Cutting towers of number fields

2019/01/14 by Hajir, Farshid, Maire, Christian, Ramakrishna, Ravi
#11R21 #11R29 #11R37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1901.04354

Abstract

Given a prime p, a number field \K and a finite set of places S of \K, let \K_S be the maximal pro-p extension of \K unramified outside S. Using the Golod-Shafarevich criterion one can often show that \K_S/\K is infinite. In both the tame and wild cases we construct infinite subextensions with bounded ramification using the refined Golod-Shafarevich criterion. In the tame setting we achieve new records on Martinet constants (root discriminant bounds) in the totally real and totally complex cases. We are also able to answer a question of Ihara by producing infinite asymptotically good extensions in which infinitely many primes split completely.

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