# How to approach the Mazur-Wiles paper on Iwasawa theory?

I would like to read and understand the Mazur-Wiles paper on Iwasawa theory: "Class Fields of Abelian Extensions of $\Bbb Q$". What would be the right way to approach this paper?

Currently, my background is as follows: I know a fair bit of algebraic geometry (equivalent to Hartshorne), Class Field Theory (Milne's notes), Elliptic Curves (Silverman's two books) and the basic theory of Modular forms (Diamond and Shurman) and have read parts of Washington's book "Cyclotomic Fields".

I am currently in the middle of reading about modular forms mod p through Serre's papers in the Antwerp volume ("Formes modulaires et fonctions zêta p-adiques"). I have also gone through Ribet's paper on Herbrand-Ribet.

Is this sufficient background to read the Mazur-Wiles paper? I feel like I lack some background on the reduction of modular curves - this was the only part of Ribet's paper that I couldn't follow.

I would also be interested in surveys of the paper if there are any.

• One thing that you definitely need be at ease with are finite (flat) group schemes over more or less arbitrary bases. This comes up when you approach Mazur's "Eisenstein Ideal" paper in IHES (which is abundantly quoted in Mazur–Wiles) and also in the paper itself. A good reference for this is Tate's article in the Fermat volume. Once you are at ease with this, I guess you might need to study Katz' paper in Antwerp, the appendix on the geometric interpretation of modular forms over arbitrary bases (cont...) – Filippo Alberto Edoardo Jul 11 '17 at 10:56
• (...cont) Mazur's IHES paper is not 100% necessary and it might be technically difficult, but from time to time you might need to refer to that paper, or to Katz–Mazur's book. Finally, there is a brillant survey written by Coates as Bourbaki seminar, available here numdam.org/article/SB_1980-1981__23__220_0.pdf – Filippo Alberto Edoardo Jul 11 '17 at 10:57
• Thanks, when you say I need to study Katz' paper, do you mean just the appendix or the entire paper? Do you think it is necessary to read Serre's article in the Antwerp volumes (Formes modulaires et fonctions zêta p-adiques)? – Asvin Jul 11 '17 at 11:47
• By "Katz' paper" I meant $p$-adic properties of modular schemes and modular forms, which is in the Antwerp volume (and I thought it was the appendix of the volume); the (first) appendix of Katz' paper is the link with classical complex modular forms, and (although very interesting in its own) probably not strictly necessary—but if you digest the whole paper, Appendix 1 will come for free as an easy final reading. Concerning Serre: beautifully written, technically Katz contains it but the main idea beneath Katz' paper is to "understand geometrically" Serre's. Read it, Katz will become easier. – Filippo Alberto Edoardo Jul 11 '17 at 15:52
• Thanks for clarifying. I had a quick look at the survey you mentioned, it looks really good! – Asvin Jul 11 '17 at 16:24