Modular forms

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 + b}⁄_{cz
+ 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) = &sum_{n &ge 0} a_nqⁿ

in terms of q = e^2πiz. 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.

Modular forms can be viewed as functions on the set of elliptic curves with level structure.
Modular forms can be viewed as pluricanonical forms on modular curves.
By a construction of Eichler and Shimura, modular forms of weight 2 give rise to elliptic curves and abelian varieties.
By the Taniyama-Shimura conjecture, elliptic curves give rise to modular forms of weight 2.
By a construction of Pierre Deligne, modular forms give rise to two-dimensional p-adic Galois representations.
Modular forms are a kind of automorphic form and thus give rise to automorphic representations.

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