.

\subsection{Complex uniformization}{\ell}abel{S:cpx} We describe the complex points $Sh_U(\mathbf C)$ for $U\subset G(\mathbf AFf)$. Put
\[X^+=\stt{\tau=(\tau_\sigma)_{\sigma\in{\varSigma}}\in\mathbf C^{{\varSigma}}\mid \Im \tau_\sigma >0\text{ for all } \sigma\in{\varSigma}}.\]
The action of $g=(g_\sigma)_{\sigma\in{\varSigma}}\in G(F\otimes_\mathbf Q\mathbf R)$ with $g_\sigma=\MX{a_\sigma}{b_\sigma}{c_\sigma}{d_\sigma}$ and $\det g_\sigma>0$ on $X^+$ is given by $\tau=(\tau_\sigma)\mapsto g\tau={\ell}eft(\frac{a_\sigma \tau_\sigma+b_\sigma}{c_\sigma \tau_\sigma+d_\sigma}\right)$. Let $F_+$ be the set of totally positive elements in $F$ and let $G(F)^+=\stt{g\in G(F)\mid \det g\inF_+}$. Define the complex Hilbert modular Shimura variety by
\[M(X^+,U):=G(F)^+\backslash X^+{\times} G(\mathbf AFf)/U.\]
It is well known that $M(X^+,K)\stackrel{\sim}{\to}Sh_U(\mathbf C)$ by the theory
of abelian varieties over $\mathbf C$ (\mbox{{\it cf.} }\cite[\S\,4.2]{Hida:p-adic-automorphic-forms}).

For $\tau=(\tau_\sigma)_{\sigma\in{\varSigma}}\in X^+$, we let $p_\tau$ be the isomorphism $V\otimes_\mathbf Q\mathbf R\stackrel{\sim}{\to} \mathbf C^{{\varSigma}}$ defined by
$p_\tau(ae_1+be_2)=a\tau+b$ with $a,b\in F\otimes_\mathbf Q\mathbf R=\mathbf R^{{\varSigma}}$. We can associate a AVRM to $(\tau,g)\in X^+{\times} G(\mathbf AFf)$ as follows.
\begin{itemize}
\item The complex abelian variety ${\EuScript A}_g(\tau)=\mathbf C^{{\varSigma}}/p_\tau(g*\mathscr L)$.
\item  The $F_+$-orbit of polarization
$\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\pairing}_{\mathcal a}n$ on ${\EuScript A}_g(\tau)$ is given by the Riemann form $\pairing_{\mathcal a}n:=\pairing\circ p_\tau^{-1}$.
\item The $\iota_\mathbf C:O\hookrightarrow\End {\EuScript A}_g(\tau)\otimes_\mathbf Z\mathbf Q$ is induced from the pull back of the natural $F$-action on $V$ via $p_\tau$.
\item The level structure ${algebraically closed ute{e}t}a_g:
\mathscr L\otimes_\mathbf Z\mathbf A_f \stackrel{\sim}{\to} (g*\mathscr L)\otimes_\mathbf Z\mathbf A_f=H_1({\EuScript A}_g(\tau),\mathbf A_f)$ is defined by  ${algebraically closed ute{e}t}a_g(v)= g*v$.\varepsilonsilonnd{itemize}
Let $\ul{{\EuScript A}_g(\tau)}$ denote the $\mathbf C$-quadruple $({\EuScript A}_g(\tau),\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\pairing}_{\mathcal a}n,\iota_\mathbf C,K{algebraically closed ute{e}t}a_g)$. Then the map $[(\tau,g)]\mapsto [\ul{{\EuScript A}_g(\tau)}]$ gives rise to
an isomorphism $M(X^+,U)\stackrel{\sim}{\to} Sh_U(\mathbf C)$.

For a positive integer $n$, the exponential map gives the isomorphism $\varepsilonsilonxp(2\pi i -):p^{-n}\mathbf Z/\mathbf Z\simeq\boldsymbol{\mu}_{p^n}$ and thus induces a level $p^n$-structure ${\ell}p(g_p)$:
\begin{align*}{\ell}p(g_p)\colon&\mathcal D_F^{-1}\otimes_\mathbf Z\boldsymbol{\mu}_{p^n}\stackrel{\sim}{\to} \mathcal D_F^{-1}e_2\otimes_\mathbf Z p^{-n}\mathbf Z/\mathbf Z\hookrightarrow \mathscr L\otimes_\mathbf Z p^{-n}\mathbf Z/\mathbf Z\stackrel{g*}\stackrel{\sim}{\to} {\EuScript A}_g(\tau)[p^n].

\varepsilonsilonnd{align*}
Put
\[{\ell}sgN:=\stt{g\inK\mid g_p\equiv\MX{1}{*}{0}{1}\pmod{p^n}}.\]
We have a non-canonical isomorphism:
\begin{align*}
M(X^+,{\ell}sgN)&\stackrel{\sim}{\to} Iusa(\mathbf C)\\
 [(\tau,g)]&\mapsto  [({\EuScript A}_g(\tau),\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\pairing}_{\mathcal a}n,\iota_\mathbf C,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(p)}_g, {\ell}p(g_p))].\varepsilonsilonnd{align*}

Let $\ul{z}=\stt{z_\sigma}_{\sigma\in{\varSigma}}$ be the standard complex coordinates of $\mathbf C^{{\varSigma}}$ and $d\ul{z}=\stt{dz_\sigma}_{\sigma\in{\varSigma}}$. Then $\cO_{\cF}$-action on $d\ul{z}$ is given by
$\iota_\mathbf C(\alpha)^* dz_\sigma=\sigma(\alpha)dz_\sigma,\,\sigma\in{\varSigma}\simeq\Hom(F,\mathbf C)$. Let $z=z_{id}$ be the coordinate corresponding to $\iota_\infty:F\hookrightarrow\mathbf Qbar\hookrightarrow\mathbf C$.
Then
\begin{equation}{\ell}abel{E:7.N}(\cO_{\cF}\otimes_\mathbf Z\mathbf C) dz=H^0({\EuScript A}_g(\tau),\boldsymbol{\omega}ega_{{\EuScript A}_g(\tau)/\mathbf C}).\varepsilonsilonnd{equation}

\subsection{Hilbert modular forms}
Let $k=\sum_{\sigma}k_{\sigma}\sigma\in\mathbf Z_{\geq 1}[{\varSigma}]$ such that
\[k_{\sigma_1}\equiv k_{\sigma_2}\equiv\cdots\equiv k_{\sigma_d}\pmod{2}\text{ for all }\sigma_1,\cdots,\sigma_d\in{\varSigma}.\]
For
$\tau=(\tau_\sigma)_{\sigma\in{\varSigma}}\in X^+$ and $g=(\MX{a_\sigma}{b_\sigma}{c_\sigma}{d_\sigma})_{\sigma\in{\varSigma}}\in
G(F\otimes_\mathbf Q\mathbf R)$, we put
\[\ul{J}(g,\tau)^k=\indent od_{\sigma\in{\varSigma}}(c_\sigma\tau_\sigma+d_\sigma)^{k_\sigma}.\]

\begin{defn}
Let $k_{mx}=\max_{\sigma\in{\varSigma}}{k_{\sigma}}$. Denote by $\mathbf M_k({\ell}sgN,\mathbf C)$ the space of holomorphic Hilbert modular forms of weight $k$ and level ${\ell}sgN$. Each $\mathbf f\in\mathbf M_k({\ell}sgN,\mathbf C)$ is a $\mathbf C$-valued function $\mathbf f:X^+{\times} G(\mathbf AFf)\to \mathbf C$ such that the function $\mathbf f(-,g_f):X^+\to\mathbf C$ is holomorphic for each $g_f\in G(\mathbf AFf)$, and for $u\in {\ell}sgN$
and $\alpha\in G(F)^+$,
\[\mathbf f(\alpha(\tau,g_f)u)=(\det \alpha)^{-\frac{(k_{mx}-2){\varSigma}+k}{2}}\ul{J}(\alpha,\tau)^k\cdot \mathbf f(\tau,g_f).\]
Here $\det\alpha$ is considered to be the element $(\sigma(\det\alpha))_{\sigma\in{\varSigma}}$ in $(\mathbf C^{\times})^{\varSigma}$.
\varepsilonsilonnd{defn}

For every $\mathbf f\in \mathbf M_k({\ell}sgN,\mathbf C)$,we have the Fourier expansion
\[\mathbf f(\tau,g_f)=\sum_{\beta\inF_+\cup\stt{0}}W_\beta(\mathbf f,g_f)e^{2\pi i{\mathbb T}r_{F/\mathbf Q}(\beta \tau)}.\]
For a semi-group $L$ in $F$, let $L_+=F_+{\mathcal a}p L$ and $L_{\geq 0}=L_+\cup \stt{0}$. If $B$ is a ring, we denote by $B\powerseries{L}$ the set of all formal series
\[\sum_{\beta\in L}a_\beta q^\beta,\,a_\beta\in B.\]
Let $a,b\in(\mathbf AFf^{(pN)})^{\times}$ and let ${\mathfrak a}=\mathfrak i\frakl_F(a)$ and
${\mathfrak b}=\mathfrak i\frakl_F(b)$. The $q$-expansion of $\mathbf f$ at the cusp $({\mathfrak a},{\mathfrak b})$ is given by
\begin{equation}{\ell}abel{E:FC0}\mathbf f|_{({\mathfrak a},{\mathfrak b})}(q)=\sum_{\beta\in (N^{-1}{\mathfrak a}{\mathfrak b})_{\geq 0}}W_\beta(\mathbf f,\MX{a^{-1}}{0}{0}{b})q^\beta\in \mathbf C\powerseries{(N^{-1}{\mathfrak a}{\mathfrak b})_{\geq 0}}. \varepsilonsilonnd{equation}
If $B$ is a $\cW$-algebra in $\mathbf C$, we put
\begin{align*}
\mathbf M_k(\frakc,{\ell}sgN,B)&=\stt{\mathbf f\in \mathbf M_k({\ell}sgN,\mathbf C)\mid \mathbf f|_{({\mathfrak a},{\mathfrak b})}(q)\in B\powerseries{(N^{-1}{\mathfrak a}{\mathfrak b})_{\geq 0}}\text{ for all }({\mathfrak a},{\mathfrak b})
\text{ such that ${\mathfrak a}{\mathfrak b}^{-1}={\mathfrak c}$}}.\\
\varepsilonsilonnd{align*}
\subsubsection{Tate objects}
Let $\mathscr S$ be a set of $d$ linearly $\mathbf Q$-independent elements in $\Hom(F,\mathbf Q)$ such that $l(F_+)>0$ for $l\in\mathscr S$. If $L$ is a lattice in $F$ and $n$ a positive integer, let
$L_{\mathscr S,n}=\stt{x\in L\mid l(x)>-n\text{ for all }l\in\mathscr S}$ and put $B((L;\mathscr S))={\ell}im{\ell}imits_{n\to\infty} B\powerseries{L_{\mathscr S,n}}$.
To a pair $({\mathfrak a},{\mathfrak b})$ of two prime-to-$pN$ fractional ideals, we can attach the Tate AVRM $Tate_{{\mathfrak a},{\mathfrak b}}(q)={\mathfrak a}^*\otimes_\mathbf Z{\bbG}_{m}}   \def\Ga{{{\bbG}_{\rm a}}/q^{{\mathfrak b}}$ over $\mathbf Z(({\mathfrak a}{\mathfrak b};\mathscr S))$ with $O$-action $\iota_{\mathcal a}n$, where ${\mathfrak a}^*:={\mathfrak a}^{-1}\mathcal D_F^{-1}$. As described in \cite{Katz:p_adic_L-function_CM_fields}, $Tate_{{\mathfrak a},{\mathfrak b}}(q)$ has a canonical ${\mathfrak a}{\mathfrak b}^{-1}$-polarization ${\ell}am_{\mathcal a}n$ and also carries $\boldsymbol{\omega}_{\mathcal a}n$ a canonical $\cO_{\cF}\otimes\mathbf Z(({\mathfrak a}{\mathfrak b};\mathscr S))$-generator of $\boldsymbol{\omega}ega_{Tate_{{\mathfrak a},{\mathfrak b}}}$ induced by the isomorphism $\Lie(Tate_{{\mathfrak a},{\mathfrak b}}(q)_{/\mathbf Z(({\mathfrak a}{\mathfrak b};\mathscr S))})={\mathfrak a}^*\otimes_\mathbf Z\Lie({\bbG}_{m}}   \def\Ga{{{\bbG}_{\rm a}})\simeq{\mathfrak a}^*\otimes\mathbf Z(({\mathfrak a}{\mathfrak b};\mathscr S))$.
Since ${\mathfrak a}$ is prime to $p$, the natural inclusion ${\mathfrak a}^*\otimes_\mathbf Z\boldsymbol{\mu}_{p^n}\hookrightarrow{\mathfrak a}^*\otimes_\mathbf Z{\bbG}_{m}}   \def\Ga{{{\bbG}_{\rm a}}$ induces a canonical level $p^n$-structure ${algebraically closed ute{e}t}a_{p,{\mathcal a}n}\colon \mathcal D_F^{-1}\otimes_\mathbf Z\boldsymbol{\mu}_{p^n}={\mathfrak a}^*\otimes_\mathbf Z\boldsymbol{\mu}_{p^n}\hookrightarrow Tate_{{\mathfrak a},{\mathfrak b}}(q)$. Let $\mathscr L_{{\mathfrak a},{\mathfrak b}}=\mathscr L\cdot\MX{{\mathfrak b}}{}{}{{\mathfrak a}^{-1}}={\mathfrak b} e_1\oplus{\mathfrak a}^*e_2$. Then we have a level $N$-structure  ${algebraically closed ute{e}t}a_{\mathcal a}n^{(p)}:N^{-1}\mathscr L_{{\mathfrak a},{\mathfrak b}}/\mathscr L_{{\mathfrak a},{\mathfrak b}}\stackrel{\sim}{\to} Tate_{{\mathfrak a},{\mathfrak b}}(q)[N]$
over $\mathbf Z[\zeta_{N}]((N^{-1}{\mathfrak a}{\mathfrak b};\mathscr S))$ induced by the fixed primitive $N$-th root of unity $\zeta_{N}$.
We write $\ul{Tate}_{{\mathfrak a},{\mathfrak b}}$ for the Tate $\mathbf Z(({\mathfrak a}{\mathfrak b};\mathscr S))$-quadruple $(Tate_{{\mathfrak a},{\mathfrak b}}(q),{\ell}am_{\mathcal a}n,\iota_{\mathcal a}n,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(p)}_{\mathcal a}n,{algebraically closed ute{e}t}a_{p,{\mathcal a}n})$ at $({\mathfrak a},{\mathfrak b})$.

