Kartik Prasanna

Algebraic cycles and exotic Heegner points I,II

In the first talk, I will give an introduction to Rankin-Selberg L-series associated to a modular form and a theta function, focussing especially on their behaviour at the central point. In this context, I will review the results of Gross-Zagier and Zhang relating the central derivative to heights of Heegner cycles, and of Gross and Waldspurger relating central values to period integrals. I will also discuss the construction of certain generalized Heegner cycles that correspond to the vanishing of the L-function in the case of theta series of higher weight. In the second talk, I will explain how the generalized Heegner cycles may be used, via the Abel-Jacobi map, to give a new (conjectural) construction of rational points on CM elliptic curves, in a case not covered by the Gross-Zagier theorem. Everything that is new in the above is joint work with Bertolini and Darmon.