Preparation for a modular forms course

Question

I'm trying to get a sense of what type of math to brush up on in order to take a course based on the book A First Course in Modular Forms by F. Diamond and J. Shurman. The text claims to be accessible to advanced undergraduates / first year graduate students with adequate preparation in algebra and complex analysis; I'm trying to get a sense of what parts of these subjects are heavily used enough to warrant some review.

I will have approximately 4 weeks where I'll be relatively responsibility free, and so should have a significant amount of time each day to prepare. I'm told to favor group theory over complex analysis, but I'm looking for a bit more specific advice. My current plan is to go through Chapter 5 of Robert Ash's algebra notes. Keep in mind I've seen all the material in the notes before, and am just refreshing myself. I also plan to start Stein & Shakarchi's Complex Analysis — again, I've seen much of this material before also, but certainly need a refresher.

So I guess my question is — where does the heavy duty algebra / heavy duty complex analysis show up?

Thanks.

Answer 1

I'd say that one of the key things to get your head around is the theory of Riemann surfaces. D + S use this very heavily in the chapter on dimension formulae, and in various other places too. So you should make sure you're happy with:

• patching together Riemann surfaces from coordinate charts
• the Riemann-Hurwitz theorem and the genus
• differentials
• line bundles and the Riemann-Roch theorem

My perception is that serious algebra is much less necessary for a modular forms course at this level. I can't think of anything much you need other than the ideas of left, right and double cosets and familiarity with the classification of finitely generated abelian groups.

I've taught a modular forms course three times myself. The first time I did so, I followed D+S quite closely, and the students really struggled with understanding the Riemann surface structure of modular curves. So if your professor is going to follow D+S's approach I think this might be the most important thing to prepare yourself for. (On subsequent occasions I've taken a rather different approach which avoids mentioning Riemann surface theory at all, which is much more accessible; the catch is that you can prove an upper bound for the dimension of modular forms spaces but you can't give an exact formula.)