The Proof of Fermat’s Last Theorem by R.Taylor and A.Wiles (1995) [pdf]
ams.org
ams.org
Here's even a shorter summary:
Wiles proved a special case of the Taniyama-Shimura conjecture, ie establishing that for every rational elliptic curve there is a modular form with the same Dirichlet l-series. Faltings previously proved that if there was a counterexample to Fermat's Last Theorem, there'd be a certain kind of elliptic curve that is NOT modular - thus Wiles, by proving the modularity theorem for this class, proved FLT. But his proof had an essential gap, so he had to invent a certain kind of Iwasawa theory as an alternative method, to complete the proof.
In otherwords, "summarized" really is an important word.
[edit: fixed spelling]
What I find interesting though is that behind every single sentence I could I understand from the first half of this article, there were literally hours of time spent on learning the concepts. For instance:
>With this addition, the solution set has the structure of an abelian group, with ∞ as the neutral element. The inverse of (x, y) is (x, −y), and the sum of three points vanishes if they lie on a line.
This just reminds the hours I spent learning about group of Weil linear divisors, how they correspond to Cartier divisors on nonsingular varieties, why O(a) != O(b) for a != b if a, b are codimension 1 subvarieties (i.e. points) of eliptic curves, and so on, and so on. Literally every single sentence brings back memories of hours of study.
Now, reading sentences from the half I don't understand makes it completely obvious to me that just like the hours I spent to understand the first half, I need more hours to understand the second half. I feel sad I don't do math anymore.
If you're interested in the mathematics behind this, I'm not sure a direct attack on the FLT proof is the best route to take. The paper that proves a famous conjecture is normally sitting on a mountain of prior work, which means the final paper (1) assumes familiarity with that mountain and (2) is highly technical because all of the understandable things have already been tried.
So instead, why not start learning about the mountain?
The truth of FLT follows from the two claims:
(1) Taniyama--Shimura--Weil conjecture: "Every elliptic curve is modular."
(2) Ribet's Theorem: "If FLT has a counterexample, then such and such an elliptic curve is not modular."
As it happens, TSW was originally believed to be too difficult to prove, but I suppose the connection with FLT motivated people. Taylor and Wiles proved the absolute minimum of TSW that they could get by with and still get a contradiction from Ribet's theorem. (My understanding is that TSW is now fully proved -- the "modularity theorem".)
If you're wanting to "get" FLT, I'd encourage you to look into elliptic curves, modular forms, and their relationship. I wonder if working on the mathematics of elliptic curve cryptography might be a good way to get a feel for elliptic curves.
However, if you do want to take the direct route, I believe that Faltings's highly compressed article provides a syllabus. Once you can read and completely understand every sentence of that article, it is highly likely that the Wiles and Taylor and Wiles papers will make sense. I... would really not recommend this route.
Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem by Simon Singh
The proof of Fermat's last theorem has a long and intriguing history, and Singh's writing is accessible and entertaining for anyone with an interest in math and science, regardless of education level.
I am not only not one of the 'specialists', but am not one of the 'wider mathematical audience', it appears.