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

Abstract

We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are /?-adic points on E given by the values of certain /?-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.