Heights of CM cycles and derivatives of L-series

Abstract

We extend the work of S. Zhang and Yuan-Zhang-Zhang to obtain a Gross-Zagier formula for modular forms of even weight in terms of an arithmetic intersection pairing of CM-cycles on Kuga-Sato varieties over Shimura curves. Combined with a result of the first author and de Vera-Piquero adapting Kolyvagin's method of Euler systems to this setting, we bound the associated Selmer and Tate-Shafarevich groups, assuming the non-vanishing of the derivative of the L-function at the central point.