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As one Putnam Fellow to another: I took a graduate course on modular forms and I still don't really feel that I know what they are. I can't help feeling that they're the mathematical analogue of quantum mechanics: "If you think you understand modular forms, you don't understand modular forms."

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Wow! That's a lot. But are they something that it's intrinsically difficult to understand or is it more like "yeah sure I can follow the definitions and such and maybe a theorem or two, but why is this important at all?"

It's more that they're kind of magical. You think you have a handle on how they behave, then you see a theorem and them and you're like "how the hell do they do that?"

For automorphic forms, absolutely.

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