Why do we want to find rational point on a elliptic curve?

Question

Find the rational point on a curve is a important theme in math, and some famous problems like Fermat's last theorem is associated with it.

Why do we want to find rational point on a elliptic curve? May be the research of Fermat's last theorem led to it?

Thank you for sharing your mind.

Answer 1

There are few reasons we care:

1. Diophantine Problems: These are some of the oldest types of problem in math. Given an equation, we'd like to find the integer solutions, because in a lot of applications, integer solutions are the ones that make sense. We consider rational solutions because $\mathbb{Q}$ is a field and we can therefore division and can apply Field Theory and Galois Theory.

For curves, degree is not the best way to classify curves. Genus seems to be a much better metric. In the case of $g=0$, these curves are conic sections and are well understood. The next most complicated case, is $g=1$ then we have the Mordell Weil Theorem, which says the rational points are a finitely generated group. This theorem is not constructive though, so finding the generators of the group is still an area of a lot of research.

2. Other problems reduce to finding rational points on an elliptic curve. My favourite example is the congruent number problem: A number $D$ is the area of a right triangle with sides of rational length if and only if $$Dy^2 = x^3-x$$ has a rational point. Tunnell's theorem gives a quick criterion for checking this, but the sufficiency depends on the proof of the BSD conjecture.

There are others, but I don't have them off the top of my head now. Will update this post as they come to mind