for each $v\in \fp_D$.

Waldspurger’s formula {#sec:w0504}
---------------------

Suppose $E/F$ is a quadratic extension of number fields. Let $D$ be a quaternion algebra over $F$ which $E$ embeds and $\pi=\otimes_v\pi_v$ be a cuspidal automorphic representation of $D_\A^\times$ with trivial central character. We recall the explicit formula relating the central $L$-value $L(1/2, \pi)L(1/2, \pi\otimes\eta)$ and the period $\mP_E$ which was proved by Waldspurger in [@Wal2].

For each place $v$, take a $D_v^\times$-invarian non-degenerate Hermitian pairing $\l\,,\r_v$ on $\pi_v$ so that we have $$\l\,,\r=\prod_v\l\,,\r_v.$$ Here, the left hand side is the Petersson inner prouct on $\pi$. We also take a local Haar measure $\d h_v$ on $F_v^\times\bs E_v^\times$ for each $v$ so that we have $$\d h=\prod_v\d h_v,$$ where the left hand side is the Tamagawa measure on $\A_F^\times\bs\A_E^\times$. Then, the integral $$\al_{E_v}(\vp_v, \vp'_v)=\int_{F_v^\times\bs E_v^\times}\l\pi_v(h_v)\vp_v, \vp'_v\r_v\,\d h_v, \hspace{10pt} \vp_v, \vp'_v\in\pi_v$$ converges absolutely and defines an element of $\operatorname{Hom}_{E_v^\times\times E_v^\times}(\pi_v\boxtimes\bar{\pi}_v, \C)$. We define the normalized $E_v^\times$-invariant Hermitian pairing $\al_{E_v}^\#(\cdot,\cdot)$ by $$\al_{E_v}^\#(\vp_v, \vp'_v)=\frac{L(1, \pi_v, \operatorname{Ad})L(1, \eta_v)}{\zeta_{F_v}(2)L(1/2, \pi_v)L(1/2, \pi_v\otimes\eta_v)}\cdot\al_{E_v}(\vp_v, \vp'_v).$$

\[thm:Wald\] For any $\vp=\otimes_v\vp_v$ and $\vp'=\otimes_v\vp'_v\in\pi$ in $\pi$, one has $$\label{eq:Wald}
\mP_E(\vp)\ol{\mP_E(\vp')}=\frac14\cdot\frac{\zeta_F(2)L(1/2, \pi)L(1/2, \pi\otimes\eta)}{L(1, \pi, \operatorname{Ad})L(1, \eta)}\cdot\prod_v\al_{E_v}^\#(\vp_v, \vp'_v)$$ In particular, we have $\al_{E_v}^\#(\vp_v, \vp'_v)=1$ for almost all $v$.

This explicit formula is used in the 