We discuss the nonvanishing of the family of central values L( , f ⊗ χ), where f is a fixed automorphic form on GL(2) and χ varies through class group characters of an imaginary quadratic field K = Q( √ −D), as D varies; we prove results of the nature that at least D 1/5000 such twists are nonvanishing. We also discuss the related question of the rank of a fixed elliptic curve E/Q over the Hilbert class field of Q( √ −D), as D varies. The tools used are results about the distribution of Heegner points, as well as subconvexity bounds for L-functions
