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arXiv · 1901.04354

Cutting towers of number fields

Farshid Hajir·Christian Maire·Ravi Ramakrishna
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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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Farshid Hajir, Christian Maire, Ravi Ramakrishna. 2019-01-14. Cutting towers of number fields. https://arxiv.org/abs/1901.04354

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