# A recommended roadmap to Fermat's Last Theorem

I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof.

In a similar vein to this question, I was hoping to get a roadmap as to the required topics, with either suggested books or papers to read, I would be required to learn undertake this task. I am, in particular, looking for expository papers on Galois representations of elliptic curves and deformations of Galois representations.

As for my background I am currently a first year graduate student with the usual algebra, analysis, and topology prerequisites. I also have a course in algebraic number theory (up to the proof of the finiteness of class numbers), modular forms, and algebraic curves (up to Riemann-Roch) under my belt. I am also currently working through Silverman's AEC.

• This should be community wiki. – Igor Rivin May 24 '12 at 13:49
• Let me make the following "philosophical" point. If you read these books like Cornell-Silverman-Stevens, then probably you'll be happy, but in some sense all you will have learnt is how to deduce FLT from stuff that was regarded as standard in the early 90s. So, for example, you will have to take on trust the modularity of $E[3]$ (proved by Langlands and Tunnell using a lot of very complicated analysis, e.g. analytic continuation of Eisenstein series, cyclic base change for GL(2), non-Galois cubic base change...). As another example... – Kevin Buzzard May 24 '12 at 14:50
• ...you'll have to believe in the Neron model of the Jacobian of a curve over a $p$-adic field, the relationship between the reduction of the curve and the reduction of the model. You'll have to believe in local-global for modular forms, a hard theorem of Carayol involving some very delicate vanishing cycles calculations. You'll have to believe in SGA7. You'll have to believe in the reduction of Shimura curves at primes dividing the discriminant (to follow Ribet's work) and this is very technical... – Kevin Buzzard May 24 '12 at 14:54
• ..., and you'll have to believe in Fontaine's work on $p$-divisible groups in order to follow Ramakrishna's thesis, which is crucial. Those are just a few things that spring to mind. In books like Cornell-Silverman-Stevens a lot of these things are regarded as "standard" (because they were!) and references are given. On the other hand $R=T$ theorems are now regarded as "standard"! And FLT follows "via a standard argument" from such theorems! So in some sense it's hard to see where to logically draw the line :-) – Kevin Buzzard May 24 '12 at 14:56
• To continue on Kevin's riff, you'll also have to believe Faltings's proof of the Tate conjecture, the Hecke-Weil theorem relating modular forms and L-series with functional equations and a bunch of other stuff. – Felipe Voloch May 24 '12 at 21:56