Hu, Yan (2019) Rationality of Darmon Points Over Genus Fields of Nonmaximal Orders. [Ph.D. thesis]
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Abstract (italian or english)
Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon in , 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+ F /F). We extend these results to the situation of ramiﬁed quadratic characters; we show that Darmon points of conductor c ≥ 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.
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