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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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How far in Diamond and Shurman is the course getting? It seems to me that the easiest way to do this is to start reading Diamond and Shurman and see where you get stuck. –  Qiaochu Yuan Dec 12 '11 at 7:17
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It may be a good idea to see what modular forms are about first. Read for example, Chapter 7 of Serre's A course in arithmetic. I think that would give a very good idea of what the first 5 chapters of Diamond and Shurman are about. –  Soarer Dec 12 '11 at 7:30
    
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. –  Sid Raval Dec 12 '11 at 7:41
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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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