Heegner points on Hijikata-Pizer-Shemanske curves and the Birch and Swinnerton-Dyer conjecture
Let \(E/\mathbb Q\) be an elliptic curve of conductor \(N\), \(K\) an imaginary quadratic filed of discriminant \(-D\) and \(\chi\) an anticyclotomic character of the absolute Galois group \(G\) of \(K\) of finite order and of conductor \(c\). Let \(H\) be the fixed field of the kernel of \(\chi\). Let \(L(E/K,\chi,s)\) be the \(L\)-function of \(E\) over \(K\) with the character \(\chi\). Assume that the global root number \(\varepsilon(E/K,\chi)=-1\). Then \(L(E/K,\chi,s)\) has a zero at the point \(s=1\). The Birch-Swinnerton-Dyer conjecture implies that \(L'(E/K,\chi,s)\neq 0\Rightarrow \dim_{\mathbb C}E(H)^{\chi}=1\), where \(E(H)^{\chi}\) is the \(\chi\)-part of the \(G\)-module \(E(H)\otimes\mathbb C\). The tool to prove this implication is the Euler system of Heegner points on \(E\) arising the modular parametrisations using Shimura curves. Previous works done on this implication need some hypothesis such that \(N\) is prime to \(Dc\) and do not give the modular parametrisations explicitly. The results in \textit{J. Nekovář} [Lond. Math. Soc. Lect. Note Ser. 320, 471--547 (2007; Zbl 1152.11023)] and \textit{X. Yuan} et al. [The Gross-Zagier formula on Shimura curves. Princeton, NJ: Princeton University Press (2013; Zbl 1272.11082)] show that the implication can be deduced from proving the existence of non-zero Heegner points in \(E(H)^{\chi}\). The authors construct Heegner points in \(E(H)^{\chi}\) by giving explicit modular parametrisations using Shimura curves associated to orders of Hijikata-Pizer-Shemanske type in indefinite quaternion algebras and prove the implication, without above hypothesis, in a more general context. The authors also prove a similar but weaker results for modular abelian varieties and provide a conjecture on the existence of Heegner points on modular abelian varieties.
scientific article; zbMATH DE number 6918952