I don't get the point of the title. 4 pages + a PhD in math is needed.. lol. Here's even a shorter summary: Wiles proved a special case of the Taniyama-Shimur

I don't get the point of the title. 4 pages + a PhD in math is needed.. lol. 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]