On Diophantine properties for values of Dedekind zeta functions

TL;DR

This paper investigates the Diophantine properties of Dedekind zeta function values at real points, establishing when these values satisfy Northcott and Bogomolov properties, and constructing specific families of number fields with particular zeta value behaviors.

Contribution

It provides new results on the Northcott and Bogomolov properties for Dedekind zeta values, including constructions of number fields with small zeta values and generalizations of previous properties.

Findings

Bogomolov property fails for σ ≥ 1/2

Constructed families of number fields with small zeta values for σ > 1

Proved Northcott property for ζ_K(s) with degree for Re(s)<0

Abstract

We study the Northcott and Bogomolov property for special values of Dedekind $\zeta$-functions at real values $\sigma \in \mathbb{R}$. We prove, in particular, that the Bogomolov property is not satisfied when $\sigma \geq \frac{1}{2}$. If $\sigma > 1$, we produce certain families of number fields having arbitrarily large degrees, whose Dedekind $\zeta$-functions $\zeta_K(s)$ attain arbitrarily small values at $s = \sigma$. On the other hand, if $\frac{1}{2} \leq \sigma \leq 1$, we construct suitable families of quadratic number fields, employing either Soundararajan's resonance method, which works when $\frac{1}{2} \leq \sigma < 1$, or results on random Euler products by Granville and Soundararajan, and by Lamzouri, which work when $\frac{1}{2} < \sigma \leq 1$. We complete the study by proving that the Dedekind $\zeta$ function together with the degree satisfies the Northcott property for every complex $s\in{\mathbb{C}}$ such that $\mathrm{Re}(s) <0$, generalizing previous work of G\'en\'ereux and Lal\'in.