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I want to study FLT (Fermat's Last Theorem), and now I'm studying moduli of elliptic curves.
I've heard that Deligne-Rapoport, Katz-Mazur, Mazur's "Modular curves...", and Katz's "p-adic..." are very good for this topic.

But I don't know what's the difference between these papers. For me it seems that these treat almost same topics.

So what should I read at first? (Now I'm reading Katz-Mazur. It's easy to read even for me, a beginner of arithmetic geometry. So I think I should read it at first. And glancing through Mazur, it seems to use many results from other 3 papers.)

And would you recommend other good papers which I read understand for understanding FLT?

Any help will be much appreciated!

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  • $\begingroup$ Every answer is very great! But does anyone know what's the difference of these papers, and which should I read at first? $\endgroup$
    – k.j.
    Commented Aug 22, 2019 at 9:00

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I would have a look at Cornell--Silverman--Stevens's book called "Modular forms and Fermat's last theorem". IMO it covers the material really well and you can always chase through the references there for any further details you need.

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Not having been mentioned before, I would recommend the two books "Fermat's Last Theorem, Basic Tools" and "Fermat's Last Theorem, The Proof" by Takeshi Saito. https://bookstore.ams.org/mmono-243 and https://bookstore.ams.org/mmono-245.

In particular, moduli of elliptic curves appear in chapter 2.2 but only over $\mathbb{Q}$. Then the second volume starts with chapter 8 "Modular curves over $\mathbb{Z}$" which covers the topics on the moduli spaces of elliptic curves needed for Fermat's Last Theorem.

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I recommend Dick Hain's beautiful Lectures on Moduli of Elliptic Curves for a classical complex analytic and topological perspective, although farther from arithmetic geometry, so less helpful for Fermat's Last Theorem.

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As another reference, there is Milne's notes. This starts from basics and builds fastly and exhaustively to reach the fermat's Last theorem, meanwhile meeting some principles involved in Birch-Swinnerton-Dyer also on the way. It also proves the Riemann hypothesis for elliptic curves. Truly a number theorist and algebraic geometer's haven!

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