  Let $\pi$ be a cuspidal automorphic representation of PGL($2n$) over a number
field $F$, and $\eta$ the quadratic idele class character attached to a
quadratic extension $E/F$. Guo and Jacquet conjectured a relation between the
nonvanishing of $L(1/2,\pi)L(1/2, \pi \otimes \eta)$ for $\pi$ of symplectic
type and the nonvanishing of certain GL($n,E$) periods. When $n=1$, this
specializes to a well-known result of Waldspurger. We prove this conjecture,
and related global results, under some local hypotheses using a simple relative
trace formula.
  We then apply these global results to obtain local results on distinguished
supercuspidal representations, which partially establish a conjecture of Prasad
and Takloo-Bighash.
