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

Attach to the 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.

William Stein 2012-09-24