Finitely Ramified Iterated Extensions

Let p be a prime number, K a number field, and S a finite set of places of K. Let KS be the compositum of all extensions of K (in a fixed algebraic closure K) which are unramified outside S, and put GK,S = Gal(KS/K) for its Galois group. These arithmetic fundamental groups play a very important role in number theory. Algebraic geometry provides the most fruitful known source of information concerning these groups. Namely, given a smooth projective variety X/K, the p-adic étale cohomology groups of X are finitedimensional vector spaces over Qp equipped with an action of GK,S where S consists of the primes of bad reduction for X/K together with the primes of K of residue characteristic p. The richness of this action can be judged, for example, by the intimate relationships between algebraic geometry and the theory of automorphic forms which it mediates. For this and many other reasons, it would be difficult to overstate the importance of these p-adic Galois representations. Nonetheless, linear p-adic groups simply form too restrictive a class of groups to capture all Galois-theoretic information, and some important conjectures in the subject, notably the Fontaine-Mazur conjecture [10] (to mention only one, see the discussion in Section 7), point specifically toward the kind of information inside arithmetic fundamental groups which cannot be captured by finitedimensional p-adic representations.