Rationality of Darmon points over genus fields of non-maximal orders

Stark–Heegner points, also known as Darmon points, were introduced by Darmon in [5] as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of global points defined over the strict Hilbert class field HF+ of the real quadratic field F and twisted by (unramified) quadratic characters of Gal(Hc+/F). We extend these results to the situation of ramified quadratic characters; more precisely, we show that Darmon points of conductor c≥ 1 twisted by quadratic characters of Gc+=Gal(Hc+/F), where Hc+ is the strict ring class field of F of conductor c, come from rational points on the elliptic curve defined over Hc+.