\subsubsection{Geometric modular forms}{\ell}abel{S:GME}We collect here definitions and basic facts of geometric modular forms. The whole theory can be found in \cite{Katz:p_adic_L-function_CM_fields} and \cite{Hida:p-adic-automorphic-forms}.
Let $T$ be the algebraic torus over $\cW$ such that $T(R)=(\cO_{\cF}\otimes_\mathbf Z R)^{\times}$ for every $\cW$-algebra $R$. Let $k\in\Hom(T,{\bbG}_{m}}   \def\Ga{{{\bbG}_{\rm a}}{}_{/\cW})$. Let $B$ be a $\cW$-algebra.
Consider $[(\ul{A},{\ell}p)]=[(A,{\ell}am,\iota,{\ell}pp,{\ell}p)]\inIusa(\frakc)(C)$ (resp. $[(\ul{A},{\ell}p)]=[(A,\bar{{\ell}am},\iota,{\ell}pp,{\ell}p)]\inIusa(C)$) for a $B$-algebra $C$ with a differential form $\boldsymbol{\omega}$ generating
$H^0(A,\boldsymbol{\omega}ega_{A/C})$ over $\cO_{\cF}\otimes_\mathbf Z C$. A geometric modular form $f$ over $B$ of weight $k$ on $Iusa(\frakc)$ (resp. $Iusa$) is a functorial rule of
assigning a value $f(\ul{A},{\ell}p,\boldsymbol{\omega})\in C$ satisfying the following axioms.
\begin{mylist}
\item[(G1)] $f(\ul{A},{\ell}p,\boldsymbol{\omega})=f(\ul{A}',{\ell}p',\boldsymbol{\omega}')\in C$ if $(\ul{A},{\ell}p,\boldsymbol{\omega})\simeq (\ul{A}',{\ell}p',\boldsymbol{\omega}')$ over $C$,
\item[(G2)]For a $B$-algebra homomorphism $\varphi:C\to C'$, we have
\[f((\ul{A},{\ell}p,\boldsymbol{\omega})\otimes_C C')=\varphi(f(\ul{A},{\ell}p,\boldsymbol{\omega})),\]
\item[(G3)]$f((\ul{A},{\ell}p,a\boldsymbol{\omega})=k(a^{-1})f(\ul{A},{\ell}p,\boldsymbol{\omega})$ for all $a\in T(C)=(\cO_{\cF}\otimes_\mathbf Z C)^{\times}$,
\item[(G4)]$f(\ul{Tate}_{{\mathfrak a},{\mathfrak b}},\boldsymbol{\omega}_{\mathcal a}n)\in B\powerseries{(N^{-1}{\mathfrak a}{\mathfrak b})_{\geq 0}}\text{ at all cusps }({\mathfrak a},{\mathfrak b})$ in $Iusa(\frakc)$ (resp. $Iusa$).
\varepsilonsilonnd{mylist}
For each $k\in\mathbf Z[{\varSigma}]$, we regard $k\in\Hom(T,{\bbG}_{m}}   \def\Ga{{{\bbG}_{\rm a}}{}_{/{\mathcal W}})$ as the character $x\mapsto x^k,\,x\in (\cO_{\cF}\otimes_\mathbf Z\cW)^{\times}$.
We denote by $\mathcal M_k(\frakc,{\ell}sgN,B)$ (resp. $\mathcal M_k({\ell}sgN,B)$) the space of geometric modular forms over $B$ of weight $k$ on $Iusa(\frakc)$ (resp. $Iusa$).
For $f\in\mathcal M_k({\ell}sgN,B)$, we write $f|_\frakc\in\mathcal M_k(\frakc,{\ell}sgN,B)$ for the restriction $f|_{Iusa(\frakc)}$.

For each $f\in \mathcal M_k({\ell}sgN,\mathbf C)$, we regard $f$ as a holomorphic Hilbert modular form of weight $k$ and level ${\ell}sgN$ by
\[f(\tau,g_f)=f({\EuScript A}_g(\tau),\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{\ell}am_{\mathcal a}n},\iota_\mathbf C,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}_g,2\pi i dz),\]
where $dz$ is the differential form in \varepsilonsilonqref{E:7.N}. By GAGA this gives rise to an isomorphism $\mathcal M_{k}({\ell}sgN,\mathbf C)\stackrel{\sim}{\to}\mathbf M_k({\ell}sgN,\mathbf C)$ and $\mathcal M_{k}(\frakc,{\ell}sgN,\mathbf C)\stackrel{\sim}{\to}\mathbf M_k(\frakc,{\ell}sgN,\mathbf C)$.
Moreover, as discussed in \cite[\S 1.7]{Katz:p_adic_L-function_CM_fields}, we have the following important identity which bridges holomorphic modular forms and geometric modular forms
\[\mathbf f|_{({\mathfrak a},{\mathfrak b})}(q)=\mathbf f(\ul{Tate}_{({\mathfrak a},{\mathfrak b})},\boldsymbol{\omega}_{\mathcal a}n)\in\mathbf C\powerseries{(N^{-1}{\mathfrak a}{\mathfrak b})_{\geq 0}}.\]
By the $q$-expansion principle, if $B$ is $\cW$-algebra in $\mathbf C$ and $\mathbf f \in\mathbf M_k(\frakc,{\ell}sgN,B)=\mathcal M_{k}(\frakc,{\ell}sgN,\mathbf C)$, then $\mathbf f|_\frakc\in\mathcal M_{k}(\frakc,{\ell}sgN,B)$.

\subsubsection{\text{$p$-adic } modular forms}Let $B$ be a \text{$p$-adic } $\cW$-algebra in $\mathbf Cp$. Let $V(\frakc,K,B)$ be the space of Katz \text{$p$-adic } modular forms over $B$ defined by
\[V(\frakc,K,B):=\indent olim_m\varinjlim_n H^0(Iusa(\frakc){}_{/B/p^mB},\mathcal O_{Iusa}).\]
In other words, Katz \text{$p$-adic } modular forms consist of formal functions on the Igusa tower.

Let $C$ be a $B/p^mB$-algebra. For each $C$-point $[(\ul{A},{\ell}p)]=[(A,{\ell}am,\iota,{\ell}pp,{\ell}p]\inI_K(\frakc)(C)=\indent olim_nIusa(\frakc){}(C)$, the $p^\infty$-level structure ${\ell}p$ induces an isomorphism ${\ell}p_*:\mathcal D_F^{-1}\otimes_\mathbf Z C\simeq \Lie A$ which in turn gives rise to  a generator $\boldsymbol{\omega}({\ell}p)$ of $H^0(A,\boldsymbol{\omega}ega_A)$ as a $\cO_{\cF}\otimes_\mathbf Z C$-module.
Then we have a natural injection\begin{equation}{\ell}abel{E:padicavatar.N}
\begin{aligned}\mathcal M_k(\frakc,{\ell}sgN,B)&\hookrightarrow V(\frakc,K,B)\\
f&\mapsto \widehat{f}(\ul{A},{\ell}p):=f(\ul{A},{\ell}p,\boldsymbol{\omega}({\ell}p))
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
which preserves the $q$-expansions in the sense that $\widehat f|_{({\mathfrak a},{\mathfrak b})}(q):=\widehat f(\ul{Tate}_{{\mathfrak a},{\mathfrak b}})=f|_{({\mathfrak a},{\mathfrak b})}(q)$. We call $\widehat{f}$ the \text{$p$-adic } avatar of $f$.

\subsection{CM points}{\ell}abel{S:CMpoint}
Recall that we have fixed $\mathbf CMP\inE$ in \subsecref{SS:choiceofcmpt} satisfying (d1-3) and the associated embedding $\iota:E\hookrightarrow M_2(F)$ in \varepsilonsilonqref{E:imb.W}.
Let $q_{\mathbf CMP}:E\stackrel{\sim}{\to} V=F e_1\oplus F e_2$ be the isomorphism given by $q_{\mathbf CMP}(a\mathbf CMP+b)=ae_1+be_2$. Then \[q_{\mathbf CMP}(x\alpha)=q_{\mathbf CMP}(x)\iota(\alpha)\text{ for all }x,\alpha\inE,\] and $p_{\mathbf CMP}:=q_{\mathbf CMP}^{-1}:V\otimes_\mathbf Q\mathbf R\stackrel{\sim}{\to}E\otimes_\mathbf Q\mathbf R\simeq\mathbf C^{\varSigma}$ is the period map associated to the point $\mathbf CMP_{\varSigma}:=(\sigma(\mathbf CMP))_{\sigma\in{\varSigma}}\in X^+$.

Let $\varsigma=\indent od_v\varsigmav\in G(\mathbf AF)$ where $\varsigmav\in G_v$ for each place $v$ is defined in \varepsilonsilonqref{E:cmptv.W}. Let $\varsigma_f\in G(\mathbf AFf)$ be the finite part of $\varsigma$. According to our choices of $\varsigmav$, we have
\[\varsigma_f*(\mathscr L\otimes_\mathbf Z\mathbf Zhat)=(\mathscr L\otimes_\mathbf Z\mathbf Zhat)\cdot\varsigma_f'=q_{\mathbf CMP}(\mathcal O_E\otimes_\mathbf Z\mathbf Zhat).\]
Define
$x:\mathbf AK^{\times}\to X^+{\times} G(\mathbf AFf)$ by
\[a=(a_\infty,a_f)\mapsto x(a):=(\mathbf CMP_{\varSigma},\iota(a_f)\varsigma_f).\]
Let $a\in (\mathbf AKf^{(pN)})^{\times}$ and let
\[(\ul{A}(a),{\ell}p(a))_{/\mathbf C}=({\EuScript A}_{\iota(a)\varsigma_f}(\mathbf CMP_{\varSigma}),\pairing_{\mathcal a}n,\iota_{\mathcal a}n,{\ell}pp(a),{\ell}p(a))\] be the $\mathbf C$-quintuple associated to $x(a)$ as in \secref{S:cpx}.
The alternating pairing $\pairing:E{\times}E:\toF$
defined by $\pair{x}{y}=(\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{x}y-x\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{y})/(\mathbf CMP-\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathbf CMP})$ induces an isomorphism
$\cO_\cK\wedge_{\cO_{\cF}} \cO_\cK={\mathfrak c}(\cO_\cK)^{-1}\mathcal D_F^{-1}$ for the fractional
ideal ${\mathfrak c}(\cO_\cK)=\mathcal D_F^{-1}((\mathbf CMP-\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\mathbf CMP}) \mathcal D_{E/F}^{-1})$. The hypothesis (d2) on $\mathbf CMP$ implies that \[\text{${\mathfrak c}(\cO_\cK)$ is prime to $p{\mathfrak c}_{\mathbb I}i\mathfrak n D_{E/F}$}.\]
Note that ${\mathfrak c}(\cO_\cK)$ descends to a fractional ideal of $\cO_{\cF}$ and that ${\mathfrak c}(\cO_\cK)$ is the polarization of $x(1)=(\ul{A}(1),{\ell}p(1))$.
In addition, $x(a)=(\ul{A}(a),{\ell}p(a))_{/\mathbf C}$ is an
abelian variety with CM by $\mathcal O_E$ with the polarization ideal of $x(a)$ given by \[{\mathfrak c}(a):={\mathfrak c}(\cO_\cK)\mathrm N({\mathfrak a})^{-1}\quad({\mathfrak a}=\mathfrak i\frakl_E(a)).\]
It thus gives rise to a complex point $[x(a)]$ in $I_K(\frakc(a))(\mathbf C)$. Let $\mathbf CMring$ be the \text{$p$-adic } completion of the maximal unramified extension of $\mathbf Zp$ in $\mathbf Cp$. The general theory of CM abelian varieties shows that $[x(a)]$ indeed descends to a point in $I_{K}({\mathfrak c}(a))(\mathbf CMring)\hookrightarrowI_{K}(\mathbf CMring)$, which is still denoted by $x(a)$. The collection $\stt{[x(a)]}_{a\in (\mathbf AKf^{(pN)})^{\times}}\subset I_{K}(\mathbf CMring)$ are called \varepsilonsilonmph{CM points} in Hilbert modular Shimura varieties.

