On Diophantine properties for values of Dedekind zeta functions

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 $\sigma \ge \frac{1}{2}$. 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 $\frac{1}{2}\le \sigma \le 1$, we construct suitable families of quadratic number fields, employing either Soundararajan's resonance method, which works when $\frac{1}{2}\le \sigma <1$, or results on random Euler products by Granville and Soundararajan, and by Lamzouri, which work when $\frac{1}{2}<\sigma \le 1$. 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 Généreux and Lalín.

Final version, to appear in Mathematische Annalen