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Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures

David Hokken, Mahya Mehrabdollahei, Berend Ringeling

Abstract

Let $χ_{-f}$ be the odd quadratic Dirichlet character of conductor $f$, and let $\mathrm{m}(P)$ denote the Mahler measure of a polynomial $P$. In 1984, Chinburg conjectured that for any such $χ_{-f}$ there exist an integral bivariate rational function $P$ (and, in the strong form, an integral polynomial) such that $\mathrm{m}(P)$ is a rational multiple of $L'(χ_{-f},-1)$. The strong form of the conjecture was previously proven for $18$ values of $f$. We double the number of numerical examples, giving $8$ new instances of the strong and $18$ new instances of the weak conjecture. Our examples arise from an explicit approach, which also captures almost all of the previously known results, and is based on work of Boyd and Rodriguez-Villegas. Moreover, we prove Chinburg's weak conjecture if we allow cyclotomic coefficients.

Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures

Abstract

Let be the odd quadratic Dirichlet character of conductor , and let denote the Mahler measure of a polynomial . In 1984, Chinburg conjectured that for any such there exist an integral bivariate rational function (and, in the strong form, an integral polynomial) such that is a rational multiple of . The strong form of the conjecture was previously proven for values of . We double the number of numerical examples, giving new instances of the strong and new instances of the weak conjecture. Our examples arise from an explicit approach, which also captures almost all of the previously known results, and is based on work of Boyd and Rodriguez-Villegas. Moreover, we prove Chinburg's weak conjecture if we allow cyclotomic coefficients.
Paper Structure (5 sections, 1 theorem, 34 equations, 3 tables)

This paper contains 5 sections, 1 theorem, 34 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Permissible polynomials
  3. Algorithm and discussion of tables of new and old results
  4. Chinburg's weak conjecture for rational functions with cyclotomic coefficients
  5. Tables

Key Result

Theorem 4.1

Let $\chi=\chi_{-f}$ be an odd quadratic character of conductor $f$, and put Then

Theorems & Definitions (21)

  • Conjecture 1: Chinburg's conjectures
  • Remark 2
  • Definition 3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Conjecture 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 11 more