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!
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7$\begingroup$ From the form of the equation, you see that the curve has multiplicative reduction for all odd primes as we may assume that $a$ and $b$ are coprime. So the curve is semistable at all primes except maybe $2$. This information is useful as it tells you that the level of a newform attached to the hypothetical curve is a power of $2$ times a square-free integer. Then level-lowering will bring it down to $2$ and to the contradiction. If there were a square involved in the starting level this would be harder. $\endgroup$– Chris WuthrichCommented Jul 18, 2022 at 20:32
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1$\begingroup$ I cannot resist pointing out that the question, interpreted literally, is trivial: we know that FLT is true, hence Frey curves do not exist. So the statement "all Frey curves are semistable" is vacuously true, as is the statement "all Frey curves are non-semistable". Ex falso quodlibet! (This is a facetious comment, I have nothing to add to the excellent answers already posted explaining how to prove semistability of Frey curves without using FLT.) $\endgroup$– David LoefflerCommented Jul 19, 2022 at 9:49
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1 Answer
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See Proposition 1 in http://www.fen.bilkent.edu.tr/~franz/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 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).
