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Recently this question caught my eye. Is there a relation to the modularity problem of elliptic curves over $\mathbb{Q}(\zeta_m)$ and this problem? Namely, if all elliptic curves over $\mathbb{Q}(\zeta_m)$ are modular, then there are no nontrivial solutions to the Diophantine equation $$ x^n + y^n = z^n $$ in $\mathbb{Z}[\zeta_m]$ for $n > 2$?

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2 Answers 2

up vote 6 down vote accepted

No, it does not, because FLT does not hold for general cyclotomic fields. Here's a totally stupid counterexample: take a Pythagorean triple, e.g. $3^2 + 4^2 = 5^2$. Then the number field generated by $\sqrt{3}, \sqrt{4} = 2$, and $\sqrt{5}$ is an abelian extension of $\mathbb{Q}$ and hence contained in a cyclotomic field $\mathbb{Q}(\zeta_m)$ for some $m$ ($m = 60$ will do, I guess). So there is a solution to $x^4 + y^4 = z^4$ in $\mathbb{Q}(\zeta_m)$ for some $m$ with $xyz \ne 0$.

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David's answer shows that the answer is no. On the other hand, it is certainly conjectured that elliptic curves over any number field will be modular (more precisely, automorphic) in an appropriate sense.

To explain why this is possible (i.e. why modularity of the appropriate Frey curve is not incompatible with the existence of FLT solutions over certain number fields $F$), remember that the proof of FLT depends on various facts:

Firstly, we need modularity of elliptic curves. In fact, if we are looking at FLT for the prime $p$, what we really need is not modularity of the whole Frey curve, but modularity of the finite flat group scheme of $p$-torsion on the Frey curve. And in fact, what we really need is modularity for this finite flat group scheme at the optimal level. (This is what Ribet's proof of the epsilon conjecture gave.)

In the case of FLT over $\mathbb Q$, this optimal level turns out be to $N = 2$ (!), and so the final ingredient needed is that there be no non-trivial wt. $2$ cuspforms of level $2$.

However, over other number fields, it may (and indeed will!) turn out that there are non-trivial cuspforms, even at small levels like $2$. So the first part of the proof of FLT should go through, but it won't lead to a contradiction, because the cuspform at small level that is constructed is actually allowed to exist!

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