Periods and nonvanishing of central L -values for GL(2 n )

Let π be a cuspidal automorphic representation of PGL(2 n ) over a number field F , and η the quadratic idèle class character attached to a quadratic extension E/F . Guo and Jacquet conjectured a relation between the nonvanishing of L (1/2, π ) L (1/2, π ⊗ η ) for π 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.