Special values of \(L\)-functions and the arithmetic of Darmon points (Q2865906)

The subject paper generalizes previous work of Darmon and Greenberg to introduce a family of points on Jacobians of Shimura curves, which are called Darmon points, and then prove an avatar version of the Gross-Zagier formula.NEWLINENEWLINEMore specifically, let \(K\) be a real quadratic field of discriminant \(\delta_K\), and \(E_{/\mathbb{Q}}\) an elliptic curve without complex multiplication of conductor prime to \(\delta_K\). Let \(L_K(E,s)\) be the complex \(L\)-function of \(E\) over \(K\) and assume the sign of its functional equation is \(+1\). Fix a complex-valued character \(\chi\) of \(\text{Gal}(H_c/K)\) where \(H_c\) is the narrow ring class field of \(K\) of conductor \(c\) and write \(\mathbb{Z}[\chi]\) the cyclotomic subring of \(\mathbb{C}\) generated over \(\mathbb{Z}\) by the values of \(\chi\). The authors introduce the algebraic part \(\mathcal{L}_K(E,\chi,1) \in \mathbb{Z}[\chi]_S\) of the special value of \(L_K(E,\chi,1)\), where \(S\) is a certain auxiliary finite set of primes. It can be shown that \(L_K(E, \chi,1) \neq 0\) if and only if \(\mathcal{L}_K(E,\chi,1) \neq 0\).NEWLINENEWLINEFor a sign \(\varepsilon \in \{\pm\}\) and a suitable so-called \(p\)-admissible prime \(\ell\), the authors introduce a map NEWLINE\[NEWLINE\delta_{\ell} : J_{\varepsilon}^{(\ell)}(K_{\ell}) \otimes \mathbb{Z}[\chi]_S \to \mathbb{Z}[\chi]/p\mathbb{Z}[\chi]_SNEWLINE\]NEWLINE and a twisted sum of Darmon points \(P_{\chi}^{\varepsilon} \in J_{\varepsilon}^{(\ell)}(K_{\ell}) \otimes \mathbb{Z}[\chi]_S\). Here \(J_{\varepsilon}^{(\ell)}\) is an abelian variety over \(\mathbb{Q}\) and its existence is based on a conjecture of a previous work of the authors [Am. J. Math. 134, No. 5, 1197--1246 (2012; Zbl 1294.11099)]. The main theorem of the paper is as follows:NEWLINENEWLINETheorem: The equality \(\delta_{\ell}(P^{\varepsilon}_{\chi}) = t_{\ell} \cdot [\mathcal{L}_K(E,\chi,1)]\) holds in \(\mathbb{Z}[\chi]/p\mathbb{Z}[\chi]_S\) where \(t_{\ell}\) is the exponent of the abelianization of \(\Gamma_{\ell}\).NEWLINENEWLINEUsing the main theorem, the authors are able to prove a vanishing result for certain twisted Selmer groups of elliptic curves over narrow ring class fields of real quadratic fields assuming a conjecture that predicts the Darmon points are rational over suitable narrow ring class fields of \(K\).