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**On the Mordell-Weil and Shafarevich-Tate groups for elliptic Weil curves.**
*(English.
Russian original)*
Zbl 0681.14016

Math. USSR, Izv. 33, No. 3, 473-499 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 6, 1154-1180 (1988).

This article presents an important step towards better understanding of the arithmetic of an elliptic curve E over \({\mathbb{Q}}\). Let \(E({\mathbb{Q}})\cong F\times {\mathbb{Z}}^ g\) be the Mordell-Weil group, with F being a finite abelian group, g the rank of E, \(F=E({\mathbb{Q}})_{tors}\), and \(\text{Russian{Sh}}(E)=Ker(H^ 1({\mathbb{Q}},E)\to \prod_{v}H^ 1({\mathbb{Q}}_ v,E) )\) be the Tate-Shafarevich group of E, where v runs over all primes and \(\infty\). Assuming that E is a Weil curve, i.e. it is parametrized by modular functions, and satisfies a certain non-vanishing condition for L-function \(L(E,s)\) of E at 1, the author proved in his previous work the famous conjecture about finiteness of the group Russian{Sh}\((E)\) [Math. USSR, Izv. 32, No.3, 523-541 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No.3, 522-540 (1988; Zbl 0662.14017)]. An analogous result for elliptic curves with complex multiplication was established by K. Rubin [Invent. Math. 89, 527- 560 (1987; Zbl 0628.14018)]. According to the Birch and Swinnerton-Dyer conjecture, there is a conjectural formula for the order of \(\text{Russian{Sh}}(E,{\mathbb{Q}})\) in terms of the special value \(L^{(g)}(E,1)\), which asserts also that this number is a square.

In the paper under review the last statement is proven for a certain class of Weil curves, including curves of the form \(DY^ 2=4X^ 3- 4X+1\), with \(D\neq -3, -4\), \((D,37)=1\), \(-D<500\), \((-D)\neq 95, 104, 107, 139, 184, 215, 248,| 255, 344, 39\) 1. For 23 curves with \((-D)=\)7, 11, 47, 71, 83, 84, 127, 159, 164, 219, 231, 263, 271, 287, 292, 303, 308, 359, 371, 404, 443, 447, 471 the triviality of Russian{Sh}\((E)\) is established. The key point of the proof is a construction of a certain universal relation, which annulates the Selmer group \(S_ M\) (due to the exact sequence \[ 0\quad \to \quad E(Q)/M\quad \to \quad S_ M\quad \to \quad \text{Russian{Sh}}(E)_ M\quad \to \quad 0\quad). \] This relation comes from a family of Heegner points on E. Some important generalizations of the method to other cases were outlined (to cyclotomic and elliptic units).

In the paper under review the last statement is proven for a certain class of Weil curves, including curves of the form \(DY^ 2=4X^ 3- 4X+1\), with \(D\neq -3, -4\), \((D,37)=1\), \(-D<500\), \((-D)\neq 95, 104, 107, 139, 184, 215, 248,| 255, 344, 39\) 1. For 23 curves with \((-D)=\)7, 11, 47, 71, 83, 84, 127, 159, 164, 219, 231, 263, 271, 287, 292, 303, 308, 359, 371, 404, 443, 447, 471 the triviality of Russian{Sh}\((E)\) is established. The key point of the proof is a construction of a certain universal relation, which annulates the Selmer group \(S_ M\) (due to the exact sequence \[ 0\quad \to \quad E(Q)/M\quad \to \quad S_ M\quad \to \quad \text{Russian{Sh}}(E)_ M\quad \to \quad 0\quad). \] This relation comes from a family of Heegner points on E. Some important generalizations of the method to other cases were outlined (to cyclotomic and elliptic units).

Reviewer: A.A.Panchishkin

### MSC:

14H25 | Arithmetic ground fields for curves |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14H45 | Special algebraic curves and curves of low genus |

14G25 | Global ground fields in algebraic geometry |

14H52 | Elliptic curves |