p-adic Galois Representations and pro-p Galois Groups

Abstract

We study the intimate interactions between the theory of p-adic Galois representations and the structure of pro-p Galois groups. In particular, information passes in both directions. Algebraic geometry, for instance in the guise of elliptic curves and modular forms, yields naturally occurring Galois representations, whereas on the other side, co-homological techniques and variants on class field theory tell us about the generators and relations of the pro-p Galois groups. In the case of pro-p extensions ramified at (primes above) p, this combination works together rather well to elucidate the structure of the set of Galois representations. In the case of pro-p extensions unramified at p,both sides are poorly understood, but there is the fundamental conjecture of Fontaine—Mazur claiming that such representations should have finite image (since algebraic geometry can produce no others). This has very interesting consequences for the corresponding pro-p Galois groups, possibly producing a new family of just-infinite pro-p groups.