# Algebraic approaches to modular forms

I'd like to learn about modular forms. My background is mostly computational algebra and group theory, and I've had little-to-no training in complex analysis. I've briefly seen modular forms in a short literature review I did on Monstrous Moonshine. I've been scouting out various books, and most have a reasonably strong analytic focus.

While I'm happy to learn enough analysis to get into modular forms via the standard recommended textbooks, I was wondering:

Is there a good reference for learning modular forms that had a particularly algebraic or computational bent?

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How about Modular Forms: A Computational Approach, by William A. Stein ? It is available on his home page at williamstein.org. –  Chandan Singh Dalawat Jan 12 '12 at 11:59
Another good reference is the springer book \textit{A first course in modular forms} of Diamond and Shurman. Or if you wish something more algebraic, you can have a look at the paper of Diamond and Im entitled \textit{Modular forms and modular curves}. –  Nicolas B. Jan 12 '12 at 13:21
What do you mean by "group theory" in your context? If this includes things like Lie groups and algebraic groups, then there is another nice approach to modular forms and a much wider class of objects called automorphic forms. Analysis still plays a role, but the approach has a very different flavor that might appeal to you if you're comfortable with stuff like the structure theory of reductive groups and such. Dan Bump's book on automorphic forms is a very readable introduction to this approach if you've had some experience with classical modular forms. –  Ramsey Jan 12 '12 at 15:21
@Chandan Singh Dalawat: Oh nice! I was unaware there was a free version. –  BrettW Jan 12 '12 at 21:33
@Ramsey: In my context, finite group theory. Specifically p-groups and the classification of finite simple groups. I've had exposure to Lie groups and algebraic groups, so Dan Bump's book also sounds good. –  BrettW Jan 12 '12 at 21:35

You can do a great deal with no analysis whatsoever, by defining modular forms of weight $k$ to be sections of the line bundle $\omega^{\otimes k}$ over the elliptic moduli stack. That sounds quite scary, but it can be made very elementary and concrete after a couple of pages of preparatory discussion. Deligne's "Courbes elliptiques: formulaire" is a good place to look, and quite a lot of that paper is also discussed in Appendix B to "Elliptic spectra, the Witten genus and the Theorem of the Cube" by Mike Hopkins, Matthew Ando and myself. Note that this approach gives the ring $$MF_\ast = \mathbb{Z}[c_4,c_6,\Delta]/(1728\Delta-c_4^3+c_6^2)$$ of modular forms over the integers, not over $\mathbb{C}$. However, if you are interested in Moonshine you may want to construct the $q$-expansion homomorphism $MF_\ast\to\mathbb{Z}[[q]]$. I don't know a fully satisfactory treatment of that without using any analysis.
A treatment can be given using the Tate Curve, which you propably know. That is an elliptic curve $y^2 + + xy = x^3 + a_4 x + a_6$ over the ring $\mathbb{Z}[[q]]$ with coefficients $a_6, a_6$ multiples of the Eisensteinseries $E_4$ and $E_6$. It carries a canonical nowhere vanishing differential (as every affine elliptic curve) so you can evaluated a modular form on it and obtain an element in $\mathbb{Z}[[q]]$. –  Thomas Nikolaus Jan 12 '12 at 20:07
Moonshine will at some point require consideration of level greater than one, and poles at cusps. Also, it might be worth pointing out that the $\omega$ you use is not the usual sheaf of differentials. –  S. Carnahan Jan 13 '12 at 3:34