Central values of Rankin L-series over real quadratic fields.

摘要

We study the Rankin L-series of a cuspidal automorphic representation of GL(2) 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 B. Gross and S. W. Zhang in the imaginary case. The proof uses the Rankin-Selberg representation of the L-series as an adelic integral, together with the Weil-Siegel identity of S. Kudla and S. Rallis. An essential ingredient is the theory of theta liftings, especially the Shimizu correspondence between automorphic forms on GL(2) and automorphic forms on an inner form of GL(2).