Summary: We discuss the nonvanishing of the family of central values \(L(\tfrac12 , f \otimes)\), where \(f\) is a fixed automorphic form on \(\text{GL}(2)\) and \(\chi\) varies through class group characters of an imaginary quadratic field \(K = \mathbb Q(\sqrt{-D})\), 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 \(E/\mathbb Q\) over the Hilbert class field of \(\mathbb Q(\sqrt{-D})\), as \(D\) varies. The tools used are results about the distribution of Heegner points, as well as subconvexity bounds for \(L\)-functions.
For the entire collection see [Zbl 1121.11003].