Zhang, SW. Sci. China Math. (2010) 53: 573. doi:10.1007/s11425-010-0046-2
This is the note for a series of lectures that the author gave at the Centre de Recerca Matemática (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.