On the SL(2) period integral

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Let E/F be a quadratic extension of number fields. For a cuspidal representation π of SL(2,Aₑ), we study the non-vanishing of the period integral on SL(2,F)(2,AF). We characterise the non-vanishing of the period integral of π in terms of π being generic with respect to characters of Eₑ which are trivial on AF. 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 SL(2,Aₑ) whose period integral vanishes identically while each local constituent admits an SL(2)-invariant linear functional. Finally, we construct an automorphic representation π on SL(2,Aₑ) which is abstractly SL(2,AF) distinguished but none of the elements in the global L-packet determined by π is distinguished by SL(2,AF).