In the immortal words of Barry Mazur,
Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence seem like accidents. But they do exist.
More precisely, in its first incarnation, a modular form of weight k and level N is an analytic function f from the complex upper half plane to the complex plane such that
f (az + bcz + d) = (cz + d)k · f(z)
for all integers a, b, c, d with ad - bc = 1 and c divisible by N. (There is also a growth condition as z approaches infinity or the real axis.) Taking a = b = d = 1, c = 0 we see in particular that
f(z + 1) = f(z)
so that f(z) admits a Fourier expansion
f(z) = &sumn &ge 0 anqn
in terms of q = eiz. The simplest example of a modular form is the Ramanujan modular form &Delta of weight 12 and level 1.

It is probably not at all clear from the above definition why modular forms are of such importance in arithmetic geometry. We offer here a few of the reasons.

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