## Modularity of Elliptic Curves over

Fix an elliptic curve over  . In this section we will explain what it means for to be modular, and note the connection with Conjecture 10.2.3 from the previous section.

First, we give the general definition of modular form (of weight ). The complex upper half plane is    Im A cuspidal modular form of level  (of weight ) is a holomorphic function such that and for every integer matrix with determinant  and , we have

For each prime number of good reduction, let . If is a prime of bad reduction let , depending on how singular the reduction  of  is over . If has a cusp, then , and or if has a node; in particular, let if and only if the tangents at the cusp are defined over  .

Extend the definition of the to for all positive integers  as follows. If let . If is a power of a prime  of good reduction, let

If is a prime of bad reduction let .

Attach to the function

It is an extremely deep theorem that is actually a cuspidal modular form, and not just some random function.

The following theorem is called the modularity theorem for elliptic curves over  . Before it was proved it was known as the Taniyama-Shimura-Weil conjecture.

Theorem 10.2.4 (Wiles, Brueil, Conrad, Diamond, Taylor)   Every elliptic curve over is modular, i.e, the function is a cuspidal modular form.

Corollary 10.2.5 (Hecke)   If is an elliptic curve over  , then the -function has an analytic continuous to the whole complex plane.

William Stein 2012-09-24