Heegner points and the arithmetic of elliptic curves over ring class extensions
Let \(E\) be an elliptic curve defined over \(\mathbb Q\) and let \(K\) be a quadratic imaginary field such that all primes dividing the conductor \(N\) of \(E\) split in \(K\). Then \(N\mathcal O_{K}=\mathcal N\bar{\mathcal N}\) for an ideal \(\mathcal N\) of \(\mathcal O_{K}\). For each positive squarefree product \(c\) of rational primes that are inert in \(K\) and coprime to \(N\), let \(K_{c}\) denote the ring class field of conductor \(c\) and let \(G\) denote the abelian group \(\mathrm{Gal}(K_{c}/K)\). Note that \(\#G\) equals the class number \(h_{c}\) of \(\mathcal O_{c}:=\mathbb Z+c\,\mathcal O_{K}\). The pair \((\mathbb C/\mathcal O_{c},(\mathcal N\cap\mathcal O_{c})^{-1}/\mathcal O_{c})\) defines a CM elliptic curve equipped with a cyclic subgroup of order \(N\), and the isomorphism class of this pair defines a point \(x_c\in X_{0}(N)(K_{c})\). Let \(y_{c}=\phi_{E}(x_c)\in E(K_c)\) be the corresponding Heegner point, where \(\phi: X_{0}(N)\to E\) is a fixed modular parametrization of minimal degree, and let \(W=\mathbb Z[G]y_{c}\).NEWLINENEWLINEThe main result of the paper is the following: assuming the truth of the Birch and Swinnerton-Dyer formula, the validity of another conjecture involving the value at 1 of a certain \(L\)-function attached to a character and a newform and certain other conditions which insure, in particular, that \(E(K)+W\) is a subgroup of finite index in \(E(K_{c})\), the authors derive a formula for the index \([E(K_{c})/\text{tors}: (E(K)+W)/\text{tors}]\) in terms of the class number \(h_{c}\) and various other familiar arithmetical constants, e.g., orders of Tate-Shafarevich groups and Tamagawa factors. A specific example is given involving the curve \(y^{2}+y=x^{3}+x^{2}-2x\). As a result of their work on the index formula mentioned above, the authors realize that previously published explicit generalizations of the Gross-Zagier to ring class field appear to be wrong and offer a conjecturally correct formula.
scientific article; zbMATH DE number 6042414
scientific article