# Texts on moduli of elliptic curves

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!

• Every answer is very great! But does anyone know what's the difference of these papers, and which should I read at first? – k.j. Aug 22 '19 at 9:00

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.

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.

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!

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.