Wide moments of L-functions I : Twists by class group characters of imaginary quadratic fields

We calculate certain "wide moments" of central values of Rankin-Selberg L-functions L ˝ ; 1 2 where is a cuspidal automorphic representation of GL 2 over ‫ޑ‬ and is a Hecke character (of conductor 1) of an imaginary quadratic field.This moment calculation is applied to obtain "weak simultaneous" nonvanishing results, which are nonvanishing results for different Rankin-Selberg L-functions where the product of the twists is trivial.The proof relies on relating the wide moments of L-functions to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger's formula.To achieve this, a classical version of Waldspurger's formula for general weight automorphic forms is derived, which might be of independent interest.A key input is equidistribution of Heegner points (with explicit error terms), together with nonvanishing results for certain period integrals.In particular, we develop a soft technique for obtaining the nonvanishing of triple convolution L-functions.