A note on the arithmetic of an elliptic curve over \({\mathbb Z}_ p\)-extensions (Q1109841)

This is a short announcement of a theorem on a conjecture of Mazur. For a Weil parametrized elliptic curve \(X_ 0(N)\to E\), defined over the rationals, \textit{B. Mazur} defined the Heegner module \({\mathcal E}(K_{\infty})\) [see Proc. Int. Congr. Math., Warsaw 1983, Vol. 1, 185--211 (1984; Zbl 0597.14023)], which is conjecturally of rank one as a \(\Lambda\)-module, where \(\Lambda\) is the Iwasawa algebra for the anticyclotomic \({\mathbb Z}_ p\)-extension \(K_{\infty}\) of a quadratic imaginary field \(K\). The author proves this conjecture for the elliptic modular curve \(X_ 0(19)\) under the following conditions: (0) the discriminant of \(K\) is less than \(-4\), and \(p>3\) a prime not equal to \(19\); (1) \(19\) splits in \(K\), (2) class number \(h(p-\varepsilon (p))\) is not divisible by \(3\), where \(\varepsilon(p)\) is \(0,\;1\) or \(-1\) according as \(p\) ramifies, splits or stays prime in \(K\).