motivation for talking about torsion points on an elliptic curve

Question

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points are interesting.

Question: From the point of view of solving diophantine equations, what is the motivation for looking at the torsion points on an elliptic curve?

I understand that the elliptic curve (as a finitely generated abelian group -- by the Mordell-Weil Theorem) has a torsion part and a free part, so besides for the rank of the free part, the only group-theoretically interesting thing is the torsion part. But I'm asking from a number theory point of view, not an algebra point of view, if you catch my drift.

Answer 1

I hope I catch your drift.

From the point of view of diophatine equations, I would think that you would consider elliptic curves over $k = \mathbb Q$ (or a finite extension of $\mathbb Q$). So as a diophantine equation you would like to know all of its rational solutions.

If the curve has rank $r$ then by Mordell-Weil each such a solution is, uniquely, a sum of $r+1$ points, namely the sum of a point in the torsion subgroup and $r$ points in the free part. Thus, to answer the most obvious diophantine question, the complete set of rational solutions, you need to compute the torsion subgroup as well as $r$ generators for the free part.

The torsion group is easy to compute, the $r$ generators not always possible in many cases by current machinery.

In conclusion, the group theoretic structure of the curve is not merely an exercise in abstract algebra, but a powerful tool in describing the complete set of solutions.

Answer 2

If you start with an elliptic curve $E$ over a number field $k$, for example $k=\mathbb{Q}$, there are many reasons why one would want to study all of the torsion points $E_{\text{tors}}$ defined over an algebraic closure $\overline{k}$. A fancy modern viewpoint is that $E_{\text{tors}}$ serves as a number theoretic version of homology. But in terms of motivation, it's very helpful to consider the simpler case of the multiplicative groups $\mathbb{G}_m$ and work by analogy. Thus the torsion subgroup of $\mathbb{G}_m(\overline{k})=\overline{k}^*$ consists of roots of unity, and roots of unity are (among other things) the basic building blocks for constructing abelian extensions. A striking example of this is the Kronecker-Weber theorem, which says that every abelian extension of $\mathbb Q$ is contained in a field generated by roots of unity. In attempting to generalize this to quadratic imaginary fields, one is lead to the theorem that every abelian extension of $\mathbb{Q}(\sqrt{-d})$ is contained in a field that is generated by torsion points on a suitably chosen elliptic curve.