A natural question that arises is why the Mordell-Weil theorem is important. Apparently, it provides us with a very useful information about the structure of the group of rational points of an elliptic curve, but is there any deeper reason, that makes the Mordell-Weil theorem worth-studying?
Louis Joel. Mordell (1888-1972;
The answer to that question is simple enough, if we think of an elliptic curve as a geometric object defined by a diophantine equation. One of the goals of number theory, even from ancient times, is the study of diophantine equations, and the determination of their integral or rational solutions, the so called diophantine problems. The geometric analogous of that, is the finding of integral or rational points of the curve that a diophantine equation defines. So, the general problem that we are interested in, is the following:
Given any curve, are we able to describe the set of the rational points on it, or even better, to determine it explicitly?
The formulation of this general problem is rather imprecise, because the term "curve" has not been specified. Historically, the first curves that were studied, were defined over Q. For simplicity, we stick to the case of smooth curves. Given any smooth curve C, the problem is to determine the set C(Q) of the rational points on C. It turns out that we may consider the projective model of the curve, since it differs only in a finite set of points, i.e. the singularities and the point at infinity. Of course, there is always the possibility for C not to have any rational points, i.e. C(Q) = 0. If otherwise, using the classification of curves according to their genus, we obtain the following brief solution to our problem.
• If C is curve of genus 0, then the set C(Q) is isomorphic to the projective line P1 (Q). In other words, we may give a parameterization of C(Q) in terms of one-variable rational functions.
• If C is a curve of genus 1, then it is an elliptic curve. Mordell (1922) proved that in this case the set C(Q) is a finitely generated abelian group.
• Finally, if C is a curve of genus > 2, then the set C(Q) is finite, which is a result of great importance due to Faltings.
Weil (1929) extended the result of Mordell for elliptic curves defined over arbitrary num¬ber field.
Strictly speaking, an elliptic curve E defined over a number field K is a nonsingular pro¬jective algebraic curve of genus 1, with at least one K-rational point. The elliptic curve E is defined by
X3 + aX + β,
with α, β e K. The interesting is that we are able to define the operation of addition on E, as it seems in the following figure.
It turns out that (E(K"), +) is an abelian group, and so is (E(K), +). The theorem of Mordell and Weil states that the group (E(K), +) is finitely generated.