en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
1 correction found
Ribet's proof of the epsilon conjecture in 1986 accomplished the first of the two goals proposed by Frey.
This reverses the two steps in Frey’s strategy. Ribet proved the link that semistable modularity would imply Fermat’s Last Theorem; the remaining step was Wiles’s later proof of semistable modularity.
Full reasoning
Ribet’s own 1993 AMS article says that Frey outlined the implication “Taniyama ⇒ Fermat” and that Ribet succeeded in proving those conjectures in 1986. In other words, Ribet proved the link from the Taniyama–Shimura conjecture to Fermat’s Last Theorem, not the modularity theorem itself.
Specifically, Ribet writes that Frey’s curve led to the implication “Taniyama ⇒ Fermat,” and then: “I succeeded in proving the conjectures in July, 1986 … My announcement that I had proved ‘Taniyama ⇒ Fermat’…” He then explains that Wiles later announced a proof of Taniyama’s conjecture for the semistable case.
So if the two goals are:
- prove modularity (at least for semistable elliptic curves), and
- prove that a counterexample to Fermat would force a non-modular semistable elliptic curve,
then Ribet accomplished the second goal, not the first. The sentence in the post swaps their order.
2 sources
- NEWS ITEM FOR THE “NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY” — Kenneth A. Ribet
Frey outlined an incomplete proof that his curve was not modular, i.e., that one has the implication “Taniyama ⇒ Fermat.” ... I succeeded in proving the conjectures in July, 1986 ... My announcement that I had proved “Taniyama ⇒ Fermat” convinced the mathematical community that Fermat’s Last Theorem must be true.
- Andrew John Wiles | Office of the Dean of the Faculty, Princeton University
In the early 1980s, work of Gerhard Frey, Jean-Pierre Serre and Kenneth Ribet showed that Fermat’s Last Theorem would follow if one knew the fundamental Shimura-Taniyama Conjecture in the theory of modular forms and elliptic curves.