I've studied some fundamentals of algebraic geometry and number theory, and now I want to read papers which seem to be the "main stream" of frontier research on arithmetic.

I've heard that Mazur's "Modular curves and the Eisenstein ideal" is one of such papers (and I've also heard that it is good for people who have finished reading Hartshorne here), so I'm about to read it. But glancing through it, I feel that it goes far beyond Hartshorne, and that it needs the modular forms of moduli stack (I don't know what this is at all). And many papers on arithmetic seem to need this theory. However these papers refer to Katz's paper and Deligne, Rapoport's paper on the theory of modular forms, I feel these two papers are very difficult (and too long to start reading, not knowing if these are really good).

So my question is: Please suggest me some references on the theory of modular forms (of moduli stack? Sorry, I know nothing, except that this modular forms are not one which I'm studying in Diamond, Shurman.). If Katz and Deligne, Rapoport are the best, or these are enough to read Mazur, I tried these.

Thank you very much!


1 Answer 1


I don't know what you mean by Modular forms of moduli stack, I think maybe you mean modular forms on moduli stacks. Either way, you should probably have a look at the book by Katz--Mazur titled "Arithmetic Moduli of Elliptic Curves". It should explain how to think about modular curves in the correct setting you want. In order to understand the modular forms rephrased in this language, the reference to Katz's paper "p-adic properties..." is a really good one, but nowadays there are lots of notes online which explain the "Katz definition of modular forms". I think some really good notes on this are Frank Calegari's AWS notes which you can find here: http://swc.math.arizona.edu/aws/2013/ They should take you from the modular forms you know and love to Katz's version.

Also, if its your first time seeing stacks and are scared of such things (like I am) then you are probably fine just ignoring the word stack and just think about a scheme, since, in most cases you can just rig it so that the moduli problem is representable by some scheme (i.e. by making sure you have a nice enough level (a crucial term being sufficiently small))

Lastly, I don't know what you kids these days mean by main stream, but looking at books like Cornell--Silverman--Stevens on the proof of Fermat's Last Theorem would give you an idea of the popular tools are used in number theory and arithmetic geometry (although some might even say this is now old stuff and if you want to see the future then you need to look at things people like Scholze are doing..)


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