Modular Forms and Fermat's Last Theorem

What Hacker News says

1. dhosekAug 2020
   On the flip side, the computer proof at least resolves the question of whether a given conjecture is true. The proof of the four color theorem by computer in 1976–89 opened the door to simplifications and clarifications of the proof in the decade following. This is not unlike a lot of the difficult proofs of the last century. The receommended path to understanding Wiles's proof of Fermat's Last Theorem begins with Yves Hellegouach's book, Invitation to the Mathematics of Fermat Wiles http://www.amazon.com/exec/obidos/ASIN/0123392519/donhosek and then Gary Cornell's Modular Forms and Fermat's Last Theorem http://www.amazon.com/exec/obidos/ASIN/0387946098/donhosek which assumes a graduate student's understanding of number theory and then the reader is ready to actually read Wiles.

   When I was in grad school, I remember reading a comment from someone talking about looking at her roommate's materials for her PhD in English and realizing that while what she had was comprehensible to her even if she didn't necessarily understand any of it, her own materials for her math PhD were completely unintelligible to outsiders.

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2. dsaccoMar 2018
   Fun fact: Wiles’ proof of Fermat’s Last Theorem took him six years to complete in isolation, during which time he essentially told no one in the mathematical community that he was working on it. To prove it, he had to utilize results from number theory, algebraic geometry and category theory. He largely proved it in 1993, save for one error that he did not correct until 1995. One of my favorite videos on YouTube is a ~30 second clip of Wiles describing the moment he discovered the solution to the last remaining error in his proof: https://youtube.com/watch?v=SccDUpIPXM0. Wiles was completely enamored with the problem since his childhood, and pursued a proof in earnest from his early 30s.

   His proof was exceptionally understated when it was presented. In 1993, he gave lectures on his proof of the Taniyama-Shimura-Weil conjecture (the modularity theorem), which was the bulk of the machinery of the proof (the modularity theorem for semistable elliptic curves implies Fermat’s Last Theorem). He simply stated at the end of the lecture that his proof also happened to imply Fermat’s Last Theorem.

   Up until Wiles did it, much of the mathematical community at the time didn’t think Fermat’s Last Theorem was “accessible” to prove with contemporary techniques. Incidentally, Fermat himself wrote that he had a “truly marvelous proof” of the theorem in the margin next to it in one of his books, but which he couldn’t write down becuase the margin was apparently too small to contain it. There’s pretty broad skepticism Fermat actually had a proof however, considering that most of Wiles’ (very long, and very complex) proof borrowed results from large swaths of mathematics not developed until the 20th century.

   If anyone is interested in trying to learn how the proof works at the formal level, the book Modular Forms and Fermat’s Last Theorem by Cornell, Silverman and Stevens provides a rigorous presentation that builds up the necessary mathematics to understand the entire thing. It’s easily a graduate-level textbook, but it’s probably accessible (with effort) to anyone who has worked through undergraduate abstract algebra (up to finite fields and Galois theory) and number theory (up to elliptic curves).

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