### Extensions of number fields with wild ramification of bounded depth, by Farshid Hajir and Christian Maire

Abstract: We consider p-extensions of number fields such that the filtration
of the Galois group by higher ramification groups is of prescribed finite
length. We extend well-known properties of tame extensions to this more
general setting; for instance, we show that these towers, when infinite,
are ``asymptotically good'' (an explicit bound for the root discriminant
is given). We study the difficult problem of bounding the relation-rank
of the Galois groups in question. Results of Gordeev and Wingberg imply that
the relation-rank can tend to infinity when the set of ramified primes
is fixed but the length of the ramification filtration becomes large.
We show that all p-adic representations of these Galois groups are
potentially semistable; thus, a conjecture of Fontaine and Mazur on the
structure of tamely ramified Galois p-extensions extends to our case.
Further evidence in support of this conjecture is presented.

Farshid Hajir and Christian Maire <fhajir@csusm.edu,maire@math.u-bordeaux.fr>