$\DeclareMathOperator\Mp{Mp}$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!



Global Waldspurger packet is finite or infinite?
**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.


