# On the relationship between Fermats Last Theorem and Elliptic Curves

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.

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This question is also posted on mathoverflow: mathoverflow.net/questions/103577/… –  William Jul 31 '12 at 3:31
I may be wrong, but I do not think there is a simple explanation. –  Alex Becker Jul 31 '12 at 3:33

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.

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