"Extensions of Q with minimal ramification and self-dual automorphic forms"
Very little is known about one basic object of number theory, the absolute Galois group G = Gal(Qbar/Q). In particular, if p is a prime, there is an analogous local Galois group. Does it imbed in G? (Yes.) Does it embed in GS where GS is like G, but we impose some ramification conditions (which will be explained)? The answer, as Chenevier and I proved recently, is yes. It is difficult to explain the proof, but I will try to show the geometric analogues.