Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about modular forms (except they're intrinsically sections of powers of the canonical bundle over some moduli space of elliptic curves, and transcendentally differentials on the upper half plane invariant w.r.t. some specific subgroup of $SL(2,\mathbb{Z})$), I have the curiosity -that many other non experts might have- to understand a bit why that is considered a so vast and important topic in mathematics. The wikipedia page doesn't help: on the contrary, it makes this topic appear as quite narrow and merely technical.

I would roughly divide the question into three (though maybe not neatly distinct) parts:

1) Why are modular forms

per seinteresting?

That is, do they "generate" some piece of rich self-contained mathematics? To make an analogy: cohomology functors were born as applied tools for studying spaces, but have then evolved to a very rich theory in itself; can the same be said about m.f.'s?

2) How are modular forms deeply

relatedto other, possibly quite distant, mathematical areas?

For example: I've heard about deep relations to some generalized cohomology theories (elliptic cohomology) via formal group laws coming from elliptic curves; and I've heard about the so called moonshine conjecture; there should also be some more classical relations to the theory of integral quadratic forms and diophantine equations, and of course to elliptic curves; and people here always mention Galois representations...

3) Why are modular forms

usefulas "applied" technical tools?

In this last question I'm ideally expecting indications of cases (or actual theorems) in which some questions that do *not* involve modular forms are asked about some mathematical objects, and an answer that does *not* involve m.f.'s is given, *but* the method used to obtain that answer/proof makes consistent *use* of m.f.'s.

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