Upper bounds of root discriminant lower bounds

Siman Wong

Vol. 277 (2015), No. 1, 241–255

DOI: 10.2140/pjm.2015.277.241

Abstract

For any rational number $t\in \left[\mathfrak{0},\mathfrak{1}\right]$, define the logarithmic Martinet function $\beta \left(t\right)$ to be the liminf of the logarithm of the root discriminant of number fields $K$ with $r_{\mathfrak{1}}\left(K\right)∕\left[K:\mathbb{Q}\right]=t$ as $\left[K:\mathbb{Q}\right]$ goes to infinity. Under the generalized Riemann hypothesis for Dedekind zeta functions of number fields, we show that $\beta \left(t\right)<\mathfrak{1}\mathfrak{4}.\mathfrak{5}\mathfrak{5}$ for a dense subset of rational numbers $t\in \left[\mathfrak{0},\mathfrak{1}\right]$. We also study unconditional estimates of the growth of root discriminants by studying how the polynomial discriminant behaves under perturbation of coefficients, and by using Pisot numbers.

Keywords

Chebotarev density theorem, class field towers, Pisot numbers, root discriminants

Mathematical Subject Classification 2010

Primary: 11R29

Secondary: 11R37, 11R21

Milestones

Received: 23 January 2014

Revised: 13 February 2015

Accepted: 13 February 2015

Published: 6 August 2015