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

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 .

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

William Stein 2012-09-24