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

TL;DR

This paper extends the rationality results of Darmon points on elliptic curves over real quadratic fields to include cases with ramified quadratic characters, showing these points originate from rational points over specific class fields.

Contribution

It generalizes the known rationality results of Darmon points from unramified to ramified quadratic characters, linking these points to rational points over ring class fields.

Findings

Darmon points twisted by ramified quadratic characters are rational over ring class fields.

The results connect local Darmon points with global rational points on elliptic curves.

Extension of rationality conjecture to ramified cases broadens understanding of Darmon points.

Abstract

Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon 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 $H^+_F$ of the real quadratic field $F$ and twisted by (unramified) quadratic characters of $Gal(H_c^+/F)$. We extend these results to the situation of ramified quadratic characters; more precisely, we show that Darmon points of conductor $c\geq 1$ twisted by quadratic characters of $G_c^+=Gal(H_c^+/F)$, where $H_c^+$ is the strict ring class field of $F$ of conductor $c$, come from rational points on the elliptic curve defined over $H_c^+$.