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.

  • $\begingroup$ All too complicated and not effective. As we will define this field $k$? Why it should define the distribution of integer solutions? And if we are trying to solve the equation which are not yet known factors, as we will decide if you do not know what is behind the curve. For example this simple type: $ax^2+by^2=cz^2$ ? You know the formula which factors gives a value rank this curve? $\endgroup$ – individ Jun 19 '14 at 5:25
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    $\begingroup$ By Nagell-Lutz theorem, on an elliptic curve with integer coefficients, rational point with finite order has integer coordinates. So finding torsion points help you to find integral instead of just rational solution. $\endgroup$ – achille hui Jun 19 '14 at 17:33

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.

  • $\begingroup$ You did indeed catch my drift. It's a shame that it's difficult to compute the generators of the free part. Thanks for the answer. $\endgroup$ – nigel Jun 21 '14 at 17:07

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.


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