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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).
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
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