The Mordell-Weil Theorem

Abstract

In 1901 Poincare conjectured (or, rather, had tacitly assumed) [6] that all the rational points on any elliptic curve defined over ℚ are obtained from only finitely many by adding them in all possible ways. This was proved by Mordell in 1922 [3]. Soon afterwards in 1928 Weil, in his thesis [7], extended this result to the case of an “abelian variety” defined over a number field K. An abelian variety X defined over K is, roughly speaking, the set of common zeros in projective space of a finite number of homogeneous polynomial equations in several variables with coefficients in K, together with an abelian group law giving the coordinates of P 1 ± P 2 as rational functions of the coordinates of P 1 and P 2 [5]. Weil proved that the group X(K) of the K-rational points of an abelian variety X, i.e., the solutions of these polynomial equations with coordinates in K, is finitely generated. For details see Ref. 2 or 7. Later on he also gave a simpler proof, using the concepts he had introduced in his thesis, for the special case of elliptic curves. It is this proof that we shall be following (cf. Ref. 4 or 8). A very interesting account is in Cassels [1].