Central L-Values and Toric Periods for GL(2)

Let π be a cusp form on GL(2) over a number field F and let E be a quadratic extension of F. Denote by π E the base change of π to E and by Ω a unitary character of A × E / E × . We use the relative trace formula to give an explicit formula for L(1/2, π E ⊗ Ω) in terms of period integrals of Gross-Prasad test vectors. We give an application of this formula to equidistribution of geodesics on a hyperbolic 3-fold.