Heegner cycles are the higher weight analogues to Heegner points. Those points and cycles play an important role in the theory of complex multiplication and of the arithmetic of elliptic curves of rank 1. Stark-Heegner points are conjectural points on elliptic curves which would be the real-quadratic counterparts to Heegner points in the emerging theory of real multiplication. In this theory, Darmon-Vonk’s rigid meromorphic cocycles seem to be the real-quadratic analogue of singular moduli.

I will present a generalization of rigid meromorphic cocycles to higher weight and use it to define a p-adic higher Green’s functions on real-quadratic points. This construction is motivated by the recently resolved conjecture by Gross and Zagier  on the algebraicity of values of complex higher Green’s functions. I will present a conjecture on the algebraicity of values of the p-adic Green’s functions that has been numerically verified. The values of the p-adic Green’s function are best envisioned as p-adic local intersection numbers of certain conjectured cycles associated to RM-points, the so-called Stark-Heegner cycles.