Certainly about thirty years. Sort of in this period the ideas have been becoming more and more fixed. The first indications maybe go back to maybe the late fifties. But the ideas didn't really start becoming definite until maybe 1970. These dates are very rough.
Modular forms are functions which transform in a simple way under the action of that group.
I've done various things, but they're all rather difficult to explain on this sort of level. In one direction, things have been known for twenty-five years or something. If one starts with a suitable modular form, the way to construct a representation of the Galois group has been known for twenty-five years. Now the big problem has been to start with a representation of the Galois group and try to produce a modular form. In fact, there's one result that's rather old, due to Langlands and Tunnell.