Extensions of Number Fields with Wild Ramification of Bounded Depth

Fix a prime number p, a number field K, and a finite set S of primes of K. Let Sp be the set of all primes of K of residue characteristic p. Inside a fixed algebraic closure K of K, let KS be the maximal p-extension (Galois extension with pro-p Galois group) of K unramified outside S, and put GS = Gal(KS/K). The study of these “fundamental groups” is governed by a dichotomy between the tame (S∩ Sp = ∅) and wild (S∩ Sp = ∅) cases. One feature of this dichotomy is the following. In the tame case, every open subgroup of GS has finite abelianization (following Lubotzky, we say GS is FAb). On the other hand, if Sp ⊆ S, then GS has a surjection onto Z2 p (induced by the Zp-extensions of K), where r2 is the number of imaginary places of K. (For surjections of GS to Zp when S ⊂ Sp, see [19].) Indeed, the difference between the tame and wild cases is highlighted by a conjecture of Fontaine and Mazur [8] which predicts that, in the tame case, GS is “p-adically finite,” meaning it has no infinite p-adic analytic quotients. A second, and subtly related, feature is the following: for p ∈ S−Sp, the filtration D(KS/K, p) ⊇ D(KS/K, p) ⊇ · · · of GS by higher ramification groups at p (in the upper numbering) has length at most 2, that is, D(KS/K, p) vanishes, whereas in the case of wild ramification in an infinite p-extension, it is often the case that the higher ramification groups of all indices are nontrivial; the latter condition is called “deeply ramified,” [5], the archetypal example being a Zp-extension.