Mordell-Weil Theorem

For elliptic curves over the rationals , the group of rational points is always finitely generated (i.e., there always exists a finite set of group generators). This theorem was proved by Mordell (1922-23) and extended by Weil (1928) to Abelian varieties over number fields.

REFERENCES:

Ireland, K. and Rosen, M. "The Mordell-Weil Theorem." Ch. 19 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 319-338, 1990.

Mordell, L. J. "On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees." Proc. Cambridge Philos. Soc. 21, 179-192, 1922-23.

Nagell, T. "Rational Points on Plane Algebraic Curves. Mordell's Theorem." §69 in Introduction to Number Theory. New York: Wiley, pp. 253-260, 1951.

Serre, J. P. Lectures on the Mordell-Weil Theorem, 3rd ed. Braunschweig, Germany: Vieweg, 1997.

Weil, A. "L'arithmétique sur les courbes algébriques." Acta Math. 52, 281-315, 1928.