# Proof of the Weak Mordell-Weil Theorem

Suppose is an elliptic curve over a number field , and fix a positive integer . Just as with number fields, we have an exact sequence

Then we have an exact sequence

From this we obtain a short exact sequence

 (12.1)

Now assume, in analogy with Section 12.1, that , i.e., all -torsion points are defined over . Then

and the sequence (12.2.1) induces an inclusion

 (12.2)

Explicitly, this homomorphism sends a point to the homomorphism defined as follows: Choose such that ; then send each to . Given a point , we obtain a homomorphism , whose kernel defines an abelian extension  of that has exponent . The amazing fact is that can be ramified at most at the primes of bad reduction for and the primes that divide . Thus we can apply theorem 12.1.1 to see that there are only finitely many such .

Theorem 12.2.1   If is a point, then the field  obtained by adjoining to all coordinates of all choices of is unramified outside and the primes of bad reduction for .

Proof. [Sketch of Proof] First one proves that if is a prime of good reduction for , then the natural reduction map is injective. The argument that  is injective uses formal groups'', whose development is outside the scope of this course. Next, as above, for all . Let be the inertia group at . Then by definition of interia group, acts trivially on . Thus for each we have

Since is injective, it follows that for , i.e., that is fixed under all . This means that the subfield of  generated by the coordinates of is unramified at . Repeating this argument with all choices of  implies that is unramified at  .

Theorem 12.2.2 (Weak Mordell-Weil)   Let be an elliptic curve over a number field , and let be any positive integer. Then is finitely generated.

Proof. First suppose all elements of have coordinates in . Then the homomorphism (12.2.2) provides an injection of into

By Theorem 12.2.1, the image consists of homomorphisms whose kernels cut out an abelian extension of  unramified outside and primes of bad reduction for . Since this is a finite set of primes, Theorem 12.1.1 implies that the homomorphisms all factor through a finite quotient of . Thus there can be only finitely many such homomorphisms, so the image of is finite. Thus itself is finite, which proves the theorem in this case.

Next suppose  is an elliptic curve over a number field, but do not make the hypothesis that the elements of have coordinates in . Since the group is finite and its elements are defined over  , the extension  of  got by adjoining to  all coordinates of elements of is a finite extension. It is also Galois, as we saw when constructing Galois representations attached to elliptic curves. By Proposition 11.3.1, we have an exact sequence

The kernel of the restriction map is finite, since it is isomorphic to the finite group cohomology group . By the argument of the previous paragraph, the image of in under

is finite, since it is contained in the image of . Thus is finite, since we just proved the kernel of is finite.

William Stein 2012-09-24