Extensions of number fields with wild ramification of bounded depth

TL;DR

This paper investigates p-extensions of number fields with bounded wild ramification depth, extending tame extension properties, analyzing Galois group structures, and supporting conjectures on p-adic representations.

Contribution

It generalizes properties of tame extensions to wild ramification of bounded depth and explores the structure and bounds of associated Galois groups.

Findings

Infinite towers are asymptotically good with explicit root discriminant bounds.

Relation-rank of Galois groups can tend to infinity with increasing ramification depth.

All p-adic representations are potentially semistable, supporting Fontaine-Mazur conjecture.

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.