Finiteness of $E(\mathbf Q)$ and $\textit{Ш}(E,\mathbf Q)$ for a~subclass of Weil curves

V. A. Kolyvagin

Izvestiya. Mathematics , Tome 32 (1989) no. 3, pp. 523-541

Résumé

Let $E$ be an elliptic curve over $\mathbf Q$, admitting a Weil parametrization $\gamma\colon X_N\to E$, $L(E,\mathbf Q,1)\ne0$. Let $K$ be an imaginary quadratic extension of $\mathbf Q$ with discriminant $\Delta\equiv\textrm{square}\pmod{4N})$, and let $y_K\in E(K)$ be a Heegner point. We show that if $y_K$ has infinite order ($K$ must not belong to a finite set of fields that can be described in terms of $\gamma$), then the Mordell–Weil group $E(\mathbf Q)$ and the Tate–Shafarevich group $Ш(E,\mathbf Q)$ of the curve $E$ (over $\mathbf Q$) are finite. For example, $Ш(X_{17},\mathbf Q)$ is finite. In particular, $E(\mathbf Q)$ and $Ш(E,\mathbf Q)$ are finite if $(\Delta,2N)=1$ and $L_f'(E,K,1)\ne0$, where $f=\infty$ or $f$ is a rational prime such that $\bigl(\frac fK\bigr)=1$ and $(f,Na_f)=1$, where $a_f$ is the coefficient of $f^{-s}$ in the $L$-series of $E$ over $\mathbf Q$. We indicate in terms of $E$, $K$, and $y_K$ a number annihilating $E(\mathbf Q)$ and $Ш(E,\mathbf Q)$. Bibliography: 11 titles.