\newcommand\mfFC[1]{\bfa_\beta(\mathbf f_{#1}^*,{\mathfrak c})}
\def{\mathbb I}pi{\varPsi_{\pi,{\ell}am,v}}

\section{Anticyclotomic \text{$p$-adic } Rankin-Selberg $L$-functions}{\ell}abel{S:anticy_padic_L}
\subsection{Toric forms}
\def\mathbf CLKF{Cl_-}
\begin{defn}[The toric form]{\ell}abel{D:toricform}We define the complex Hilbert modular form $\mathbf f_{\mathbb I}i:X^+{\times} G(\mathbf AFf)\to\mathbf C$ associated to $\varphi_{\mathbb I}i$ by
\begin{equation}\begin{aligned}\mathbf f_{\mathbb I}i(\tau,g_f)=&
\varphi_{\mathbb I}i(g)\cdot \ul{J}(g_\infty,\mathbf i)^k(\det g_\infty)^{-\frac{(k_{mx}-2){\varSigma}+k}{2}}abelian variety s{\det g}^{k_{mx}/2-1}_{\mathbf AF},\\
&\quad (\mathbf i=(\sqrt{-1})_{\sigma\in{\varSigma}},\,g=(g_\infty,g_f),\,g_\infty \mathbf i=\tau,\,\det g_\infty>0).
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
Here $\det g_\infty=(\det g_{\sigma})_{\sigma\in{\varSigma}}\in (\mathbf R^{\times})^{\varSigma}$ and $\det g_\infty>0$ means $\det g_\sigma>0$ for all $\sigma\in{\varSigma}$. 

Let $\mathbf f_{\mathbb I}i^{*}$ be the normalization of $\mathbf f_{\mathbb I}i$ given by
\[\mathbf f_{\mathbb I}i^{*}= N(\pi,{\mathbb I}i)^{-1}abelian variety s{\det\varsigma_f}_{\mathbf AFf}^{1-k_{mx}/2}\cdot \mathbf f_{\mathbb I}i.\]
Let $\delta_k^m$ be the Shimura-Maass differential operator (\mbox{{\it cf.} }\cite[(1.21)]{HidaTilouine:KatzPadicL_ASENS}). Then the normalized differential operator $\widetilde V_+^m$ defined in \varepsilonsilonqref{E:Shi_Maass} is the representation theoretic avatar of $\delta_k^m$ in the following sense:

\[\delta_k^m\mathbf f_{\mathbb I}i(\tau,g_f)=(\widetilde V_+^m\varphi_{\mathbb I}i)(g_\infty,g_f)\ul{J}(g_\infty,i)^{k+2m}(\det g_\infty)^{1-\frac{k_{mx}{\varSigma}+k}{2}-m}abelian variety s{\det g}^{k_{mx}/2-1}_{\mathbf AF}\]
(\mbox{{\it cf.} } \cite[\S 4.5]{Hsieh:Hecke_CM}). We call $\delta_k^m\mathbf f^*_{\mathbb I}i$ the normalized toric form of character ${\mathbb I}i$.
\varepsilonsilonnd{defn}
Similarly, for each $u\in (\mathcal O_F\otimes_\mathbf Z\mathbf Zp)^{\times}$, we let $\mathbf f_{{\mathbb I}i,u}^*$ be the normalized modular form associated to the $u$-component $\varphi_{{\mathbb I}i,u}$ (\mbox{{\it cf.} } ${\ell}ocalW{{\mathbb I}i,u}$ in \varepsilonsilonqref{E:Wu.W}). It is clear from \varepsilonsilonqref{E:ucompnent.W} that
\begin{equation}{\ell}abel{E:13.W}\mathbf f_{\mathbb I}i^{*}=\sum_{u\in\mathcal U_p}\mathbf f^*_{{\mathbb I}i,u}.\varepsilonsilonnd{equation}
Let ${\ell}sgN$ be the open-compact subgroup defined in \varepsilonsilonqref{E:opncpt.W}. Then $\mathbf f_{\mathbb I}i^*$ and $\stt{\mathbf f_{{\mathbb I}i,u}^*}_{u\in\mathcal U_p}$ belong to $\mathbf M_k({\ell}sgN,\mathbf C)$ for sufficiently large $n$.

 For $a\in(\mathbf AKf^{(p)})^{\times} {\times}(\mathcal O_E\otimes\mathbf Zp)^{\times}$, we consider the Hecke action $|[a]$ given by
\begin{align*}
|[a]:\mathbf M_k({\mathfrak c}(a),{\ell}sgN,\mathbf C)&\to\mathbf M_k({\mathfrak c},{}_a{\ell}sgN,\mathbf C)\quad({}_a{\ell}sgN:=\iota_\varsigma(a){\ell}sgN\iota_\varsigma(a^{-1})), \\
\mathbf f&\mapsto\mathbf f|[a](\tau,g_f):=\mathbf f(\tau,g_f\iota_\varsigma(a)).\varepsilonsilonnd{align*}
The Hecke action $|[a]$ can be extended to the spaces of $p$-integral modular forms (\mbox{{\it cf.} }\cite[\S 2.6]{Hsieh:VMU}). It follows from {\ell}mref{L:4.W} immediately that
\begin{equation}{\ell}abel{E:17.W}\mathbf f^*_{{\mathbb I}i,u}|[a]={\mathbb I}i^{-1}\mathbf Abs^{k_{mx}/2-1}_{\mathbf AK}(a)\cdot \mathbf f^*_{{\mathbb I}i,u.a^{1-c}}\text{ for all }a\in\mathcal T_f\quad(u.a^{1-c}:=ua_{{\varSigma}_p}a_{{\varSigma}bar_p}^{-1}).\varepsilonsilonnd{equation}
\subsection{The toric period integral}
Next we consider the toric period integral of $\mathbf f_{\mathbb I}i^*$. Let $U_E=(E\otimes_\mathbf Q\mathbf R)^{\times}{\times}(\cO_\cK\otimes_\mathbf Z\mathbf Zhat)^{\times}$ be a subgroup of $\mathbf AK^{\times}$ and let $\mathbf CLKF=E^{\times}\mathbf AF^{\times}\backslash \mathbf AK^{\times}/U_E$. Let ${\mathcal R}$ be the subgroup of $\mathbf AK^{\times}$ generated by $E_v^{\times}$ for all ramified places $v$ and let $\mathbf CLKF^\alphag$ be the subgroup of $\mathbf CLKF$ generated by the image of ${\mathcal R}$ By {\ell}mref{L:4.W} and the fact that $\mathcal T=\mathbf AF^{\times} U_E{\mathcal R} $, we have
\begin{equation}{\ell}abel{E:5.W}P_{\mathbb I}i(\pi(\varsigma)\widetilde V_+^m\varphi_{\mathbb I}i)
=\vol(U_E,dt)\sharp(\mathbf CLKF^\alphag)\cdot \sum_{[t]\in\mathbf CLKF/\mathbf CLKF^\alphag}\widetilde V_+^m\varphi_{\mathbb I}i(\iota(t)\varsigma){\mathbb I}i(t).
\varepsilonsilonnd{equation}
Let $\mathcal D_1$ be a set of representatives of $\mathbf CLKF/\mathbf CLKF^\alphag$ in $(\mathbf AKf^{(pN)})^{\times}$. We define the ${\mathbb I}i$-isotypic toric period by
\[P_{\mathbb I}i(\delta_k^m\mathbf f_{\mathbb I}i^{*}):=\sum_{a\in \mathcal D_1}\delta_k^m\mathbf f_{\mathbb I}i^{*}(x(a)){\mathbb I}i\mathbf Abs_{\mathbf AK}^{1-k_{mx}/2}(a).\]
\begin{prop}{\ell}abel{P:1.W}Let $D_{E/F}$ be the discriminant of $E/F$. We have
\[P_{\mathbb I}i(\delta_k^m\mathbf f_{\mathbb I}i^{*})^2=[\mathcal O_E^{\times}:\mathcal O_F^{\times}]^2\cdot\frac{\Gamma_{\varSigma}(k+m)\Gamma_{\varSigma}(m+1)}{(\Im\mathbf CMP)^{k+2m}(4\pi)^{2m+k+1}}\cdot L(\mathbf 1half,\pi_E\otimes{\mathbb I}i)\cdot E_{{\varSigma}_p}(\pi,{\mathbb I}i)\cdot C(\pi,{\mathbb I}i),\]
where
\[C(\pi,{\mathbb I}i)= C'(\pi,{\mathbb I}i)\cdot 4^{-[F:\mathbf Q]}abelian variety s{\mathrm N_{F/\mathbf Q}(D_{E/F})}_\mathbf R^{\mathbf 1half}{\ell}eft(\frac{\sharp(\mathbf CLKF)h_F}{\sharp(\mathbf CLKF^\alphag)h_E}\right)^2\in\mathbf Zbar_{(p)}^{\times}.\]
\varepsilonsilonnd{prop}
\begin{proof}
By definition, we have
\[\mathbf f_{\mathbb I}i(x(a))=\varphi_{\mathbb I}i(\iota(a_f)\varsigma)(\Im \mathbf CMP)^{-k/2}\cdotabelian variety s{\mathrm N(a)\det\varsigma_f}_{\mathbf AF}^{k_{mx}/2-1}.\]
By \varepsilonsilonqref{E:5.W}, we find that
\begin{align*}
\vol(U_E,d^\x t)\sharp(\mathbf CLKF^\alphag)\cdot P_{\mathbb I}i(\delta_k^m\mathbf f_{\mathbb I}i^{*})
&=\frac{1}{N(\pi,{\mathbb I}i)\cdot (\Im\mathbf CMP)^{k/2+m}}\cdot P_{\mathbb I}i(\pi(\varsigma)\widetilde V_+^m\varphi_{\mathbb I}i).
\varepsilonsilonnd{align*}
From the well-known formula
\[2L(1,\tau_{E/F})=(2\pi)^{[F:\mathbf Q]}\cdot\frac{h_{E}/h_F}{abelian variety s{D_E}_\mathbf R^\mathbf 1halfabelian variety s{D_F}_\mathbf R^{-\mathbf 1half}\cdot [\mathcal O_E^{\times}:\mathcal O_F^{\times}]},\]
we see that
\begin{align*}\vol(U_E,dt)=&\vol(E^{\times}\mathbf AF^{\times}\backslash \mathbf AK^{\times},dt)\cdot \sharp(\mathbf CLKF)^{-1}\\
=&2\pi^{-[F:\mathbf Q]}L(1,\tau_{E/F})\cdot \sharp(\mathbf CLKF)^{-1}=\frac{2^{[F:\mathbf Q]}abelian variety s{D_F}_\mathbf R^\mathbf 1half}{abelian variety s{D_{E}}_\mathbf R^\mathbf 1half [\mathcal O_E^{\times}:\mathcal O_F^{\times}]}\cdot\frac{h_E}{h_F\sharp(\mathbf CLKF)}.\varepsilonsilonnd{align*}
The proposition follows form \thmref{T:main1.W} immediately. Note that the ratio $\frac{\sharp(\mathbf CLKF)h_F}{\sharp(\mathbf CLKF^\alphag)h_E}$ is a power of $2$, so the constant $C(\pi,{\mathbb I}i)$ is a \text{$p$-adic } unit.
\varepsilonsilonnd{proof}

\subsection{The Fourier expansion of $\mathbf f_{{\mathbb I}i,u}^*$}
Let $u=(u_v)\in\mathcal U_p$. We give an expression of the Fourier expansion of $\mathbf f_{{\mathbb I}i,u}^{*}$. Let ${\ell}ocalW{{\mathbb I}i,u,f}$ be the finite part of ${\ell}ocalW{{\mathbb I}i,u}$. By the definition of $\mathbf f_{{\mathbb I}i,u}$, we have
\begin{equation}{\ell}abel{E:12.W}\begin{aligned}
\mathbf f_{{\mathbb I}i,u}(\tau,g_f)&=\sum_{\beta\inF}{\ell}ocalW{{\mathbb I}i,u,f}(\DII{\beta}{1}g_f){\ell}ocalW{{\mathbb I}i,\infty}(\DII{\beta}{1}\MX{y_\infty}{x_\infty}{0}{1})\cdot y_\infty^{-k/2}\\
&=\sum_{\beta\inF_+}{\ell}ocalW{{\mathbb I}i,u,f}(\DII{\beta}{1}g_f)\beta^{k/2}e^{2\pi i{\mathbb T}r_{F/\mathbf Q}(\beta\tau)}.\\
&\quad(\tau=x_\infty+iy_\infty=(x_\sigma+iy_\sigma)_{\sigma\in{\varSigma}}\in X^+)
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
The second equality follows from the choice of Whittaker functions at the archimedean places \varepsilonsilonqref{E:archiWhittaker.W}.

