On the Northcott property for special values of L-functions

TL;DR

This paper investigates the Northcott property for special values of L-functions, establishing conditions under which these values exhibit finiteness properties, with specific results for Dedekind zeta functions.

Contribution

It introduces an axiomatic framework for Northcott, Bogomolov, and Lehmer properties and proves new results for special values of L-functions, especially in the context of pure motives and number fields.

Findings

Northcott property holds for special values left of the critical strip for pure motives.

The property does not hold at the value one of Dedekind zeta functions.

The property holds at zero of Dedekind zeta functions, with a quantitative estimate.

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.