On~the Mordell--Weil and Shafarevich--Tate groups for Weil elliptic curves

V. A. Kolyvagin

Izvestiya. Mathematics , Tome 33 (1989) no. 3, pp. 473-499

Résumé

Let $E$ be a Weil elliptic curve over the field $\mathbf Q$ of rational numbers, $L(E,\mathbf Q,s)$ the $L$-function over $\mathbf Q$, $\varepsilon=(-1)^{g+1}$, where $g$ is the order of the zero of $L(E,\mathbf Q,s)$ at $s=1$. Let $K$ be the imaginary quadratic extension of $\mathbf Q$ with discriminant $D\equiv\textrm{square}\pmod{4N}$, $y\in E(K)$ the Heegner point, $A=E$ or the nontrivial form of $E$ over $K$ according as $\varepsilon=-1$ or $1$. It is proved that if $y$ has infinite order (which is so if $(D,2N)=1$, $L'(E,K,1)\ne0)$, then the groups $A(\mathbf Q)$ and $Ш(A)$ are annihilated by a positive integer $C$ (in particular the groups are finite) determined by $y$. When $\varepsilon=1$ it is proved that $C^2$ coincides with the conjectured finite order of $Ш(A)$ for some $A$ with $L(A,\mathbf Q,1)\ne0$. It is also proved that $Ш$ is trivial for 23 elliptic curves. Bibliography: 21 titles.