On the Northcott property for special values of L-functions

Abstract

We propose an investigation on the Northcott, Bogomolov and Lehmer properties for special values of $L$-functions.We first introduce an axiomatic approach to these three properties. We then focus on the Northcott property for special values of $L$-functions. In the case of $L$-functions of pure motives, we prove a Northcott property for special values located at the left of the critical strip, assuming that the $L$-functions in question satisfy some expected properties. Inside the critical strip, focusing on the Dedekind zeta function of number fields, we prove that such a property does not hold for the special value at one, but holds for the special value at zero, and we give a related quantitative estimate in this case.