The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros

We describe a number of results and techniques concerning the nonvanishing of automorphic L-functions at s = 1/2. In particular we show that as N --> infinity at least 50% of the values L(1/2, f), with f varying among the hol omorphic new forms of a fixed even integral weight for Gammao(N) and whose functional equations are even, are positive. Furthermore, we show that any improvement of 50% is intimately connected to Landau-Siegel zeros. These re sults may also be used to show that X-0(N) = Gamma (0)(N)\H has large quoti ents with only finitely many rational points. The results below were announ ced at the conference "Exponential sums" held in Jerusalem, January 1998. T he: complete proofs, which were presented in courses at Princeton (1997), a re being prepared for publication.