Rational points on elliptic curves and cycles on Shimura varieties

The theory of CM (Complex Multiplication) points on modular and Shimura curves provides one of the most fruitful approaches to the Birch and Swinnerton-Dyer conjecture, leading, in particular, to the best known results for elliptic curves of analytic rank one. In this survey I will explore the possibility of more general constructions in which, loosely speaking, CM points are replaced by (topological, or algebraic) cycles on higher-dimensional generalisations of modular curves, called Shimura varieties.