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Izv. Akad. Nauk SSSR Ser. Mat., 1988, Volume 52, Issue 6, Pages 1154–1180 (Mi izv1225)  

This article is cited in 3 scientific papers (total in 3 papers)

On the Mordell–Weil and Shafarevich–Tate groups for Weil elliptic curves

V. A. Kolyvagin


Abstract: 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.

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English version:
Mathematics of the USSR-Izvestiya, 1989, 33:3, 473–499

Bibliographic databases:

UDC: 519.4
MSC: Primary 11G40, 11D25, 11F67; Secondary 11G05, 14K07, 14G10, 11F33
Received: 04.02.1988

Citation: V. A. Kolyvagin, “On the Mordell–Weil and Shafarevich–Tate groups for Weil elliptic curves”, Izv. Akad. Nauk SSSR Ser. Mat., 52:6 (1988), 1154–1180; Math. USSR-Izv., 33:3 (1989), 473–499

Citation in format AMSBIB
\Bibitem{Kol88}
\by V.~A.~Kolyvagin
\paper On~the Mordell--Weil and Shafarevich--Tate groups for Weil elliptic curves
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1988
\vol 52
\issue 6
\pages 1154--1180
\mathnet{http://mi.mathnet.ru/izv1225}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=984214}
\zmath{https://zbmath.org/?q=an:0681.14016}
\transl
\jour Math. USSR-Izv.
\yr 1989
\vol 33
\issue 3
\pages 473--499
\crossref{https://doi.org/10.1070/IM1989v033n03ABEH000853}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. A. Kolyvagin, D. Yu. Logachev, “Finiteness of Ш over totally real fields”, Math. USSR-Izv., 39:1 (1992), 829–853  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Pete L Clark, “There are genus one curves of every index over every number field”, crll, 2006:594 (2006), 201  crossref  mathscinet  isi
    3. N. V. Medvedev, S. S. Titov, “O topologii ellipticheskikh krivykh”, Tr. IMM UrO RAN, 18, no. 1, 2012, 242–250  mathnet  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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