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.