Central values of Rankin L-series over real quadratic fields

Abstract

We study the Rankin L-series of a cuspidal automorphic representation of GL(2) of even weight over the rational numbers, twisted by a character of a real quadratic field. When the sign of the functional equation is +1, we give an explicit formula for the central value of the L-series, analogous to the formulae obtained by Gross, Zhang, and Xue in the imaginary case. The proof uses a version of the Rankin-Selberg method in which the theta correspondence plays an important role. We give two applications, to computing the order of the Tate-Shafarevich group of the base change to a real quadratic field of an elliptic curve defined over the rationale, and to proving the equidistribution of individual closed geodesics on modular curves. © Foundation Compositio Mathematica 2006.