zbMATH — the first resource for mathematics

Computing the Mordell-Weil group of an elliptic curve over \(\mathbb{Q}\). (English) Zbl 0809.14024
Kisilevsky, Hershy (ed.) et al., Elliptic curves and related topics. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 4, 61-83 (1994).
This paper describes an implementation of Manin’s “conditional” algorithm for computing the Mordell-Weil group of an elliptic curve defined over the rationals. (The algorithm is so called because it is conditional on the truth of the standard conjectures of elliptic curve theory: Birch-(Swinnerton-Dyer), Hasse-Weil, Shimura-Tamagawa-Weil.) Tables obtained by applying the algorithm to some CM curves arising from congruence numbers are given; two curves of rank 5 are found. The implementation will be made available as part of the computer algebra system SIMATH.
The paper gives an essentially self-contained description of the algorithm (including a proof of the theorem of Manin which is the basis of the algorithm), with comments on specific implementation of the various steps. These implementations encompass a wide variety of techniques of computational elliptic curve theory. The authors mention that some similar techniques are described in a book by H. Cohen [“A course in computational algebraic number theory” (Berlin 1993; Zbl 0786.11071)].
For the entire collection see [Zbl 0788.00052].

14H52 Elliptic curves
14H25 Arithmetic ground fields for curves
14Q05 Computational aspects of algebraic curves
11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)