In Chapter 1 of the text, Stevens provides an overview of the proof of Fermat’s Last Theorem. In this overview, he shows how one needs to use a corollary of Langlands-Tunnell that says that if $E/\mathbb Q$ is an elliptic curve and $\overline{\rho}_{E,3}$ is its mod $3$ residual representation, then if this representation is irreducible, it must be modular.
As we continue through the text, we find a wonderful outline of the proof of Langlands–Tunnell and this corollary in Chapter 6, due to Gelbart. Given the proof overview in Chapter 1 and the results of Chapter 6, one might think we are finished with the issue.
Then something puzzling happens. We move into Chapter 7 on Serre’s Conjecture, authored by Edixhoven. The main result of the chapter, arguably, is the following:
Theorem 3.1. Let $p ≥ 3$ be prime. Let $ρ \colon G_{\mathbb{Q}} \to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be irreducible and modular of some type $(N, 2, 1)_{\overline{\mathbb{Q}}}$ with $N$ square free. If $\rho$ is finite at $p$, it is modular of type $(N(\rho), 2, 1)_{\overline{\mathbb{Q}}}$. If $\rho$ is not finite at $p$, it is modular of type $(p N(\rho), 2, 1)_{\overline{\mathbb{Q}}}$.
This is essentially Ribet’s Theorem, and it is proven throughout the course of Chapter 7, Section 3. One might think, given the overview of the proof of FLT given in Chapter 1, that this section would end the chapter. However, to my surprise at least, there is another section in Chapter 7, which seems to come out of nowhere, and which aims to prove the following result:
Let $E$ be a semi-stable elliptic curve over $\mathbb{Q}$, with $\rho_3$ irreducible. It follows, Proposition 2.1, that $ρ_3 \colon G_{\mathbb{Q}} \to \mathrm{GL}_2(\mathbb{F}_3)$ is surjective. The aim of this section is to show that $\rho_3$ is modular of type $(N(\rho_3), 2, 1)_{\overline{\mathbb{Q}}}$ if $\rho_3$ is finite at $3$ and of type $(3 N(\rho_3), 2, 1)_{\mathbb{\overline{Q}}}$ if $\rho_3$ is not finite at $3$.
When I first read this over a year ago, I immediately had two questions, and despite several discussions with experts, I have yet to have these questions answered in a way that I find understandable.
- How does this result not follow immediately from Theorem 3.1 above? If it does, why is there such an extensive proof of it given when Theorem 3.1 is proven in Section 3? Is this part of that proof or something?
- More importantly, if we are using the narrative of the textbook (rather than the narrative of Wiles/Taylor, whose argument this book improves upon) why do we even need this result? It seems Chapter 1 does not actually use it; instead, it uses the weaker corollary that I gave above, unless the specificity of this result (namely, the focus on the level) is buried somewhere in the argument in Chapter 1, out of plain sight. So my second question is, where specifically, in Chapter 1 of the text, is this specific result needed? (again, by specific, I mean all the details about the level are used).
For 1, you need to note the important hypothesis of Theorem 3.1: namely, the mod $3$ representation must already be modular with square-free level, weight $2$ and trivial character. But Tunnell-Langlands applies to the lift to $GL_2(\mathbb{Z}[\sqrt{-2}])$ of $\rho_3$, whose conductor is very much not square-free, and yields a form of weight $1$ and thus odd (hence nontrivial) character.
I’m not too sure about your second question, I can’t see precisely what you’re talking about: the conjunction of 7.8, 7.9 in Chapter I, maybe? My guess would be that the word “modular” does a lot of work here (without much precision – restriction on weight? level? character?), and this section of Chapter VII fills in the technical details.