$\DeclareMathOperator\Mp{Mp}$Let $F$ be a number field and $\pi$ be an irreducible cuspidal automorphic representation of $\operatorname{PGL}_2(\mathbb{A}_F)$.
Then we can think a submodule $L_{\pi}^2$ of $L_{\mathrm{disc}}^2(\Mp_2)$, the discrete spectrum of automorphic functions on $\Mp_2(F) \backslash \Mp_2(\mathbb{A})$, as
$L_{\pi}^2:=\sum_{a \in F^{\times} \backslash F^{\times^2}} \Theta(\pi \otimes \chi_a)$,
where $\chi_a$ is the quadratic character of $\mathbb{A}^{\times}/F^{\times}$ associated to the quadratic extension $F(\sqrt{a})/F$ by global class field theory and $\Theta$ is a theta lift from $\operatorname{SO}_3 \simeq \operatorname{PGL}_2$ to $\Mp_2$.
Then Shimura—Waldspurger correspondence asserts that $L_{\pi}^2$ is the full near equivalence class in $L_{\mathrm{disc}}^2(\Mp_2)$ such that each irreducible summand is in the global Waldspurger packet $A_{\pi}$ of $\pi$ and the number of irreducible summand of $L_{\pi}^2$ is half the number of $A_{\pi}$.
Since $\pi$ is irreducible cuspidal of $\operatorname{PGL}
_2$, the number of $A_{\pi}$ should be finite. Therefore, I think there are only finitely many Hecke quadratic characters $\chi$'s such that $L(\frac{1}{2},\pi \times \chi) \ne 0$ because $\Theta(\pi \otimes \chi) \ne 0$ is equal to $L(\frac{1}{2},\pi \times \chi) \ne 0$.
However, the paper of Friedberg and Hoffstein (Theorem B in https://www.jstor.org/stable/2118638) claims that there are infinitely many quadratic characters $\chi$ such that $L(\frac{1}{2},\pi \times \chi) \ne 0$. So I think it contradicts to $\sharp A_{\pi} <\infty$.
What is wrong in this reasoning?
Any comments are welcome!
Best Answer
Revised. The global Waldspurger packet of $\pi$ is indeed finite, as you say in your comment. It's elements are metaplectic representations which are in bijection with the Vogan packet of $\pi$, i.e., me the set of all cuspidal representations $\pi_D$ of a quaternion algebra $D/F$ such that $\pi_D$ corresponds to $\pi$ in the sense of Jacquet-Langlands. There are finitely many relevant $D$'s, which are determined by the finite set of places at which $\pi$ is discrete series.
However this finiteness is unrelated to the question of how many twists of $\pi$ will have nonvanishing central $L$-value. Let $\chi_E$ be the quadratic character associated to a quadratic extension $E/F$. Waldspurger relates $L(1/2, \pi)L(1/2, \pi \otimes \chi_E)$ to a period $P_E$ of some $\pi_D$ in the Vogan packet. As $E$ varies, the period changes, as does the "right" choice for the $D$ on which to choose the $\pi_D$. In particular, infinitely many periods (and thus $L$-values) may vanish and infinitely may be nonzero on the same $\pi_D$.
So there is no contradiction between finiteness of the Vogan packet and the nonvanishing of infinitely many twists. I think the confusion is arising from an assumption that the theta lifts $\Theta(\pi \otimes \chi_a)$ give different automorphic representations for different $a \in F^\times/F^{(2)}$, which is not true.