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The Mahler measure of exact polynomials in three variables

Thu Ha Trieu

TL;DR

The paper shows that, under explicit Beilinson conjecture inputs for genus 1 curves, the Mahler measure of a three-variable polynomial P can be expressed in terms of elliptic-curve L-values $L'(E,-1)$ and Bloch-Wigner dilogarithm values, with Dirichlet L-values arising in special cases. The method builds a Deligne cohomology class from the Maillot genus-1 curve via Goncharov and De Jeu polylogarithmic complexes and their regulator maps, then constructs a motivic cohomology class whose Beilinson regulator matches the Deligne class; Beilinson’s conjecture then yields the stated regulator–L-value formulas. The framework is used to derive numerous conjectural Mahler-measure identities (pure and mixed) attributed to Boyd and Brunault, and includes unconditional Dirichlet-based results via Lalín’s approach. Overall, the work links Mahler measures to explicit regulator computations on genus-1 curves, advancing a regulator-based, Beilinson-guided understanding of three-variable Mahler identities and their arithmetic content.

Abstract

We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve $L$-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In some cases, these dilogarithmic values simplify to Dirichlet $L$-values. The proof involves a construction of an element in $K_4^{(3)}$ of a smooth projective curve over a number field. This generalizes a result of Lalín for the polynomial $z + (x+1)(y+1)$. We apply our method to several other Mahler measure identities conjectured by Boyd and Brunault.

The Mahler measure of exact polynomials in three variables

TL;DR

The paper shows that, under explicit Beilinson conjecture inputs for genus 1 curves, the Mahler measure of a three-variable polynomial P can be expressed in terms of elliptic-curve L-values and Bloch-Wigner dilogarithm values, with Dirichlet L-values arising in special cases. The method builds a Deligne cohomology class from the Maillot genus-1 curve via Goncharov and De Jeu polylogarithmic complexes and their regulator maps, then constructs a motivic cohomology class whose Beilinson regulator matches the Deligne class; Beilinson’s conjecture then yields the stated regulator–L-value formulas. The framework is used to derive numerous conjectural Mahler-measure identities (pure and mixed) attributed to Boyd and Brunault, and includes unconditional Dirichlet-based results via Lalín’s approach. Overall, the work links Mahler measures to explicit regulator computations on genus-1 curves, advancing a regulator-based, Beilinson-guided understanding of three-variable Mahler identities and their arithmetic content.

Abstract

We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve -values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In some cases, these dilogarithmic values simplify to Dirichlet -values. The proof involves a construction of an element in of a smooth projective curve over a number field. This generalizes a result of Lalín for the polynomial . We apply our method to several other Mahler measure identities conjectured by Boyd and Brunault.
Paper Structure (23 sections, 14 theorems, 273 equations, 10 figures, 3 tables)

This paper contains 23 sections, 14 theorems, 273 equations, 10 figures, 3 tables.

Table of Contents

  1. The Beilinson regulator map
  2. Deligne cohomology
  3. Chow motives
  4. Motivic cohomology
  5. The Beilinson regulator map
  6. Beilinson's conjecture for genus 1 curves
  7. Goncharov's polylogarithmic complexes
  8. Goncharov's complexes
  9. The residue homomorphism of complexes
  10. Goncharov's regulator maps
  11. De Jeu's polylogarithmic complexes
  12. De Jeu's polylogarithmic complexes
  13. De Jeu's maps
  14. Relation to Goncharov's complexes
  15. Regulator maps
  16. ...and 8 more sections

Key Result

Theorem 1

Assume that $P\in \mathbb Q[x, y, z]$ is irreducible and satisfies the following conditions: where $R_2(\bar{\mathbb Q})$ is the subgroup generated by the five-term relations (see equation (130)), and $v_p(g_j)$ is the vanishing order at $p$ of $g_j$ seen as a function on $E$. Then under Beilinson's conjecture,

Figures (10)

  • Figure 1: The Deninger chain $\Gamma$.
  • Figure 2: The Deninger chain $\Gamma$.
  • Figure 3: The Deninger chain $\Gamma$.
  • Figure 4: The Deninger chain $\Gamma$ corresponding to \ref{['391']}.
  • Figure 5: The Deninger chain $\Gamma$.
  • ...and 5 more figures

Theorems & Definitions (44)

  • Theorem 1: Lal15
  • Theorem 2
  • Definition 1.1: Bur97
  • Definition 1.2
  • Definition 1.3: Deligne Cohomology
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6: Chow motives
  • Proposition 1.7
  • proof
  • ...and 34 more