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This article is cited in 53 scientific papers (total in 53 papers)
Finiteness of $E(\mathbf Q)$ and $Ш(E,\mathbf Q)$ for a subclass of Weil curves
V. A. Kolyvagin
Abstract:
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.
Received: 25.06.1987
Citation:
V. A. Kolyvagin, “Finiteness of $E(\mathbf Q)$ and $Ш(E,\mathbf Q)$ for a subclass of Weil curves”, Math. USSR-Izv., 32:3 (1989), 523–541
Linking options:
https://www.mathnet.ru/eng/im1191https://doi.org/10.1070/IM1989v032n03ABEH000779 https://www.mathnet.ru/eng/im/v52/i3/p522
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