We describe a number of results and techniques concerning the non-vanishing of automorphic L-functions at s = 1/2. In particular we show that as N → ∞ at least 50% of the values L(1/2, f), with f varying among the holomorphic new forms of a fixed even integral weight for Γ0(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 results may also be used to show that X0(N) = Γ0(N)\ℍ has large quotients with only finitely many rational points. The results below were announced at the conference "Exponential sums" held in Jerusalem, January 1998. The complete proofs, which were presented in courses at Princeton (1997), are being prepared for publication.
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