Algebraic approaches to modular forms

2012-01-12T11:51:22Z

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?

Answer by Neil Strickland for Algebraic approaches to modular forms

2012-01-12T15:10:39Z

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.

Algebraic approaches to modular forms - MathOverflow

Algebraic approaches to modular forms

Answer by Neil Strickland for Algebraic approaches to modular forms