Consequences of the Gross-Zagier formulae: Stability of average L-values, subconvexity, and non-vanishing mod p

Abstract

Applying the celebrated results of Gross and Zagier for central values of L-series of holomorphic forms of prime level, we deduce an exact average formula for suitable twists of such L-values, with a relation to the class number of associated imaginary quadratic fieds, thereby strengthening a result of Duke. We also obtain a stability result, as well as subconvexity (in this setting), and certain non-vanishing assertions.