We define the global prime-to-$p$ Fourier coefficient ${\ell}ocalK{{\mathbb I}i}^{(p)}:(\mathbf AFf^{(p)})^{\times}\to\mathbf C$ by
\begin{equation}{\ell}abel{E:FC.W}\begin{aligned}{\ell}ocalK{{\mathbb I}i}^{(p)}(a):=&N(\pi,{\mathbb I}i)^{-1}\cdot {\ell}ocalW{{\mathbb I}i,f}^{(p)}(\DII{a}{1})\quad (a=(a_v)\in\mathbf AFf^{\times})\\
=&\indent od_{v\in B({\mathbb I}i)}\frac{1}{n_vL(1,\tau_{E_v/F_v})}{\ell}ocalW{{\mathbb I}i,v}(\DII{a_v}{1})\indent od_{v\not\in B({\mathbb I}i),v\nmid p}{\ell}ocalW{{\mathbb I}i,v}(\DII{a_v}{1})\\
=&\indent od_{v\in\mathbf h,v\nmid p}{\ell}ocalK{{\mathbb I}i,v}^*(a_v).
\varepsilonsilonnd{aligned}
\varepsilonsilonnd{equation}
Here ${\ell}ocalK{{\mathbb I}i,v}^*$ are the local Fourier coefficients defined in \defref{D:localFourier.W}.
\begin{prop}{\ell}abel{P:2.W} Let ${\mathfrak c}$ be a prime-to-$p$ ideal of $F$ and let $\mathbf c\in (\mathbf AKf^{{(p)}})^{\times}$ such that $\mathfrak i\frakl_F(\mathbf c)={\mathfrak c}$. Then the Fourier expansion of $\mathbf f_{{\mathbb I}i,u}^{*}$ at the cusp $(\cO_{\cF},{\mathfrak c})$ is given by
\[\mathbf f_{{\mathbb I}i,u}^{*}|_{(\cO_{\cF},{\mathfrak c})}(q)=\sum_{\beta\in (N^{-1}{\mathfrak c})_+}\bfa_\beta(\mathbf f_{{\mathbb I}i,u}^{*},{\mathfrak c})q^\beta,\]
where
\[\bfa_\beta(\mathbf f_{{\mathbb I}i,u}^{*},{\mathfrak c})=\beta^{k/2}{\ell}ocalK{{\mathbb I}i}^{(p)}(\beta\mathbf c^{-1})\indent od_{w\in{\varSigma}bar_p,v|w}{\mathbb I}i_{w}(\beta^{-1})\mathbb I_{u_v(1+\varpi_v\mathcal O_{F_v})}(\beta).\]
In particular, $\mathbf f_{{\mathbb I}i,u}^*\in \mathbf M_k({\ell}sgN,{\EuScript O})$ by \indent opref{P:4.W}, and the Fourier expansion of $\mathbf f_{\mathbb I}i^*$ at the cusp $(\cO_{\cF},{\mathfrak c})$ is given by
\[\mathbf f_{{\mathbb I}i}^{*}|_{(\cO_{\cF},{\mathfrak c})}(q)=\sum_{\beta\in(N^{-1}{\mathfrak c})_+}\bfa_\beta(\mathbf f_{{\mathbb I}i}^{*},{\mathfrak c})q^\beta,\]
where
\[\bfa_\beta(\mathbf f_{{\mathbb I}i}^{*},{\mathfrak c})=\beta^{k/2}{\ell}ocalK{{\mathbb I}i}^{(p)}(\beta\mathbf c^{-1})\indent od_{w\in{\varSigma}bar_p}{\mathbb I}i_{w}(\beta^{-1})\cdot \mathbb I_{\mathcal O_{F,{(p)}}^{\times}}(\beta).\]
\varepsilonsilonnd{prop}
\begin{proof}It follows from the definition of ${\ell}ocalW{{\mathbb I}i,u}$ that
\begin{align*}{\ell}ocalW{{\mathbb I}i,u,f}(\DII{\beta\mathbf c^{-1}}{1})=&{\ell}ocalW{{\mathbb I}i,f}^{(p)}(\DII{\beta\mathbf c^{-1}}{1})\cdot\indent od_{v|p}{\ell}ocalW{{\mathbb I}i,u_v,v}(\DII{\beta}{1})\\
=&{\ell}ocalW{{\mathbb I}i,f}^{(p)}(\DII{\beta\mathbf c^{-1}}{1})\cdot \indent od_{w\in{\varSigma}bar_p,v|w}{\mathbb I}i_{w}^{-1}(a_v)\mathbb I_{u_v(1+\varpi_v\mathcal O_{F_v})}(\beta).
\varepsilonsilonnd{align*}
The proposition follows from \varepsilonsilonqref{E:12.W} immediately. The Fourier expansion of $\mathbf f_{\mathbb I}i^*$ follows from \varepsilonsilonqref{E:13.W}.
\varepsilonsilonnd{proof}
\defO_p{O_p}
\subsection{\text{$p$-adic } $L$-functions}{\ell}abel{SS:padicL}
We go back to the setting in the introduction. Let $E^-_{p^\infty}$ be the maximal anticyclotomic $\mathbf Zp^{[F:\mathbf Q]}$-extension of $E$ and let $\Gamma^-=\Gal(E^-_{p^\infty}/E)$. The reciprocity law $\rec_E$ at ${\varSigma}bar_p$ induces a morphism \[\rec_{{\varSigma}bar_p}\colon (F\otimes_\mathbf Q\mathbf Qp)^{\times}\simeq\indent od_{w\in{\varSigma}bar_p}E_{w}^{\times}\stackrel{\rec_E}{\ell}ongrightarrow \Gamma^-.\]
Let $ksp$ be the set of critical specializations, consisting of \text{$p$-adic } characters $\phi:\Gamma^-\to\mathbf Cp^{\times}$ such that for some $m\in\mathbf Z_{\geq 0}[{\varSigma}]\simeq\mathbf Z_{\geq 0}[{\varSigma}_p]$,
\[\phi(\rec_{{\varSigma}bar_p}(x))=x^m\text{ for all $x\in(\mathcal O_F\otimes\mathbf Zp)^{\times}$ sufficiently close to $1$.}\]
Let $\phi$ be an anticyclotomic Hecke character of $p$-power conductor and of infinity type $(m,-m)$ with $m\in\mathbf Z_{\geq 0}[{\varSigma}]$. Then $\phi$ is unramified outside $p$ and $\phi|_{\mathbf AF^{\times}}=1$.
The \text{$p$-adic } avatar $\widehat\phi$ of $\phi$ belongs to $ksp$. To be precise, let $\phi_{{\varSigma}bar_p}:=\indent od_{w\in{\varSigma}bar_p}\phi_w$. Then we have \begin{equation}{\ell}abel{E:4.W}\widehat\phi(\rec_{{\varSigma}bar_p}(x))=\phi_{{\varSigma}bar_p}(x)x^{-m}\text{ for every }x\in (F\otimes_\mathbf Q\mathbf Qp)^{\times}.\varepsilonsilonnd{equation} Hereafter, we let ${\ell}am$ be a Hecke character of $E^{\times}$ and assume that \hypref{H:HypA} and \varepsilonsilonqref{sf} hold for $(\pi,{\ell}am)$. Note that \hypref{H:HypA} and \varepsilonsilonqref{sf} also hold for $(\pi,{\ell}am\phi)$.
We will apply our calculations in \subsecref{S:toric} to the pair $(\pi,{\mathbb I}i)=(\pi,{\ell}am\phi)$.
\begin{lm}{\ell}abel{L:2.W} Let $\phi$ be as above. Then
\begin{mylist}
\item ${\ell}ocalK{{\ell}am\phi}^{(p)}={\ell}ocalK{{\ell}am}^{(p)}$.
\item $C'(\pi,{\ell}am\phi)=C'(\pi,{\ell}am)\phi(\mathbf Csplit)$.
\varepsilonsilonnd{mylist}
\varepsilonsilonnd{lm}
\begin{proof}If $v\nmid p$ is split, we have remarked that ${\ell}ocalW{{\mathbb I}i_v\phi_v^{-1},v}={\ell}ocalW{{\mathbb I}i_v,v}$. If $v$ is inert or ramified, then $\phi_v=1$ as $\phi_v$ is unramified and $p>2$. Therefore, we have ${\ell}ocalW{{\mathbb I}i\phi,f}^{(p)}={\ell}ocalW{{\mathbb I}i,f}^{(p)}$. Part (1) follows from the definition of ${\ell}ocalK{{\ell}am\phi}^{(p)}$ \varepsilonsilonqref{E:FC.W} immediately.
Next, recall that we have defined $C'(\pi,{\mathbb I}i)$ for a Hecke character ${\mathbb I}i$ in \varepsilonsilonqref{E:constantC.W}. Since $\phi$ is anticyclotomic and unramified outside $p$, part (2) follows from the well-known fact that
\[\varepsilonsilonp(\mathbf 1half,\pi_v\otimes{\ell}am_w\phi_w,\psi_v)=\varepsilonsilonp(\mathbf 1half,\pi_v\otimes{\ell}am_w,\psi)\phi_w(\mathcal D_F^2\mathbf Csplit)\quad(v=w\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{w},\,w|\mathbf Csplit).\qedhere\]
\varepsilonsilonnd{proof}
Let $O_p:=\mathcal O_F\otimes_\mathbf Z\mathbf Zp$ and let $\Gamma':=\rec_{{\varSigma}bar_p}(1+pO_p)$ be an open subgroup of $\Gamma^-$. Let $\stt{\theta(\sigma)}_{\sigma\in{\varSigma}}$ be the Dwork-Katz \text{$p$-adic } differential operators (\cite[Cor.\,(2.6.25)]{Katz:p_adic_L-function_CM_fields}) and let $\theta^m:=\indent od_{\sigma\in{\varSigma}}\theta(\sigma)^{m_\sigma}$.
\begin{prop}{\ell}abel{P:3.W}There exists a unique $V({\mathfrak c},K,\mathbf Zbarp)$-valued \text{$p$-adic } measure ${\EuScript F}_{{\ell}am,{\mathfrak c}}$ on $\Gamma^-$ such that
\begin{itemize}\item[(i)] ${\EuScript F}_{{\ell}am,{\mathfrak c}}$ is supported in $\Gamma'$,
\item[(ii)] for every $\widehat\phi\inksp$ of weight $(m,-m)$, we have
\[\int_{\Gamma^-}\widehat\phi d{\EuScript F}_{{\ell}am,{\mathfrak c}}=\theta^m\widehat\mathbf f_{{\ell}am\phi,{\mathfrak c}}^*.\]
\varepsilonsilonnd{itemize}
\varepsilonsilonnd{prop}
\begin{proof}
We denote by ${\EuScript F}_{{\ell}am,{\mathfrak c}}(q)$ the \text{$p$-adic } measure with values in the space of formal $q$-expansions such that for every $\varphi\in C(\Gamma^-,\mathbf Zbarp)$,
\[\int_{\Gamma^-}\varphi d{\EuScript F}_{{\ell}am,{\mathfrak c}}(q)=\sum_{\beta\in(N^{-1}{\mathfrak c})_+}\mfFC{{\ell}am}\varphi(\rec_{{\varSigma}bar_p}(\beta^{-1}))q^\beta.\]
Note that $\mfFC{{\ell}am}=0$ unless $\beta\in\mathcal O_{F,{(p)}}^{\times}$ and that ${\EuScript F}_{{\ell}am,{\mathfrak c}}$ has support in $\Gamma'$ by definition.

Let $\widehat\phi$ be the \text{$p$-adic } avatar of a Hecke character $\phi$ of infinity type $(m,-m)$. By \cite[(2.6.27)]{Katz:p_adic_L-function_CM_fields} (\mbox{{\it cf.} }\cite[\S 1.7 p.205]{HidaTilouine:KatzPadicL_ASENS}), the $q$-expansion of $\theta^m\widehat\mathbf f_{{\ell}am\phi}^{*}$ is given by
\begin{align*}\theta^m\widehat\mathbf f_{{\ell}am\phi}^{*}|_{(\cO_{\cF},{\mathfrak c})}(q)=&\sum_{\beta\in (N^{-1}{\mathfrak c})_+}\mfFC{{\ell}am\phi}\phi_{{\varSigma}bar_p}(\beta^{-1})\beta^m q^\beta. \varepsilonsilonnd{align*}
Therefore, by {\ell}mref{L:2.W} and \varepsilonsilonqref{E:4.W} we find that
\begin{equation}{\ell}abel{E:53.N}\int_{\Gamma^-}\widehat{\phi} d{\EuScript F}_{{\ell}am,{\mathfrak c}}(q)=\theta^m\widehat\mathbf f_{{\ell}am\phi,{\mathfrak c}}^*(q).\varepsilonsilonnd{equation}
By the $q$-expansion principle, this measure descends to the \text{$p$-adic } measure ${\EuScript F}_{{\ell}am,{\mathfrak c}}$ with values in the space of \text{$p$-adic } modular forms $V({\mathfrak c},K,\mathbf Zbarp)$.
\varepsilonsilonnd{proof}
Let ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ be the \text{$p$-adic } measure on $\Gamma^-$ such that for each $\varphihi\in\mathcal C(\Gamma^-,\mathbf Zbarp)$,
\begin{equation}{\ell}abel{E:period_measure.E}\int_{\Gamma^-}\varphihi d{\mathcal P}_{\varSigma}(\pi,{\ell}am)=\sum_{a\in\mathcal D_1}\widetilde{\ell}am(a)\int_{\Gamma^-}\varphihi|[a] d{\EuScript F}_{{\ell}am,{\mathfrak c}(a)}(x(a))\quad(\widetilde{\ell}am={\ell}am\cdot\mathbf Abs_{\mathbf AK}^{1-k_{mx}/2}).\varepsilonsilonnd{equation}
Here $\varphihi|[a](x):=\varphihi(x\rec_{E^-_\infty/E}(a))$. Let $(\boldsymbol{\omega}ega_\infty,\boldsymbol{\omega}ega_p)\in(\mathbf C^{\times})^{\varSigma}{\times}(\mathbf Zbarp^{\times})^{\varSigma}$ be the complex and \text{$p$-adic } CM periods of $(E,{\varSigma})$ introduced in \cite[(4.4 a,b) p.211]{HidaTilouine:KatzPadicL_ASENS} (\mbox{{\it cf.} } $(\boldsymbol{\omega}ega,c)$ in \cite[(5.1.46), (5.1.48)]{Katz:p_adic_L-function_CM_fields}) and let $\boldsymbol{\omega}ega_E=(2\pi i)^{-1}\boldsymbol{\omega}ega_\infty$. We have the following evaluation formula of ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$.
\begin{thm}{\ell}abel{T:padicL.W} Suppose that \hypref{H:HypA} and \varepsilonsilonqref{sf} hold. Then for each \text{$p$-adic } character $\widehat\phi\inksp$ of weight $(m,-m)$, we have the evaluation formula
\begin{align*}{\ell}eft(\frac{1}{\boldsymbol{\omega}ega_p^{k+2m}}\int_{\Gamma^-}\widehat\phi d{\mathcal P}_{\varSigma}(\pi,{\ell}am)\right)^2=&[\mathcal O_E^{\times}:\mathcal O_F^{\times}]^2\cdot \frac{\Gamma_{\varSigma}(k+m)\Gamma_{\varSigma}(m+1)}{(\Im\mathbf CMP)^{k+2m}(4\pi)^{k+2m+1}}\\
&\times E_{{\varSigma}_p}(\pi,{\ell}am\phi)\cdot \frac{L(\mathbf 1half,\pi_E\otimes{\ell}am\phi)}{\boldsymbol{\omega}ega_E^{2(k+2m)}}\cdot \phi(\mathbf Csplit)C(\pi,{\ell}am).\varepsilonsilonnd{align*}
\varepsilonsilonnd{thm}
\begin{proof}
From \cite[(2.4.6), (2.6.8), (2.6.33)]{Katz:p_adic_L-function_CM_fields} we can deduce that
\[\frac{1}{\boldsymbol{\omega}ega_p^{k+2m}}\theta^m\widehat\mathbf f_{{\ell}am\phi}^{*}(x(a))=\frac{1}{\boldsymbol{\omega}ega_E^{k+2m}}\delta_k^m\mathbf f^*_{{\ell}am\phi}(x(a)).\]
Therefore, we have
\begin{align*}
\frac{1}{\boldsymbol{\omega}ega_p^{k+2m}}\cdot \int_{\Gamma^-}\widehat\phi d{\mathcal P}_{\varSigma}(\pi,{\ell}am)=&\sum_{a\in\mathcal D_1}\widetilde{\ell}am(a)\phi(a)\cdot\frac{1}{\boldsymbol{\omega}ega_p^{k+2m}}\theta^m\widehat\mathbf f_{{\ell}am\phi}^{*}(x(a))\\
=&\frac{1}{\boldsymbol{\omega}ega_E^{k+2m}}\cdot\sum_{a\in\mathcal D_1}\widetilde{\ell}am\phi(a)\delta_k^m\mathbf f^*_{{\ell}am\phi}(x(a))=\frac{1}{\boldsymbol{\omega}ega_E^{k+2m}}\cdot P_{{\ell}am\phi}(\delta_k^m\mathbf f^*_{{\ell}am\phi}).
\varepsilonsilonnd{align*}
Combined with \indent opref{P:1.W} and {\ell}mref{L:2.W} (2), the above equation yields the proposition.
\varepsilonsilonnd{proof}

