On the relationship between Fermats Last Theorem and Elliptic Curves

Question

I have to give a presentation on elliptic curves in general. It does not have to be very in depth. I have a very basic understanding of elliptic curves (The most I understand is the concept of ranks). I was wondering if anyone could explain to me simply what is the connection between the equation $x^n +y^n=z^n$ and elliptic curves.

Answer 1

Given non-zero integers $A,$ $B$, and $C$, such that $A + B = C$, we can form the so-called Frey curve (named after the mathematician Frey, who first considered elliptic curves in the context of FLT)

$$E: y^2 = x(x-A)(x+B),$$

which has discriminant (up to some power of $2$ which one can compute precisely, but which I will ignore here) equal to $ABC$.

Suppose now that $A = a^p$, $B = b^p$, and $C = c^p$ (so that we have a solution to the Fermat equation of exponent $p$). Then the elliptic curve $E$ has a discriminant which (up to the power of $2$ that I am ignoring) is a perfect $p$th power.

This means that the group of $p$-torsion points $E[p]$ on $E$ (which is a two-dimensional vector space over the field $\mathbb F_p$ of $p$-elements, equipped with an action of the Galois group of $\overline{\mathbb Q}$ over $\mathbb Q$) has very special properties --- in algebraic number theory terms, it is very close to being unramified. (More specifically, but more technically, it is unramified except possibly at $2$ and $p$, and at $p$ the ramification is very mild --- it is finite flat.)

Now the Shimura--Taniyama conjecture, which is what Wiles (together with Taylor) proved, shows that $E$, and so $E[p]$, arises from a weight two modular form. Ribet's earlier results on Serre's epsilon conjecture imply that this modular form must actually be of level $2$. (This is where we use the above information about the ramification.) But there are no non-zero cuspforms of weight $2$ and level $2$, and we get a contradiction.

Although it is much harder (in that the only way we know to rule out the existence of $E[p]$ is by the --- quite difficult --- Shimura--Taniyama conjecture, or else by related more recent results such as Khare and Wintenberger's work on Serre's conjecture), one can think of the non-existence of $E[p]$ as being analogous to Minkowski's theorem in algebraic number theory, which says that an everywhere-unramified extension of $\mathbb Q$ cannot exist.

Answer 2

There are certainly people on this site that are much more capable than I am of explaining the connection competently, so I'll stick to providing a reference. The last chapter of Diamond-Shurman's A First Course in Modular Forms gives an overview of how this story goes, at the level of a graduate-level textbook (of course, given the scope of the result, they have to black-box many, many things). They explain how to construct the Frey curve, which is an elliptic curve associated to a hypothetical solution to Fermat's equation, and they show how the theory of modular forms (in particular the Taylor-Wiles result that all elliptic curves are modular in the sense that they have an "associated" modular form) allows one to show that no such Frey curves can exist (very briefly, if a Frey curve existed, its associated modular form would have to live in a particular space of forms which one can show directly to be empty).