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
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## References

- 1.B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves II,
*J. Reine Angew. Math.***218**, 79–108 (1965).MathSciNetMATHGoogle 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Scholar - 4.J. T. Tate, Rational points on elliptic curves, Phillips Lectures given at Haverford College, 1961 (unpublished).Google Scholar

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© Springer Science+Business Media New York 1988