\section{The $\mu$-invariant of \text{$p$-adic } $L$-functions}{\ell}abel{S:mu_invariant}

\def\mathbb Fadsu{\varphi_{\pi,{\ell}am,u}}
In this section, we use the explicit computation of Fourier coefficients of $\stt{\mathbf f_{{\ell}am,u}^*}_{u\in\mathcal U_p}$ to study the $\mu$-invariant of the \text{$p$-adic } measure ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ by the approach of Hida \cite{Hida:mu_invariant}.
\subsection{The $t$-expansion of \text{$p$-adic } modular forms} We begin with a brief review of the $t$-expansion of \text{$p$-adic } modular forms. A functorial point in $I_K({\mathfrak c})$ can be written as $[(\ul{A},{\ell}p)]=[(A,{\ell}am,\iota,\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{algebraically closed ute{e}t}a}^{(p)},{\ell}p)]$. Enlarging $\mathbf CMring$ if necessary, we let $\mathbf CMring$ be the \text{$p$-adic } ring generated by the values of ${\ell}am$ on finite ideles over the Witt ring $W(\mathbb Fpbar)$. Let $\mathfrak m_\mathbf CMring$ be the maximal ideal of $\mathbf CMring$ and fix an isomorphism $\mathbf CMring/\mathfrak m_\mathbf CMring\stackrel{\sim}{\to}\mathbb Fpbar$.
Let $T:=\cO_{\cF}^*\otimes_\mathbf Z\boldsymbol{\mu}_{p^\infty}$ and let $\widehat T=\varinjlim_m T_{/\mathbf CMring/\mathfrak m_\mathbf CMring^m}=\cO_{\cF}^*\otimes_\mathbf Z\formal{\bbG}_m$. Let $\stt{{\times}i_1\cdots,{\times}i_d}$ be a basis of $\cO_{\cF}$ over $\mathbf Z$ and let $t$ be the character $1\in \cO_{\cF}=X^*(\cO_{\cF}^*\otimes_\mathbf Z{\bbG}_{m}}   \def\Ga{{{\bbG}_{\rm a}})=\Hom(\cO_{\cF}^*\otimes_\mathbf Z{\bbG}_{m}}   \def\Ga{{{\bbG}_{\rm a}},{\bbG}_{m}}   \def\Ga{{{\bbG}_{\rm a}})$. Then we have $\mathcal O_{\widehat T}\stackrel{\sim}{\to} \mathbf CMring\powerseries{t^{{\times}i_1}-1,\cdots t^{{\times}i_d}-1}$.
For $y=(\ul{A}_y,{algebraically closed ute{e}t}a_y)\inI_K({\mathfrak c})(\mathbb Fpbar)\subset I_K(\mathbb Fpbar)$, it is well known that the deformation space $\widehat S_{y}$ of $y$ is isomorphic to the formal torus $\widehat T$ by the theory of Serre-Tate coordinate (\cite{Katz:ST}). The $p^\infty$-level structure ${\ell}p_y$ of $A_y$ induces a canonical isomorphism $\varphihi_{y}\colon\widehat{T}_{/\cW}\stackrel{\sim}{\to} \widehat{S}_{y}=\Spf \widehat\mathcal O_{I_K({\mathfrak c}),y}$ (\mbox{{\it cf.} } \cite[(3.15)]{Hida:mu_invariant}).

Now let $\bfx:=x(1)_{/\mathbf CMring}\inI_K({\mathfrak c})(\mathbf CMring)$ be a fixed CM point of type $(E,{\varSigma})$ and let $x_0=\bfx\otimes_{\mathbf CMring}\mathbb Fpbar=(\ul{A}_0,{\ell}p_0)$.
For a deformation $z=(\ul{A},{\ell}p)_{/{\mathcal R}}$ of $x_0$ over an artinian local ring ${\mathcal R}$ with the maximal ideal $\mathfrak m_{\mathcal R}$ and the residue field $\mathbb Fpbar$, we let $t(\ul{A},{\ell}p):=t(\varphihi_{x_0}^{-1}((\ul{A},{\ell}p)_{/{\mathcal R}}))\in 1+\mathfrak m_{\mathcal R}$. Then $\bfx$ is the canonical lifting of $x_0$, i.e.  $t(\bfx)=1$. For $f\in V({\mathfrak c},K,\mathbf CMring)$, we define \[f(t):=\varphihi_{x_0}^*(f)\in\mathcal O_{\widehat T}=\mathbf CMring\powerseries{T_1,\cdots T_d}\quad(T_i=t^{{\times}i_i}-1).\]
The formal power series $f(t)$ is called the \varepsilonsilonmph{$t$-expansion} around $x_0$ of $f$. For each $u\inO_p^{\times}$, let $uz:=(\ul{A},u{\ell}p)$ is a deformation of $ux_0$. Then we have $t(uz)=t(z)^u$ and hence $\varphihi_{ux_0}^*(f)(t)=\varphihi_{x_0}^*(f)(t^u)=f(t^u)$.
\subsection{The vanishing of the $\mu$-invariant}
Let $\pi_-:(\mathbf AKf^{(pN)})^{\times}\to \Gamma^-$ be the natural map induced by the reciprocity law. Let $Z'=\pi_-^{-1}(\Gamma')$ be a subgroup of $(\mathbf AKf^{(pN)})^{\times}$ and let $Cl'_-\supset Cl^\alphag_-$ be the image of $Z'$ in $Cl_-$. Let $\mathcal D_1'$ (resp. $\mathcal D_1''$) be a set of representative of $Cl_-'/Cl_-^\alphag$ (resp. $Cl_-/Cl_-'$) in $(\mathbf AKf^{(pN)})^{\times}$. Let $\mathcal D_1:=\mathcal D_1'\mathcal D_1''$ be a set of representative of $Cl_-/Cl_-^\alphag$. Recall that $\mathcal U_p$ is the torsion subgroup of $O_p^{\times}$. Let $\mathcal U$ be the torsion subgroup of $E^{\times}$ and let $\mathcal U^\alphag=(E^{\times})^{1-c}{\mathcal a}p \cO_\cK^{\times}$ be a subgroup of $\mathcal U$. We regard $\mathcal U^\alphag$ as a subgroup of $O_p$ by the imbedding induced by ${\varSigma}bar_p$. Let $\mathcal D_0$ be a set of representatives of $\mathcal U_p/\mathcal U^\alphag$ in $\mathcal U_p$.

Fix ${\mathfrak c}:={\mathfrak c}(\cO_\cK)$ to be the polarization ideal of the CM point $x(1)$. The following theorem reduces the calculation of the $\mu$-invariant $\mu^-_{\pi,{\ell}am,{\varSigma}}$ to the determination of the $q$-expansion of $\mathbf f^*_{{\ell}am,u}$.
\begin{thm}{\ell}abel{T:linear.V}Suppose that $p$ is unramified in $F$. Then
\[\mu^-_{\pi,{\ell}am,{\varSigma}}= \inf_{\substack{(a,u)\in \mathcal D_1{\times}\mathcal D_0 \\ \beta\inF_+}} v_p(\bfa_\beta(\mathbf f^*_{{\ell}am,u},{\mathfrak c}(a))).\]
\varepsilonsilonnd{thm}
\begin{proof}For every pair $(u,a)\in\mathcal U_p{\times} \mathcal D_1$, we let $\mathbf f^*_{u,a}:=\mathbf f^*_{{\ell}am,u}|_{{\mathfrak c}(a)}\in\mathcal M_k({\mathfrak c}(a),K,{\EuScript O})$. Let ${\EuScript F}_{u,a}$ be the \text{$p$-adic } avatar of $\mathbf f^*_{u,a}$. Fix a sufficient large finite extension $L$ over $\mathbf Qp$ so that $\chi$ and $\mathbf f^*_{u,a}|[a]$ are defined over $\mathcal O_L$ for all $(u,a)$, and hence ${\EuScript F}_{u,a}|[a]\in V({\mathfrak c},\mathcal O_L)$.  For each $z\in Z'$, let $\Dmd{z}$ be the unique element in $1+pO_p$ such that $\rec_{{\varSigma}bar_p}(\Dmd{z})=\pi_-(z)\in\Gamma^-$. For $(a,b)\in\mathcal D_1{\times}\mathcal D_1''$, we define
\begin{align*}\widetilde{\EuScript F}_a(t)&=\sum_{u\in \cU_p}{\EuScript F}_{u,a}(t^{u^{-1}}),\\
{\EuScript F}^b(t)&=\sum_{a\in b\mathcal D_1}\widetilde{\ell}am(ab^{-1})\widetilde{\EuScript F}_a|[a](t^{\Dmd{ab^{-1}}}).\varepsilonsilonnd{align*}
Let ${\mathcal P}^b_{{\varSigma}}(\pi,{\ell}am)$ be the \text{$p$-adic } measure on $1+pO_p\simeq\Gamma'$ obtained by the restriction of ${\mathcal P}_{\varSigma}(\pi,{\ell}am)$ to $\pi_-(b)\Gamma'$. In other words,
\begin{align*}\int_{\Gamma'}\varphi d{\mathcal P}^b_{\varSigma}(\pi,{\ell}am):=&\int_{\Gamma^-}\mathbb I_{b.\Gamma'}\cdot\varphi|[b^{-1}]d{\mathcal P}_{\varSigma}(\pi,{\ell}am)\\
=&\sum_{a\in b\mathcal D_1'}\widetilde{\ell}am(a)\int_{\Gamma^-}\varphi|[ab^{-1}]d{\EuScript F}_{{\ell}am,{\mathfrak c}(a)}(x(a)).
\varepsilonsilonnd{align*}
Here the second equality follows from the fact that ${\EuScript F}_{{\ell}am,{\mathfrak c}(a)}$ has support in $\Gamma'$ (\indent opref{P:3.W} (i)). The argument of \cite[Prop.\,5.2]{Hsieh:VMU} shows that ${\EuScript F}^b(t)$ is the power series expansion of the measure ${\mathcal P}^b_{\varSigma}(\pi,{\ell}am)$ regarded as a \text{$p$-adic } measure on $O_p$ and that
\begin{align*}\mu^-_{\pi,{\ell}am,{\varSigma}}&=\inf_{b\in\mathcal D_1''}\mu({\EuScript F}^b),\text{ where}\\
&\mu({\EuScript F}^b):=\inf\stt{r\in\mathbf Q_{\geq 0}\mid p^{-r}{\EuScript F}^b\not\equiv 0\pmod{\mathfrak m_p}}.\varepsilonsilonnd{align*}
By \varepsilonsilonqref{E:17.W} we find that
\[\widetilde{\EuScript F}_a(t)=\sharp(\mathcal U^{\alphag})\cdot \sum_{u\in\mathcal D_0}{\EuScript F}_{u,a}(t^{u^{-1}}),\]
and hence
\[{\EuScript F}^b(t)=\sharp(\mathcal U^{\alphag})\cdot \sum_{(u,a)\in \mathcal D_0{\times} b\mathcal D_1'}\widetilde{\ell}am(ab^{-1}){\EuScript F}_{u,a}|[a](t^{\Dmd{ab^{-1}} u^{-1}}).\]
Proceeding along the same lines in \cite[Thm.\,5.5]{Hsieh:VMU}, we can deduce the theorem from the above equation by the linear independence of \text{$p$-adic } modular forms modulo $p$ acted by the automorphisms in $\mathcal D_0{\times} \mathcal D_1'$ (\cite[Thm.\,3.20, Cor.\,3.21]{Hida:mu_invariant}) and the $q$-expansion principle for \text{$p$-adic } modular forms.
\varepsilonsilonnd{proof}

