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 Izv. Akad. Nauk SSSR Ser. Mat., 1988, Volume 52, Issue 3, Pages 522–540 (Mi izv1191)

Finiteness of $E(\mathbf Q)$ and $Ø(E,\mathbf Q)$ 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 $Ø(E,\mathbf Q)$ of the curve $E$ (over $\mathbf Q$) are finite. For example, $Ø(X_{17},\mathbf Q)$ is finite. In particular, $E(\mathbf Q)$ and $Ø(E,\mathbf Q)$ are finite if $(\Delta,2N)=1$ and $L_f'(E,K,1)\ne0$, where $f=\infty$ or $f$ is a rational prime such that $(\frac fK)=1$ 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 $Ø(E,\mathbf Q)$.
Bibliography: 11 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1989, 32:3, 523–541

Bibliographic databases:

UDC: 519.4
MSC: Primary 11G40, 11G05, 11F67; Secondary 14K07, 11D25, 14G10, 11R23

Citation: V. A. Kolyvagin, “Finiteness of $E(\mathbf Q)$ and $Ø(E,\mathbf Q)$ for a subclass of Weil curves”, Izv. Akad. Nauk SSSR Ser. Mat., 52:3 (1988), 522–540; Math. USSR-Izv., 32:3 (1989), 523–541

Citation in format AMSBIB
\Bibitem{Kol88} \by V.~A.~Kolyvagin \paper Finiteness of $E(\mathbf Q)$ and $\textit{Ø}(E,\mathbf Q)$ for a~subclass of Weil curves \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1988 \vol 52 \issue 3 \pages 522--540 \mathnet{http://mi.mathnet.ru/izv1191} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=954295} \zmath{https://zbmath.org/?q=an:0662.14017} \transl \jour Math. USSR-Izv. \yr 1989 \vol 32 \issue 3 \pages 523--541 \crossref{https://doi.org/10.1070/IM1989v032n03ABEH000779} 

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Kolyvagin, “On the Mordell–Weil and Shafarevich–Tate groups for Weil elliptic curves”, Math. USSR-Izv., 33:3 (1989), 473–499
2. V. A. Kolyvagin, D. Yu. Logachev, “Finiteness of Ø over totally real fields”, Math. USSR-Izv., 39:1 (1992), 829–853
3. Shin-ichi YOSHIDA, “SOME VARIANTS OF THE CONGRUENT NUMBER PROBLEM I”, Kyushu J Math, 55:2 (2001), 387
4. Dmitry Logachev, “Action of Hecke Correspondences on Heegner Curves on a Siegel Threefold”, Journal of Algebra, 236:1 (2001), 307
5. Kazuo Matsuno, “Construction of elliptic curves with large Iwasawa
$${\lambda}$$
-invariants and large Tate-Shafarevich groups”, manuscripta math, 122:3 (2007), 289
6. D. Yu. Logachev, “Reduction of a problem of finiteness of Tate-Shafarevich group to a result of Zagier type”, Dalnevost. matem. zhurn., 9:1-2 (2009), 105–130
7. D. Logachev, “Kolyvagin's trace relations for Siegel sixfolds”, Journal of Algebra, 324:6 (2010), 1177
8. Ilker Inam, “Selmer groups in twist families of elliptic curves”, Quaestiones Mathematicae, 35:4 (2012), 471
9. Bosser V. Surroca A., “Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture”, Bull. Braz. Math. Soc., 45:1 (2014), 1–23
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