Are all Frey elliptic curves semi-stable? If so, where exactly is this needed in the modularity approach, now that we know modularity for all rational elliptic curves?


Thank you!



Are Frey elliptic curves semi-stable?
See Proposition 1 in [http://www.fen.bilkent.edu.tr/~franz/ta/ta-flt.pdf](http://www.fen.bilkent.edu.tr/%7Efranz/ta/ta-flt.pdf) and the paragraph following it. You have to adjust the terms in a potential counterexample to Fermat's Last Theorem for prime exponent $p \geq 5$ to make the Frey curve semistable. Whether or not modularity is *now* known in greater generality, Wiles could not handle the general case in the early 1990s but he could handle the semistable case and that is good enough for FLT.


If you look at Ribet's article [here](https://math.berkeley.edu/%7Eribet/Articles/invent_100.pdf) that first established Frey's idea that FLT should follow from modularity of elliptic curves over $\mathbf Q$, the link appears on the second page as Corollary 1.2. In the statement of this result, Ribet assumes *all* elliptic curves over $\mathbf Q$ are modular. At that time it did not seem worth making a weaker hypothesis of modularity only for semistable elliptic curves over $\mathbf Q$ even though that is the kind of elliptic curve arising in Frey's construction and Ribet is quite explicit about that in his proof since it's essential in the argument (as Wuthrich indicates in the comment above).


