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.