Topics in Number Theory pp 127-145 | Cite as

# Computation of the Mordell-Weil Group

Chapter

## Abstract

If
is possible only with

*G*is an abelian group (written additively), the elements*g*_{1},...,*g*_{ r }of*G*are called*independent*if$$m_1 g_1 + \cdots m_r g_r = 0\;\left( {m_j \varepsilon \mathbb{Z}} \right)$$

*m*_{1}= ⋯ =*m*_{ r }= 0. Thus if one of*g*_{1},...,*g*_{ r }is of finite order,*g*_{1},...,*g*_{ r }cannot be independent. For any elliptic curve*E*defined over ℚ the group*E*(ℚ) of rational points on*E*is finitely generated. The (*Mordell-Weil*)*rank**r*_{ℚ}(*E*) of*E*is defined to be the maximum number of independent elements in*E*(ℚ). In particular,*r*_{ℚ}(*E*) = 0 if and only if*E*(ℚ) is finite (consisting of points of finite order).## Keywords

Elliptic Curve Rational Point Elliptic Curf Prime Divisor Integer Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves II,
*J. Reine Angew. Math.***218**, 79–108 (1965).MathSciNetzbMATHGoogle Scholar - 2.E. Lutz, Sur l’équation
*y*^{2}=*x*^{3}–*Ax*–*B*dans les corps*p*-adiques,*J. Reine Angew. Math.***177**, 237–247 (1937).Google Scholar - 3.B. Mazur, Rational points on modular curves,
*Modular Functions of One Variable*V,*Lecture Notes in Mathematics*Vol. 601, Springer Verlag, Berlin (1977).Google Scholar - 4.J. T. Tate, Rational points on elliptic curves, Phillips Lectures given at Haverford College, 1961 (unpublished).Google Scholar

## Copyright information

© Springer Science+Business Media New York 1988