\begin{thm}{\ell}abel{T:mu-invariant}In addition to \hypref{H:HypA} and \varepsilonsilonqref{sf}, we suppose that $p$ is unramified in $F$ and \begin{equation}cd{ai${}_E$}\text{the residual Galois representation $\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{\rho}_p(\pi_E):=\rho_p(\pi)|_{G_E}\pmod{\mathfrak m_p}$ is absolutely irreducible}.\varepsilonsilonnd{equation}cd
Then $\mu^-_{\pi,{\ell}am,{\varSigma}}=0$ if and only if
\[\sum_{v|{\mathfrak c}^-_{\ell}am}\mu_p({\mathbb I}pi)=0,\]
where $\mu_p({\mathbb I}pi)$ are the local invariants defined as in \varepsilonsilonqref{E:9.W}.
\varepsilonsilonnd{thm}
\begin{proof}
It is not difficult to deduce from the formula of $\bfa_\beta(\mathbf f^*_{{\ell}am,u},{\mathfrak c}(a))$ in \indent opref{P:2.W} and \indent opref{P:4.W} that
\[\mu_p({\mathbb I}pi)>0\text{ for some $v|{\mathfrak c}_{\ell}am^-$}\Rightarrow \bfa_\beta(\mathbf f^*_{{\ell}am,u},{\mathfrak c}(a))\equiv 0\pmod{\mathfrak m}\text{ for all }a\in \mathbf AFf^{\times},\]
and hence $\mu^-_{\pi,{\ell}am,{\varSigma}}>0$ by \thmref{T:linear.V}.

Conversely, we suppose that $\mu_p({\mathbb I}pi)=0$ for all $v|{\mathfrak c}^-_{\ell}am$. We are going to show $\mu^-_{\pi,{\ell}am,{\varSigma}}=0$ by contradiction. Assume that $\mu^-_{\pi,{\ell}am,{\varSigma}}>0$. By \indent opref{P:2.W} \thmref{T:linear.V}, for each $a\in\mathbf AKf^{(pN)}$ we find that
\begin{align*}&\bfa_\beta(\mathbf f_{{\ell}am,u}^*,{\mathfrak c}(a))\equiv 0\pmod{\mathfrak m}\text{ for all }u\in\mathcal U_p\text{ and }\beta\inF_+\\
\iff&{\ell}ocalK{{\ell}am}^{(p)}(\beta \mathbf c^{-1}\mathrm N(a^{-1}))\equiv 0\pmod{\mathfrak m}\text{ for all }\beta\in\mathcal O_{F,{(p)}}^{\times}.\varepsilonsilonnd{align*}
Therefore, as a function on $(\mathbf AFf^{(p)})^{\times}$, we have
\begin{equation}{\ell}abel{E:8.W}\begin{aligned}{\ell}ocalK{{\ell}am}^{(p)}(a)&\equiv 0\pmod{\mathfrak m}\text{ for all }\\
&a\in \mathcal O_{F,{(p)}}^{\times} \mathbf c^{-1} \det(U(N))\mathrm N((\mathbf AKf^{(pN)})^{\times})=F^{\times}\mathbf c^{-1}\mathrm N((\mathbf AKf^{(p)})^{\times}).\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
By \indent opref{P:4.W}, there exists ${algebraically closed ute{e}t}a=({algebraically closed ute{e}t}a_v)\in\indent od_{v|{\mathfrak c}^-_{\mathbb I}i}F^{\times}_v$ such that ${\ell}ocalK{{\ell}am,v}^*({algebraically closed ute{e}t}a_v)\not \equiv 0\pmod{\mathfrak m}$ for each $v|{\mathfrak c}_{\ell}am^-$. We extend ${algebraically closed ute{e}t}a$ to be the idele in $\mathbf AFf^{\times}$ such that ${algebraically closed ute{e}t}a_v=1$ at $v\nmid{\mathfrak c}_{\mathbb I}i$. Therefore,
\varepsilonsilonqref{E:8.W} together with the factorization formula of ${\ell}ocalK{{\ell}am}^{(p)}$ \varepsilonsilonqref{E:FC.W} imply that for each uniformizer $\varpi_v$ at $v\nmid p\frakr$, we have
\begin{equation}{\ell}abel{E:14.W}\begin{aligned}{\ell}ocalK{{\ell}am}^{(p)}({algebraically closed ute{e}t}a\varpi_v)&\equiv 0\pmod{\mathfrak m}\iff W_v^0(\DII{\varpi_v}{1})\equiv 0\pmod{\mathfrak m}\text{ whenever }\\
&\varpi_v\in [{algebraically closed ute{e}t}a^{-1}\mathbf c^{-1}]:=F^{\times}{algebraically closed ute{e}t}a^{-1}\mathbf c^{-1}\mathrm N((\mathbf AKf^{(p)})^{\times}).\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
On the other hand, by \varepsilonsilonqref{E:Galois.W}, we find that
\[{\mathbb T}r\rho_p(\pi)(\mathbb Frob_v)=\omega(\varpi_v)^{-1}abelian variety s{\varpi_v}^{-k_{mx}/2}W_v^0(\DII{\varpi_v}{1})\text{ for all }v\nmid p\mathfrak n.\]
Let $\rec_{E/F}:\mathbf AK^{\times}\to \Gal(E/F)$ be the surjection induced by the reciprocity law. Combined with \varepsilonsilonqref{E:14.W}, the above equation yields that
\begin{align*}{\mathbb T}r\rho_p(\pi)(\mathbb Frob_v)&\equiv 0\pmod{\mathfrak m}\text{ whenever }\\
&\mathbb Frob_v|_{E}=\rec_{E/F}(\varpi_v)=\rec_{E/F}({algebraically closed ute{e}t}a^{-1}\mathbf c^{-1}).\varepsilonsilonnd{align*}
This in particular implies that $\rec_{E/F}({algebraically closed ute{e}t}a^{-1}\mathbf c^{-1})$ must be the complex conjugation $c$, and hence we arrive at a contradiction to \varepsilonsilonqref{ai${}_E$} by the following {\ell}mref{L:3.W}.
\varepsilonsilonnd{proof}
\begin{lm}{\ell}abel{L:3.W}Let $p>2$ be a prime. Let $G$ be a finite group and $H\subset G$ be a index two subgroup. Let $\rho: G\hookrightarrow \GL_2(\mathbb Fpbar)$ be a faithful irreducible representation of $G$. Let ${\mathrm T}={\mathbb T}r\rho: G\to\mathbb Fpbar$ be the trace function. Assume that
\begin{mylist}\item There exists an order two element $c\in G-H$,
\item ${\mathrm T}(hc)=0$ for all $h\in H$.
\varepsilonsilonnd{mylist}
Then $\rho|_H$ is reducible.
\varepsilonsilonnd{lm}
\begin{proof}The assumption (2) implies that ${\mathrm T}(c)=0$, and hence $\det\rho(c)=-1$. We may assume that $\rho(c)=\MX{0}{1}{1}{0}$. Suppose that $p\nmid \sharp(G)$. By the usual representation theory of finite groups, we have
\[1=\pair{{\mathrm T}}{{\mathrm T}}:=\frac{1}{\sharp(G)}\sum_{g\in G}{\mathrm T}(g){\mathrm T}(g^{-1})=\frac{1}{2\sharp(H)}\sum_{h\in H}{\mathrm T}(h){\mathrm T}(h^{-1})=\mathbf 1half\cdot \pair{{\mathrm T}|_H}{{\mathrm T}|_H}.\]
Since $\pair{{\mathrm T}|_H}{{\mathrm T}|_H}=2$, we conclude that $\rho|_H$ is not irreducible.

Now we assume that $p\mid \sharp(H)$. For each $b\in M_2(\mathbb Fpbar)$ with $b^2=0$, define the $p$-subgroup $P_b$ of $\rho(H)$ by
\[P_b=\stt{h\in \rho(H)\mid h=1+xb\text{ for some }x\in\mathbb Fpbar}.\]
Let $h\in H$ be an element of $p$-power order. It is well known that $(\rho(h)-1)^2=0$, and hence
${\mathrm T}(h)=2$ and $\det\rho(h)=1$. Combined with ${\mathrm T}(hc)=0$, these equations imply that
\[\rho(h)\in P_{b_1}\text{ or }P_{b_2}\text{ with }b_1=\MX{1}{1}{-1}{-1},\,b_2=\MX{1}{-1}{1}{-1}.\]

Note that either $P_{b_1}$ or $P_{b_2}$ is trivial. Indeed, if $h_1\not =1\in P_{b_1}$ and $h_2\not =1\in P_{b_2}$. Then $h_1h_2\in H$ and ${\mathbb T}r(h_1h_2c)\not =0$, which is a contradiction.
In particular, we conclude that elements of $p$-power order in $H$ are commutative to each other and that there is only one $p$-Sylow subgroup of $H$, which we denote by $P$. It is clear that $H$ normalizes $P$. Since $P\not =\stt{1}$, there is a unique line fixed by $\rho(P)$, which is an invariant subspace of $\rho(H)$. We find that $\rho|_H$ is reducible if $p\mid \sharp(H)$.
\varepsilonsilonnd{proof}
\begin{Remark}The assumption (3) in \thmref{T:B.W} in the introduction implies the vanishing of $\mu_p({\mathbb I}pi)$ for all $v|{\mathfrak c}_{\ell}am^-$.
\varepsilonsilonnd{Remark}

