Chow-heegner points on CM elliptic curves and values of p-adic L-functions

We outline a new construction of rational points on CM elliptic curves, using cycles on higher-dimensional varieties, contingent on certain cases of the Tate conjecture. This construction admits of complex and p-adic analogs that are defined independently of the Tate conjecture. In the p-adic case, using p-adic Rankin L-functions and a p-adic Gross-Zagier type formula proved in our articles [2, 3], we show unconditionally that the points so constructed are in fact rational. In the complex case, we are unable to prove rationality (or even algebraicity) but we can verify it numerically in several cases.