# A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended with a proof that the Modularity Theorem implies Fermat's Last Theorem, maybe up to citing the statements of a few other big theorems (e.g. the $\epsilon$-conjecture/Ribet's Theorem).

I'm interested in finding a reference which explains the proof from this perspective. Of course, trying to learn all the arithmetic geometry necessary to understand what Wiles actually proved would be quite the harrowing feat, but this approach seems like it could shed light on what on earth equations of the form $a^n + b^n = c^n$ have to do with whether elliptic curves over $\mathbb{Q}$ are modular. In other words, imagine a universe where the Modularity Theorem was proved as soon as it was conjectured by Taniyama and Shimura in the 50's. What would books explaining "Wiles's proof" (or "Ribet's proof" or "Serre's proof" or "Frey's proof") contain?

It's fine if a reference assumes familiarity with relatively introductory algebraic number theory and algebraic geometry (like Hartshorne and a similar level ANT book such as Frolich and Taylor), but I'd rather it didn't expect I know, say, anything about deformations of Galois representations.

• There is a very nice book called "Modular forms and Fermat's last theorem" by Cornell, Silverman and Stevens, eds. – Ferra Aug 24 '15 at 9:15
• I've heard this book described as the book to read if you want to get (most) of the details of Wiles' proof - is it not too advanced for this purpose? – Dorebell Aug 24 '15 at 9:17
• You could try reading Serre's paper. It looks like what you want starts in 4.1. But if you really want to hop from big theorem to big theorem I think it is going to be hard. The objects that Ribet and Taylor-Wiles take in and spit out are complicated. I don't think anyone is seriously saying "read all of CSS", just that Stevens' summary is short and well-written. It's a good place to start. – Hoot Sep 5 '16 at 5:44
• Maybe you could try Saito, Fermat's Last Theorem, The Proof (American Mathematical Society, 2014) or (on a very basic level) : Avner Ash, Robert Gross, Fearless symmetry (Princeton University Press, 2008). – Watson Nov 12 '17 at 15:00