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.

arXiv comment44 pages. Accepted for publication in the Revista Matemática Iberoamericana

References in corpus (18)

• Triangulated categories of mixed motives
• On the Selberg class of Dirichlet series: small degrees
• Small Height and Infinite Non-Abelian Extensions
• On certain infinite extensions of the rationals with Northcott property
• Kudla--Rapoport cycles and derivatives of local densities
• On heights of motives with semistable reduction
• A Brauer-Siegel theorem for Fermat surfaces over finite fields
• Remarks on Rémond's generalized Lehmer problems
• Analogue of the Brauer-Siegel theorem for Legendre elliptic curves
• A New Northcott Property for Faltings Height
• Heights of hypersurfaces in toric varieties
• A uniqueness property of general Dirichlet series
• Points de petite hauteur sur une variété semi-abélienne de la forme ${\mathbb{G}}_m^n\times A$
• A multi-variable version of the completed Riemann zeta function and other $L$-functions
• Hyperbolic three manifolds of bounded volume and trace field degree II
• Short models of global fields
• Explicit L-functions and a Brauer-Siegel theorem for Hessian elliptic curves
• A new way to tackle a conjecture of Rémond