\def{\mathfrak L}bar{{\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{\mathfrak L}}}}
\def{(n)}{{(n)}}
\def{\mathbb I}pi{\varPsi_{\pi,{\mathbb I}i,v}}
\section{Non-vanishing of central $L$-values with anticyclotomic twists}{\ell}abel{S:NV}
In this section, we consider the problem of non-vanishing of $L$-values modulo $p$ with anticyclotomic twists. Let $\varepsilonsilonlll\not =p$ be a rational prime and let $\mathfrak l$ be a prime of $F$ above $\varepsilonsilonlll$. Let $\Gamma_{\mathfrak l}^-:=\Gal(E^-_{\mathfrak l^\infty}/E)$ be the Galois group of the maximal anticyclotomic pro-$\varepsilonsilonlll$ extension $E^-_{\mathfrak l^\infty}$ in the ray class field of $E$ of conductor $\mathfrak l^\infty$. Let $\mathfrak X_\mathfrak l^0$ be the set consisting of finite order characters $\phi:\Gamma_\mathfrak l^-\to\mu_{\varepsilonsilonlll^\infty}$. Fix a Hecke character ${\mathbb I}i$ of infinity type $(k/2+m,-k/2-m)$. For each $\phi:\Gamma_\mathfrak l^-\to\mu_{\varepsilonsilonlll^\infty}$ in $\mathfrak X_\mathfrak l^0$, we put\[L^{\alphag}(\mathbf 1half,\pi_E\otimes{\mathbb I}i\phi):=[\mathcal O_E^{\times}:\mathcal O_F^{\times}]^2\cdot \frac{\Gamma_{\varSigma}(m)\Gamma_{\varSigma} (k+m)}{(\Im\mathbf CMP)^{k+2m}(4\pi)^{k+2m+1\cdot{\varSigma}}}\cdot\frac{L(\mathbf 1half,\pi_E\otimes{\mathbb I}i\phi)}{\boldsymbol{\omega}ega_E^{2(k+2m)}}.\]
For simplicity, we assume
\[ (p\mathfrak l,\mathfrak n D_{E/F})=1.\]
In particular, $\pi$ is unramified at $\mathfrak l$ and every place above $p$. It can be shown that $L^{\alphag}(\mathbf 1half,\pi_E\otimes{\mathbb I}i\phi)\in \mathbf Zbar_{(p)}$ at least when $p\nmid D_{E}$. This section is devoted to proving the following result:
\begin{thm}{\ell}abel{T:main3.W}With the same assumptions in \thmref{T:mu-invariant}, we further assume that
\begin{mylist}\item $(p\mathfrak l,\mathfrak n{\mathfrak c}_{\mathbb I}i D_{E/F})=1$.
\item $\mu_p({\mathbb I}pi)=0$ for all $v|{\mathfrak c}_{\mathbb I}i^-$.
\varepsilonsilonnd{mylist} Then for almost all $\phi\in\mathfrak X_\mathfrak l^0$ we have \[L^{\alphag}(\mathbf 1half,\pi_E\otimes{\mathbb I}i\phi)\not\equiv 0\pmod{\mathfrak m}.\]  Here almost all means "except for finitely many $\phi\in\mathfrak X_\mathfrak l^0$" if $\dim_{\mathbf Q_\varepsilonsilonlll} F_\mathfrak l=1$ and "for $\phi$ in a Zariski dense subset of $\mathfrak X_\mathfrak l^0$" if $\dim_{\mathbf Q_\varepsilonsilonlll}F_\mathfrak l>1$ $($See \cite[p.737]{Hida:nonvanishingmodp}$)$.
\varepsilonsilonnd{thm}
When $F=\mathbf Q$, a non-primitive version of the above result under different assumptions is treated in \cite{Miljan:2}.
\subsection{}
After introducing some notation, we outline the approach of Hida \cite{Hida:nonvanishingmodp} to study this problem. We shall take $\frakr={\mathfrak c}_{\mathbb I}i\mathfrak n D_{E/F}\mathfrak l$ to be the fixed ideal in \subsecref{SS:choiceofcmpt}. For every $n\in\mathbf Z_{\geq 0}$, let $R_{\mathfrak l^n}:=\mathcal O_F+\mathfrak l^n\mathcal O_{E}$ be the order in $E$ of conductor $\mathfrak l^n$.
Let $U_{\mathfrak l^n}=(E\otimes_\mathbf Q\mathbf R)^{\times} (R_n\otimes_\mathbf Z\mathbf Zhat)^{\times}$ and let $Cl^-_{\mathfrak l^n}:=E^{\times}\mathbf AF^{\times}\backslash \mathbf AK^{\times}/U_{\mathfrak l^n}$ be the anticyclotomic ideal class group of conductor $\mathfrak l^n$. Denote by $[\cdot]_n:\mathbf AK^{\times}\to Cl^-_{\mathfrak l^n}$ the quotient map. Let $Cl^-_{\mathfrak l^\infty}=\indent olim_n Cl^-_{\mathfrak l^n}$. Let $I_\mathfrak l$ be the $\mathfrak l$-adic Iwahori subgroup of $K^0_\mathfrak l$ given by
\[I_\mathfrak l=\stt{g=\MX{a}{b}{c}{d}\in K^0_\mathfrak l\mid c\in \varpi_\mathfrak l\mathcal D_{F_\mathfrak l}}.\]
Let $K_0(\mathfrak l):=K^\mathfrak l I_\mathfrak l=\stt{g\in K\mid g_\mathfrak l\in I_\mathfrak l}$ be an open-compact subgroup of $\GL_2(\mathbf AFf)$. Recall that the $U_\mathfrak l$-operator on $\mathbf M_k(K_0(\mathfrak l),\mathbf C)$ is given by
\[F\mid U_\mathfrak l(\tau,g_f)=\sum_{u\in\mathcal O_F/\mathfrak l\mathcal O_F}F(\tau,g_f\MX{\varpi_\mathfrak l}{ud_{F_\mathfrak l}^{-1}}{0}{1}).\]
We briefly outline the approach of Hida to prove \thmref{T:main3.W} as follows:
\begin{mylist}
\item Construct a suitable \text{$p$-adic } modular form $\widehat\mathbf f^\dagger_{\mathbb I}i$ which is an eigenfunction of $U_\mathfrak l$-operator with \text{$p$-adic } unit eigenvalue.
\item Consider Hida's measure $\varphihi^\dagger_{\mathbb I}i$ on $Cl^-_{\mathfrak l^\infty}$ attached to $\widehat\mathbf f^\dagger_{\mathbb I}i$ \varepsilonsilonqref{E:lmeasure.W} and show the evaluation formula of this measure is related to central values $L^\alphag(\mathbf 1half,\pi_E\otimes{\mathbb I}i\phi)$ (\indent opref{P:evaluation.W}).
    \item The Zariski density of CM points in Hilbert modular varieties modulo $p$ reduces the proof of \thmref{T:main3.W} to the non-vanishing of certain Fourier coefficients of $\widehat \mathbf f^\dagger_{\mathbb I}i$ at some cusp (\cite[Thm.\,3.2 and Thm.\,3.3]{Hida:nonvanishingmodp}).
\varepsilonsilonnd{mylist}
We remind the reader that the proof is very close to \thmref{T:padicL.W} and \thmref{T:mu-invariant}. The essential new inputs in this section are the choice of $U_\mathfrak l$-eigenforms and the computation of local period integral at $\mathfrak l$.
\subsection{CM points of conductor $\mathfrak l^n$} Let $n\in\mathbf Z_{\geq 0}$. We choose $\varsigma_\mathfrak l^{(n)}\in G_\mathfrak l$ as follows.
If $\mathfrak l={\mathfrak L}\overline}  \nc{\opp}{\mathrm{opp}} \nc{\ul}{\underline{{\mathfrak L}}$ splits in $E$, writing $\mathbf CMP=\mathbf CMP_{\mathfrak L} e_{\mathfrak L}+\mathbf CMP_{\mathfrak L}bar e_{\mathfrak L}bar$ (so $d_{F_\mathfrak l}=\mathbf CMP_{\mathfrak L}-\mathbf CMP_{\mathfrak L}bar$ is a generator of $\mathcal D_{F_\mathfrak l}$), we put
\begin{align*}\varsigma_\mathfrak l^{(n)}=&\MX{\mathbf CMP_{{\mathfrak L}}}{-1}{1}{0}\DII{\varpi_\mathfrak l^n}{1}.
\intertext{If $\mathfrak l$ is inert, then we put}
\varsigma_\mathfrak l^{(n)}=&\MX{0}{1}{-1}{0}\DII{\varpi_\mathfrak l^n}{1}.\varepsilonsilonnd{align*}
Let $\varsigma^{(n)}:=\varsigma^{(n)}_\mathfrak l\indent od_{v\not=\mathfrak l}\varsigma_v$. According to this choice of $\varsigma_\mathfrak l^{(n)}$, we have
\[\varsigma_f^{(n)}*(\mathscr L\otimes_\mathbf Z\mathbf Zhat)=q_{\mathbf CMP}(R_{\mathfrak l^n}).\]
 Define $x_n:\mathbf AK^{\times}\to X^+{\times} G(\mathbf AFf)$ by
\[x_n(a):=(\mathbf CMP_{\varSigma},a_f\varsigma_f^{(n)}).\]
This collection $\stt{x_n(a)}_{a\in\mathbf AK^{\times}}$ of points is called CM points of conductor $\mathfrak l^n$. As discussed in \subsecref{S:CMpoint}, $\stt{x_n(a))}_{a\in(\mathbf AKf^{(p)})^{\times}}$ descend to CM points in $I_{K}(\mathbf CMring)$.
\subsection{The measures associated to $U_\mathfrak l$-eigenforms}
We construct the $U_\mathfrak l$-eigenform $\mathbf f^\dagger_{\mathbb I}i$ as follows. Write $\pi_\mathfrak l=\pi(\mu_\mathfrak l,\nu_\mathfrak l)$. Define the local Whittaker function ${\ell}ocalW{\mathfrak l}^\dagger\in{\mathcal W}(\pi_\mathfrak l,\psi_\mathfrak l)$ by
\[{\ell}ocalW{\mathfrak l}^\dagger(g)=W^0_{\mathfrak l}(g)-\mu_\mathfrak l\mathbf Abs^\mathbf 1half(\mathfrak l)W^0_\mathfrak l(g\DII{\varpi_\mathfrak l^{-1}}{1}).\]
It is not difficult to verify that 
\begin{itemize} \item ${\ell}ocalW{\mathfrak l}^\dagger$ is invariant by $I_\mathfrak l$,
\item ${\ell}ocalW{\mathfrak l}^\dagger(\DII{a}{1})=\nu_\mathfrak l\mathbf Abs^\mathbf 1half(a)\mathbb I_{\mathcal O_{F_\mathfrak l}}(a)$,
\item ${\ell}ocalW{\mathfrak l}^\dagger$ is an $U_\mathfrak l$-eigenfunction with the eigenvalue $\nu_\mathfrak l(\varpi_\mathfrak l)abelian variety s{\varpi_\mathfrak l}^{-\mathbf 1half}$.
\varepsilonsilonnd{itemize}
Define the normalized global Whittaker function ${\ell}ocalW{{\mathbb I}i}^\dagger$ by  \[{\ell}ocalW{{\mathbb I}i}^\dagger:=N(\pi,{\mathbb I}i)^{-1}abelian variety s{\det\varsigma_f}_{\mathbf AFf}^{1-k_{mx}/2}\cdot \indent od_{\sigma\in{\varSigma}}W_{k_\sigma}\cdot \indent od_{v\in\mathbf h,v\not =\mathfrak l}{\ell}ocalW{{\mathbb I}i,v}\cdot {\ell}ocalW{\mathfrak l}^\dagger,\]
 where $N(\pi,{\mathbb I}i)$ is the normalization factor in \varepsilonsilonqref{E:constantN.W}.  Let $\varphi_{\mathbb I}i^\dagger$ be the automorphic form associated to ${\ell}ocalW{{\mathbb I}i}^\dagger$ as in \varepsilonsilonqref{E:WhittakerAuto.W} and let $\mathbf f_{\mathbb I}i^\dagger$ be the associated Hilbert modular form as in \defref{D:toricform}.

 The following lemma follows from the choices of our Whittaker function ${\ell}ocalW{{\mathbb I}i}^\dagger$ and the construction of $\mathbf f_{\mathbb I}i^\dagger$.
 \begin{lm}{\ell}abel{L:6.W}Recall that ${\mathcal R}$ is the group generated by $E_v^{\times}$ for all ramified places $v$ in $\mathbf AK^{\times}$. We have
 \begin{mylist}
  \item $\mathbf f^\dagger_{\mathbb I}i$ is toric of character ${\mathbb I}i$ outside $\mathfrak l$, and
\[\mathbf f^\dagger_{\mathbb I}i(x_n(ta))=\mathbf f^\dagger_{\mathbb I}i(x_n(t)){\mathbb I}i^{-1}\mathbf Abs_{\mathbf AK}^{k_{mx}/2-1}(a)\text{ for all }a\in{\mathcal R}\cdot (R_{\mathfrak l^n}\otimes_\mathbf Z\mathbf Zhat)^{\times}.\]
 \item $\mathbf f_{{\mathbb I}i\phi}^\dagger=\mathbf f^\dagger_{{\mathbb I}i}$ for every $\phi\in\mathfrak X_\mathfrak l^0$.
\varepsilonsilonnd{mylist}
\varepsilonsilonnd{lm}
\begin{proof}Part (1) follows immediately from the fact that ${\ell}ocalW{{\mathbb I}i}^\dagger$ is a toric Whittaker function outside $\mathfrak l$ in view of {\ell}mref{L:toric.W}. In addition, for every $\phi\in\mathfrak X_\mathfrak l^0$, $\phi$ is anticyclotomic and unramified outside $\mathfrak l$. We thus have ${\ell}ocalW{{\mathbb I}i}^\dagger={\ell}ocalW{{\mathbb I}i\phi}^\dagger$, which verifies part (2) (\mbox{{\it cf.} } {\ell}mref{L:2.W}).
\varepsilonsilonnd{proof}
Following \cite[(3.9)]{Hida:nonvanishingmodp}, we define a \text{$p$-adic } $\mathbf Zbarp$-valued measure $\varphi^\dagger_{\mathbb I}i$ on $Cl^-_{\mathfrak l^\infty}$ attached to the \text{$p$-adic } avatar $\widehat\mathbf f^\dagger_{\mathbb I}i$ of $\mathbf f^\dagger_{\mathbb I}i$ as follows. For a locally constant function $\phi:Cl^-_{\mathfrak l^\infty}\to\mathbf Zbarp$ factors through $Cl^-_{\mathfrak l^n}$, we define
\begin{equation}{\ell}abel{E:lmeasure.W}\int_{Cl_{\mathfrak l^\infty}^-}\phi d\varphi^\dagger_{\mathbb I}i=\alpha_\mathfrak l^{-n}\sum_{[a]_n\in Cl_{\mathfrak l^n}^-}\theta^m\widehat\mathbf f^\dagger_{\mathbb I}i(x_n(a))\widehat{\mathbb I}i(a)\phi([a]_n),\varepsilonsilonnd{equation}
 where $\alpha_\mathfrak l=\nu_\mathfrak l(\varpi_\mathfrak l)abelian variety s{\varpi_\mathfrak l}^\frac{1-k_{mx}}{2}$ and $\widehat{\mathbb I}i$ is the \text{$p$-adic } avatar of ${\mathbb I}i\mathbf Abs^{1-k_{mx}/2}_{\mathbf AK}$. One checks that the right hand side does not depend on the choice of $n$ since $\mathbf f^\dagger_{\mathbb I}i$ is an $U_\mathfrak l$-eigenform with the eigenvalue $\alpha_\mathfrak l$.

