motivation for talking about torsion points on an elliptic curve 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.

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.
