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 $\sigma \geq \frac{1}{2}$. 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.