Are there mistakes in the proof of FLT?

Question

This semester, I teach a graduate course in epistemology of mathematics and as a case study, I assigned students a discussion on the epistemological status of Fermat's Last Theorem according to different epistemological point of views. One of the hard-line stances we discussed in the course was the one which considers an assertion as being a mathematical theorem only if it comes with a completely rigorous proof.

In 1990 and 1995, the papers On modular representations of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ arising from modular forms, Modular elliptic curves and Fermat's Last Theorem and Ring-theoretic properties of certain Hecke algebras by K.Ribet, A.Wiles and R.Taylor and A.Wiles respectively were published. Presupposing previous mathematical knowledge, they form a complete proof of Fermat's Last Theorem.

My question is the following:

Are there any mistakes in these 3 papers of some mathematical significance?

Of course, it might be hard ot judge what is of significance, so let me mention a couple of things which I believe are not significant.

• Typographical errors are not significant, even if they alter the mathematical content (so if it is written $r\leq s$ while the proof shows $s\leq r$, that is not significant).
• Reference mistakes are not significant (so if there is a sentence like "by lemma 12.8 of [6], we conclude" and lemma 12.8 of [6] allows no such conclusion, but it is easy to see that lemma 4.2 of [11] would do equally well, then that is not significant).
• For the same reason, I will not count an easily filled incomplete argument as significant, even if strictly faulty (so for instance if there is a sentence like "X is a cosy zip, and every cosy zip is a zap, therefore X is a zap" and in fact it is not true that every cosy zip is a zap, but it is true and equally well-known that every strongly cosy zip is a zap and it easy to check that X is a strongly cosy zip, I won't consider this significant).

I would count as significant anything that requires more than a few sentences of essentially novel content (compared to the content provided). If for instance, it is claimed that a certain $x$ can be of 3 different types, each of which satisfy property P, so $x$ satisfies P, but it turns out that $x$ can be of a fourth type. Then, I would count this as a significant mistake even if $x$ satisfies P in this fourth case as well, but for reasons different from the first three cases. In particular, an error does not need to threaten the complete proof nor even the local part it is contained in to be considered significant.

Basically, I'm thinking of a standard much below what formal certification of the proof would require, but still quite stringent. I think that most authors and referees typically do not achieve that standard in most published papers, though some regularly do even in proofs of deep and hard theorems (I think it is believed that Serre achieved it throughout his career with very few exceptions).

With such a standard, are there known mistakes in these 3 papers?

Answer 1

No there are not any mistakes in these papers of any interest. In the 1990s there were a bazillion study groups and seminars across the world devoted to these papers; I personally read all three of the papers you cite, back in the days when I was young and an expert in this area, and they all looked fine to me, and they all looked fine to all the people who were at the IAS with me in 1995 reading them including a whole bunch of people who were a whole lot smarter than me.

As has been pointed out the proof of FLT relies on a whole lot more stuff than just those papers, for example Langlands--Tunnell (which I have not read, and suspect I will never read, but which has been generalised out the park by other authors) and Mazur (which I read through once but which others have read through many many times; it's the kind of paper that some people get addicted to and spend many years devoted to). The full Wiles paper uses Deligne's construction of Galois representations associated to higher weight modular forms, because it proves more than FLT (e.g. it proves R=T for some Hida families) and I've also not read Deligne's construction, but I know people who I trust and who have (e.g. Brian Conrad).

Some comments which might be of interest to you:

1. I was a post-doc in Berkeley in the mid-90s and during that time I read Ribet's paper; occasionally I would find stuff which I couldn't quite follow, so I would knock on Ken's door and ask him about it, and together we would figure out what he meant. A mathematician would not call these mistakes; one could argue that sometimes there were explanations omitted which you have to be an expert to reconstruct, but I think that this is true of many many papers. In particular a mathematician would not call these "mistakes".

2. Wiles uses Gross' results on companion forms, which at the time were not unconditionally proved; Gross' arguments assumed that two "canonically"-defined Hecke operators acting on "canonically" isomorphic cohomology groups coincided; at the time nobody had any doubt that this result was correct, but there was no published proof in the literature, and indeed it looked at the time that it might be hard to check. By 1995 Taylor had discovered a workaround which avoided Gross' work so the experts knew that there was not a problem here. The fact that the Hecke operators did actually coincide was ultimately verified by a student of Conrad in the early 2000s.

3. Wiles uses etale cohomology in some places, and at the time there was a lot of noise generated by logicians about whether this meant that he had assumed Grothendieck's universe axiom, which was known to be something which one could not prove within ZFC (indeed it was known to the logicians that in theory ZFC could be consistent but that ZFC+Grothendieck's universe axiom could be inconsistent). However Deligne's SGA4.5 had, years before the FLT proof, shown that the theory of etale cohomology could be developed within ZFC.