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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.

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    The professor wants to get through Chapter 5 (Hecke Operators), and cover some additional material not in the textbook (though I don't know what). @QiaochuYuan your advice is good, and I will certainly do that; I guess what I'm trying to prevent is coming across something midway through the semester that will require a lot of chasing of old material to understand.2011-12-12

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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.)

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    @trynalearn Serre only treats the case of modular forms of level 1, which can be done explicitly without Riemann surface theory. This does not work for forms of general level.2018-08-23