RESUMEN

Let $E/F$ be a quadratic extension of number fields. For a cuspidal representation $\pi$ of ${\rm SL}_2({\Bbb A}_E)$, we study in this paper the integral of functions in $\pi$ on ${\rm SL}_2(F)\backslash {\rm SL}_2({\Bbb A}_F)$. We characterize the nonvanishing of these integrals, called period integrals, in terms of $\pi$ having a Whittaker model with respect to characters of $E\backslash {\Bbb A}_E$ which are trivial on ${\Bbb A}_F$. We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of ${\rm SL}_2({\Bbb A}_E)$ whose period integral vanishes identically while each local constituent admits an ${\rm SL}_2$-invariant linear functional. Finally, we construct an automorphic representation $\pi$ on ${\rm SL}_2({\Bbb A}_E)$ which is abstractly ${\rm SL}_2({\Bbb A}_F)$ distinguished but for which none of the elements in the global $L$-packet determined by it is distinguished by~${\rm SL}_2({\Bbb A}_F)$.