TITLE: On the relationship between Fermats Last Theorem and Elliptic Curves
QUESTION [5 upvotes]: 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.

REPLY [18 votes]: 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.