étale algebra $\cE_v$ over $F_v$ for each $v\in S$ so that there are infinitely many $E\in X(D)$ such that $\pi$ is $E^\times$-distinguished and $E_v=\cE_v$ for any $v\in S$.

0Unfortunately, this theorem is not a generalization of Theorem \[thm:FH\] because $S$-components of $E$ are not arbitrary in our theorem. Despite this defect, it has following novelty:

-   Theorem \[thm:main\] is about representations of arbitrary inner forms of $\GL_2$, while Theorem \[thm:W1\] and Theorem \[thm:FH\] only treats representations of $\GL_2$.

-   Theorem \[thm:main\] asserts nonvanishing of infinitely many periods, which is stronger than the assertions of Theorem \[thm:W1\] and Theorem \[thm:FH\] which assert nonvanishing of central values of $L$-functions for infinitely many quadratic twists.

The difference between our theorem and Theorem \[thm:FH\] is reduced to a local problem. In order to see this, let us try to deduce Theorem \[thm:main\] from Theorem \[thm:FH\]. Take a quarternion algebra $D$ and a cuspidal automorphic representation $\pi$ as in Theorem \[thm:main\]. Let $S(\pi)$ denote the finite set of places of $F$ which includes all Archimedean placs and places at which $\pi$ is ramified or $D$ is division. What we have to find is a quadratic extension $E_0$ satisfying two conditions:

-   for any $v\in S(\pi)$, 