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Question: Can Langlands Reciprocity be used to prove Fermat's Last Theorem?

Background

A few years ago I was reading a book on the Langlands Program and the introduction provided a list of motivating consequences. Among the usual suspects was the claim that it would provide a simple proof to Fermat's Last Theorem.

At the time, this tidbit had little to do with my purpose for reading the book; I stored it as a curiosity and moved on. At some point I casually brought it up to a researcher. The response was something along the lines of "Sure. Just apply reciprocity and it falls out."

I am curious again, so I have attempted to find some explanation. Unfortunately, Wiles' proof is often designated as part of the Langlands Program itself, so searching the relevant terms yields expository resources directed at that work. My question is directed at alternative approaches to FLT which have been forgotten in the wake of Wiles' success.

Unfortunately, I do not remember the book which provoked the question.

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    $\begingroup$ I think there were no viable alternatives to A. Wiles' proof and then invocation of Taniyama-Shimura. Certainly other possibilities had been vigorously pursued since at least Kummer and his contemporaries c. 1840, but without success. At the same time, really, Langlands' "Programme" did not ever really predict such a thing, but was malleable enough to accommodate it. $\endgroup$
    – paul garrett
    Commented Oct 11, 2019 at 23:45
  • $\begingroup$ This question discusses some of the points: at first the Langlands program is a correspondence between certain complex automorphic representations of $GL_n(\Bbb{Q})\setminus GL_n(\Bbb{A_Q})$ (a way to construct many new almost L-functions) and representations $Gal(\overline{\Bbb{Q}}/\Bbb{Q})\to GL_m(\Bbb{C})$, whereas the modularity theorem is about the correspondence for the Galois representation $Gal(\overline{\Bbb{Q}}/\Bbb{Q})\to End(E[p^\infty])\cong GL_2(\Bbb{Z}_p)$ $\endgroup$
    – reuns
    Commented Oct 12, 2019 at 5:04

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I think the confusion here lies in what is being called reciprocity (and perhaps the interpretation of "simple"). If by Langlands reciprocity, you mean a correspondence between classical Artin representations and automorphic representations, I think there is no known simple way to get Fermat's Last Theorem just from this. Echoing Paul Garrett's comment, I believe the only approaches to FLT known to be feasible are through elliptic curves, but there is no direct correspondence between (non-CM) elliptic curves and Artin representations.

If you are a bit more liberal, and mean a suitable correspondence between $\ell$-adic representations or mod $p$ representations (or maybe motives) and automorphic representations, then one can interpret Modularity of Elliptic Curves and Serre's Conjecture as special cases of this generalized reciprocity, from which it is relatively easy to conclude Fermat's Last Theorem.

For instance, Richard Taylor in this general overview considers the Shimura-Taniyama Conjecture to be a special case of Langlands' reciprocity conjectures.

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