Let $\phi\in\mathfrak X^0_\mathfrak l$ be a character of conductor $\mathfrak l^n$. We view $\phi$ as a character on $Cl^-_{\mathfrak l^n}$ by the reciprocity law. Following the arguments in \indent opref{P:1.W} and \thmref{T:padicL.W}, we can write the measure as a toric period integral of $\widetilde V_+^m\varphi_{{\mathbb I}i\phi}^\dagger$:
\begin{equation}{\ell}abel{E:19.W}\begin{aligned}
\frac{\boldsymbol{\omega}ega_E^{k+2m}}{\boldsymbol{\omega}ega_p^{k+2m}}\cdot \int_{Cl^-_{\mathfrak l^\infty}}\phi d\varphi_{\mathbb I}i^\dagger&=\alpha_\mathfrak l^{-n}\vol(U_{\mathfrak l^n},dt)^{-1}\frac{1}{(\Im\mathbf CMP)^{k/2+m}}\cdot P_{{\mathbb I}i\phi}(\pi(\varsigma^{(n)})\widetilde V^m_+\varphi^\dagger_{{\mathbb I}i})\\
&=\vol(U_E,dt)^{-1}\frac{1}{(\Im\mathbf CMP)^{k/2+m}}\cdot\frac{\alpha_\mathfrak l^{-n}}{L(1,\tau_{E_\mathfrak l/F_\mathfrak l})abelian variety s{\varpi_\mathfrak l}^n}\cdot P_{{\mathbb I}i\phi}(\pi(\varsigma^{(n)})\widetilde V^m_+\varphi^\dagger_{{\mathbb I}i\phi}).
\varepsilonsilonnd{aligned}\varepsilonsilonnd{equation}
Here we used the fact that
\[\vol(U_{\mathfrak l^n},dt)=\vol(U_E,dt)\cdot L(1,\tau_{E_\mathfrak l/F_\mathfrak l})abelian variety s{\varpi_\mathfrak l}^{n}.\]
We have the following evaluation formula.
\begin{prop}{\ell}abel{P:evaluation.W}Suppose that $(\mathfrak l,{\mathfrak c}_{\mathbb I}i\mathfrak n D_{E/F})=1$. For $\phi\in\mathfrak X_\mathfrak l^0$ of conductor $\mathfrak l^n$ with $n\geq 1$, we have
\[{\ell}eft(\frac{1}{\boldsymbol{\omega}ega_p^{k+2m}}\cdot \int_{Cl_{\mathfrak l^\infty}^-}\phi d\varphi_{\mathbb I}i^\dagger\right)^2=\frac{abelian variety s{\varpi_\mathfrak l}^{-n}}{\alpha_\mathfrak l^{2n}}\cdot L^{\alphag}(\mathbf 1half,\pi_E\otimes{\mathbb I}i\phi)\cdot C(\pi,{\mathbb I}i)\phi(\mathbf Csplit).\]
\varepsilonsilonnd{prop}
\begin{proof}In view of \varepsilonsilonqref{E:19.W}, it remains to compute $P_{{\mathbb I}i\phi}(\pi(\varsigma^{(n)})\widetilde V^m_+\varphi^\dagger_{{\mathbb I}i\phi})^2$, which can be written as a product of local toric period integrals as in the proof of \thmref{T:main1.W}. We have computed these local period integrals in \secref{SS:toricintegralI.W} and \subsecref{SS:toricII.W} except for the local integral at $\mathfrak l$, which will be carried out in the following {\ell}mref{L:5.W}. The desired formula is obtained by combining these calculations.
\varepsilonsilonnd{proof}

\begin{lm}{\ell}abel{L:5.W}Suppose that ${\mathbb I}i_\mathfrak l$ is unramified and $\phi\in\mathfrak X_\mathfrak l^0$ has conductor $\mathfrak l^n,\,n\geq 1$. Then
\[{\ell}ocalP{\pi(\varsigma_\mathfrak l^{(n)})W_\mathfrak l^\dagger}{{\mathbb I}i\phi}=abelian variety s{\mathcal D_{E_\mathfrak l}}_{E_\mathfrak l}^\mathbf 1half \cdot\omega_\mathfrak l(\varpi_\mathfrak l^n)abelian variety s{\varpi_\mathfrak l^n} L(1,\tau_{E_\mathfrak l/F_\mathfrak l})^2.\]
\varepsilonsilonnd{lm}
\begin{proof}
Write $F=F_\mathfrak l$ (resp. $E=E_\mathfrak l$) and $\varpi=\varpi_\mathfrak l$. For $t\in E$, we put
\[\iota_\varsigma^{(n)}(t):=(\varsigma_\mathfrak l^{(n)})^{-1}\iota(t)\varsigma_\mathfrak l^{(n)}.\]

First we suppose $\mathfrak l$ is split. A direct computation shows that
\begin{align*}\iota_\varsigma^{(n)}(t)=&\MX{1}{\varpi^{-n}d_{F}^{-1}}{0}{1}\MX{t_{\mathfrak L}}{}{0}{t_{\mathfrak L}bar}\MX{1}{-\varpi^{-n}d_{F}^{-1}}{0}{1};\\
(\varsigma_\mathfrak l^{(n)})^{-1}\boldsymbol{J}\varsigma_\mathfrak l^{(n)}=&\MX{1}{0}{\varpi^n d_F}{-1}.
\varepsilonsilonnd{align*}
We find that
\begin{align*}
&\omega^{-1}(\det\varsigma_\mathfrak l^{(n)}){\ell}ocalP{\pi(\varsigma_\mathfrak l^{(n)})W_\mathfrak l^\dagger}{{\mathbb I}i\phi}\\
=&\int_{F^{\times}}\int_{F^{\times}}W_{\mathfrak l}^\dagger(\DII{ax}{1}\MX{1}{-\varpi^{-n}d_F^{-1}}{0}{1})
W_{\mathfrak l}^\dagger(\DII{-a}{1}\MX{1}{-\varpi^{-n}d_F^{-1}}{0}{1})\omega_\mathfrak l^{-1}(a){\mathbb I}i_{{\mathfrak L}}\phi_{\mathfrak L}(x) d^\x ad^\x x\\
=&\int_{\mathcal O_F}\psi(-d_F^{-1}\varpi^{-n}x)\nu_\mathfrak l{\mathbb I}i_{\mathfrak L}\phi_{\mathfrak L}\mathbf Abs^\mathbf 1half(x)d^\x x\cdot \int_{\mathcal O_F}\psi(d_F^{-1}\varpi^{-n}a)\nu_\mathfrak l\omega_\mathfrak l^{-1}{\mathbb I}i_{\mathfrak L}^{-1}\phi_{\mathfrak L}^{-1}\mathbf Abs^\mathbf 1half(a)d^\x a\\
=&\int_{\mathcal O_F^{\times}}\psi(-d_F^{-1}\varpi^{-n}x)\phi_{\mathfrak L}(x)d^\x x\cdot \int_{\mathcal O_F}\psi(d_F^{-1}\varpi^{-n}a)\phi_{\mathfrak L}^{-1}(a)d^\x a\\
=&\varepsilonsilonp(1,\phi_{\mathfrak L}^{-1},\psi)\phi_{{\mathfrak L}}(-1)\varepsilonsilonp(1,\phi_{\mathfrak L},\psi)\cdot\zeta_F(1)^2=abelian variety s{\varpi^n\mathcal D_F}L(1,\tau_{E/F})^2.
\varepsilonsilonnd{align*}
This proves the formula in the split case.

Now we suppose that $\mathfrak l$ is inert. We shall retain the notation in \subsecref{SS:toricII.W}. Define $\bfm^\dagger:G_\mathfrak l\to\mathbf C$ by \[\bfm^\dagger(g):=\bfb_\mathfrak l(\pi(g)W_\mathfrak l^\dagger,W_\mathfrak l^\dagger).\]
Then $\bfm^\dagger(g)$ only depends on the double coset $I_\mathfrak l gI_\mathfrak l$. Put
\[\boldsymbol{P}^*:=\int_{E^{\times}/F^{\times}}\bfm^\dagger(\iota_\varsigma^{(n)}(t)){\mathbb I}i\phi(t)dt.\]
It is clear that \begin{equation}{\ell}abel{E:18.W}{\ell}ocalP{\pi(\varsigma_\mathfrak l^{(n)})W_\mathfrak l^\dagger}{{\mathbb I}i\phi}=\boldsymbol{P}^*\cdot\frac{L(1,\tau_{E/F})\omega(\det\varsigma_\mathfrak l^{(n)})}{\zeta_F(1)}.\varepsilonsilonnd{equation}
For $y\in \varpi^r\mathcal O_F^{\times}$, it is easy to verify that $\iota_\varsigma^{(n)}(1+y\cO_\cKbasis)\in I_\mathfrak l$ if $r\geq n$ and
\begin{align*}\iota_\varsigma^{(n)}(1+y\cO_\cKbasis)&\in I_\mathfrak l\mathbf w\DII{\varpi^{n-r}}{\varpi^{r-n}}I_\mathfrak l\text{ if }0{\ell}eq r<n\quad(\mathbf w=\MX{0}{-d_F^{-1}}{d_F}{0}).\varepsilonsilonnd{align*}
If $x\in\varpi\mathcal O_F$, then
\[\iota_\varsigma^{(n)}(x+\cO_\cKbasis)\in I_\mathfrak l\mathbf w\DII{\varpi^n}{\varpi^{-n}}I_\mathfrak l.\]
Note that $n=c_\mathfrak l(\phi)=c_\mathfrak l({\mathbb I}i\phi)$ as in \varepsilonsilonqref{E:alv.W}. Combined with the above observations and {\ell}mref{L:halfsum.W}, a direct computation shows that
\begin{align*}
\boldsymbol{P}^*&=X_n\cdot\bfm^\dagger(1)+(-X_n)\cdot \bfm^\dagger(\mathbf w\DII{\varpi}{\varpi^{-1}})\\
&=\bfb_\mathfrak l(W^0_\mathfrak l-\pi(\DII{\varpi^{-1}}{\varpi})W^0_\mathfrak l,W_\mathfrak l^\dagger)\cdot X_n\\
&=(\frac{\mu_\mathfrak l(\varpi)}{1-abelian variety s{\varpi}}-\frac{\nu_\mathfrak l(\varpi)}{1-\mu_\mathfrak l^{-1}\nu_\mathfrak l\mathbf Abs(\varpi)})\cdot\frac{1-\mu_\mathfrak l^{-1}\nu_\mathfrak l\mathbf Abs(\varpi)}{\mu_\mathfrak l(\varpi)-\nu_\mathfrak l(\varpi)}\cdot abelian variety s{\mathcal D_F}^\mathbf 1half X_n\\
&=\frac{1}{1-abelian variety s{\varpi}}\cdot L(1,\tau_{E/F})abelian variety s{\varpi^n}abelian variety s{\mathcal D_E}_E^\mathbf 1half.
\varepsilonsilonnd{align*}

The formula in the inert case follows from \varepsilonsilonqref{E:18.W} immediately.
\varepsilonsilonnd{proof}

\subsection{Proof of \thmref{T:main3.W}}{\ell}abel{SS:proof}
We prove \thmref{T:main3.W} in this subsection. By the evaluation formula \indent opref{P:evaluation.W}, it boils down to proving that
\begin{equation}{\ell}abel{E:20}\int_{Cl_\infty^-}\phi d\varphi^\dagger_{\mathbb I}i\not\equiv 0\pmod{\mathfrak m}\text{ for almost all $\phi\in\mathfrak X_\mathfrak l^0$}.\varepsilonsilonnd{equation}

By \cite[Thm.\,3.2, 3.3]{Hida:nonvanishingmodp} together with the toric property of $\mathbf f_{\mathbb I}i^\dagger$ {\ell}mref{L:6.W} (\mbox{{\it cf.} } \cite[Lemma 6.1 and Remark 6.2]{Hsieh:Hecke_CM}), the validity of \varepsilonsilonqref{E:20} is reduced to verifying the following condition:\begin{description}[\breaklabel\setlabelstyle{\itshape}]\item [$(\mathrm{H'})$]
For every $u\in \mathcal O_{F_\mathfrak l}$ and a positive integer $r$, there exist $\beta\in\mathcal O_{F,{(p)}}^{\times}$ and $a\in(\mathbf AKf^{(pN)})^{\times}$ such that $\beta\equiv u\pmod{\mathfrak l^r}$ and \[\bfa_\beta(\mathbf f^\dagger_{\mathbb I}i,{\mathfrak c}(a))\not\equiv 0\pmod{\mathfrak m}.\]
\varepsilonsilonnd{description}
The verification of (H$'$) under the assumptions \varepsilonsilonqref{ai${}_E$} and $\mu_p({\mathbb I}pi)=0$ for all $v|{\mathfrak c}_{\mathbb I}i^-$ follows from the same argument in \thmref{T:mu-invariant}. Note that for a polarization ideal ${\mathfrak c}(a)$ $({\mathfrak c}={\mathfrak c}(\mathcal O_E),\,a\in(\mathbf AKf^{(p\mathfrak l)})^{\times})$ and a totally positive $\beta\in\mathcal O_{F,{(p)}}^{\times}{\mathcal a}p \mathcal O_{F_\mathfrak l}$, we have $({\mathfrak c}(a),p\mathfrak l)=1$ and
\[\bfa_\beta(\mathbf f^\dagger_{\mathbb I}i,{\mathfrak c}(a))=\beta^{k/2}\indent od_{v\nmid p\mathfrak l}{\ell}ocalK{{\mathbb I}i,v}^*(\beta\mathbf c_v^{-1}\mathrm N(a_v^{-1}))\cdot \nu_\mathfrak l\mathbf Abs_{F_\mathfrak l}^\mathbf 1half(\beta)\quad(\mathfrak i\frakl_F(\mathbf c)={\mathfrak c}).\]
Let $u\in \mathcal O_{F_\mathfrak l}$ and let ${algebraically closed ute{e}t}a^u=({algebraically closed ute{e}t}a^u_v)$ be the idele in $\mathbf AF^{\times}$ such that $\bfa^*_{{\ell}am,v}({algebraically closed ute{e}t}a^u_v)\not\equiv 0\pmod{\mathfrak m}$ for all $v|{\mathfrak c}_{\mathbb I}i^-$, ${algebraically closed ute{e}t}a^u_\mathfrak l=u$ and ${algebraically closed ute{e}t}a^u_v=1$ for all $v\nmid \mathfrak l{\mathfrak c}_{\mathbb I}i^-$. To verify (H$'$), we simply proceed the Galois argument in \thmref{T:mu-invariant}, replacing ${algebraically closed ute{e}t}a$ in by ${algebraically closed ute{e}t}a^u$ therein. This completes the proof of \thmref{T:main3.W}. 

\varepsilonsilonnd{document}