The rationality of Stark-Heegner points over genus fields of real quadratic fields

We study the algebraicity of Stark-Heegner points on a modular elliptic curve E . These objects are p -adic points on E given by the values of certain p -adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field K . The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted by certain genus characters of K are defined over the predicted quadratic extensions of K . The non-vanishing of these combinations is also related to the appropriate twisted Hasse-Weil L -series of E over K , in the spirit of the Gross-Zagier formula for classical Heegner points.