Extensions of number fields with wild ramification of bounded depth
Farshid Hajir and Christian Maire
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 the basic properties of such towers (similar to the tamely ramified case but with more complications) to this more general setting; for instance, we show that these towers are "asymptotically good," (we give an explicit bound for the root discriminant). We study the difficult problem of bounding the relation-rank of the Galois groups in question; by using results of Gordeev and Wingberg, we show that they can tend to infinity when the set of ramified primes is fixed but the length of the ramification filtration becomes large. We discuss the $l$-adic representations of these Galois groups in the context of the Fontaine-Mazur and provide some examples