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, 1922-1923.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.
, the group of rational
points is always finitely generated (i.e.,
there always exists a finite set of group
generators). Equivalently,
is the finite subgroup of torsion
points and 
is the elliptic curve rank. This theorem was
proved by Mordell (1922-1923) and extended by Weil (1928) to Abelian
varieties over number fields.
