FINITENESS OF E(Q) AND \textrmØ(E,Q) FOR A SUBCLASS OF WEIL CURVES

1989/06/30 by V. A. Kolyvagin · 85 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Polynomial and algebraic computation #Mathematics #Prime (order theory) #Discriminant #Finite field #Elliptic curve #Order (exchange) #Quadratic equation #Parametrization (atmospheric modeling) #Finite set #Group (periodic table) #Prime number #Combinatorics #Pure mathematics #Mathematical analysis #Physics #Geometry

paper · doi:10.1070/im1989v032n03abeh000779

published in Mathematics of the USSR-Izvestiya 32(3), 523-541 (IOP Publishing)

openalex publication_date 1989/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/21

Abstract

Let be an elliptic curve over , admitting a Weil parametrization , . Let be an imaginary quadratic extension of with discriminant , and let be a Heegner point. We show that if has infinite order ( must not belong to a finite set of fields that can be described in terms of ), then the Mordell-Weil group and the Tate-Shafarevich group of the curve (over ) are finite. For example, is finite. In particular, and are finite if and , where or is a rational prime such that and , where is the coefficient of in the -series of over . We indicate in terms of , , and a number annihilating and . Bibliography: 11 titles.

Citations

• The arithmetic of elliptic curves
• Heegner points and derivatives ofL-series

Cited by

• On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields
• Birch and Swinnerton-Dyer conjecture [wikipedia]