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On the Northcott property for special values of L-functions

Fabien Pazuki, Riccardo Pengo

TL;DR

This work develops an axiomatic framework to study Northcott, Bogomolov, and Lehmer properties for special values of L-functions, focusing on the left of the critical strip for pure motives and on key cases inside the strip such as Dedekind zeta-values. The authors prove a Northcott property for left-of-strip special values of L-functions attached to pure motives of fixed weight, under expected motivic hypotheses, and deduce unconditional consequences for abelian varieties via modularity results. They also analyze the Dedekind zeta-function across critical-strip boundaries, proving Northcott for $\zeta_F^{\ast}(0)$ but not for $\zeta_F^{\ast}(1)$, with explicit effective bounds; they discuss analogous phenomena for L-values of abelian varieties and the BSD framework. Overall, the paper highlights deep connections between height theory, motivic L-values, and arithmetic invariants (conductors, discriminants, regulators), suggesting both how L-values can serve as heights and how these properties reflect fundamental arithmetic structures. The results pave the way for further exploration of motivic heights and their diophantine consequences, including potential extensions to mixed motives and more general L-function families.

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.

On the Northcott property for special values of L-functions

TL;DR

This work develops an axiomatic framework to study Northcott, Bogomolov, and Lehmer properties for special values of L-functions, focusing on the left of the critical strip for pure motives and on key cases inside the strip such as Dedekind zeta-values. The authors prove a Northcott property for left-of-strip special values of L-functions attached to pure motives of fixed weight, under expected motivic hypotheses, and deduce unconditional consequences for abelian varieties via modularity results. They also analyze the Dedekind zeta-function across critical-strip boundaries, proving Northcott for but not for , with explicit effective bounds; they discuss analogous phenomena for L-values of abelian varieties and the BSD framework. Overall, the paper highlights deep connections between height theory, motivic L-values, and arithmetic invariants (conductors, discriminants, regulators), suggesting both how L-values can serve as heights and how these properties reflect fundamental arithmetic structures. The results pave the way for further exploration of motivic heights and their diophantine consequences, including potential extensions to mixed motives and more general L-function families.

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.

Paper Structure

This paper contains 24 sections, 8 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Diophantine properties of the Weil height
  3. Heights of motives and special values of L-functions
  4. Main results
  5. Outlook
  6. Properties of heights
  7. Northcott property
  8. Bogomolov property
  9. Lehmer property
  10. Examples of successive infima
  11. Heights and their relations to special values of L-functions
  12. Logarithmic Weil height
  13. Mahler measure
  14. Canonical height
  15. Faltings height
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $F$ be a number field. Fix an integer $w \in \mathbb{Z}$, and let $\mathcal{M}^{(w)}(F)$ be the set introduced in def:Mw. Then, for every $\mathbf{B} = (B_1, B_2) \in \mathbb{R}_{\geq 0}^2$ and every $n \in \mathbb{Z}$ such that $2 n < w$, the set is finite. In other words, the pair of functions $\{\lvert L^\ast(\cdot,n) \rvert,\dim \colon \mathcal{M}^{(w)}(F) \to \mathbb{R}_{\geq 0} \}$ has

Theorems & Definitions (47)

  • Theorem 1.1: Northcott property at the left of the critical strip
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 37 more