Mahler's Measure and the Dilogarithm (II)

Abstract

We continue to investigate the relation between the Mahler measure of certain two variable polynomials, the values of the Bloch--Wigner dilogarithm $D(z)$ and the values ${\zeta }_F(2)$ of zeta functions of number fields. Specifically, we define a class $\A$ of polynomials $A$ with the property that $\pi m(A)$ is a linear combination of values $D$ at algebraic arguments. For many polynomials in this class the corresponding argument of $D$ is in the Bloch group, which leads to formulas expressing $\pi m(A)$ as a linear combination with unspecified rational coefficients of $V_F$ for certain number fields $F$ ($V_F:=c_F{\zeta }_F(2)$ with $c_F>0$ an explicit simple constant). The class $\A$ contains the $A$-polynomials of cusped hyperbolic manifolds. The connection with hyperbolic geometry often provides means to prove identities of the form $\pi m(A)=r{V}_F$ with an explicit value of $r\in \Q^*$. We give one such example in detail in the body of the paper and in the appendix.

37 pages. Main text by Boyd and Rodriguez-Villegas; appendix by Dunfield. V2: Improved exposition

Cited by in corpus (13)

• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• On the Mahler measure of $1+X+1/X+Y+1/Y$
• An algebraic integration for Mahler measure
• $p$-adic Mahler measure and $\mathbb{Z}$-covers of links
• Mahler Measure and the Vol-Det Conjecture
• Orderability of Homology Spheres Obtained by Dehn Filling
• Rationality of the SL(2,C)-Reidemeister torsion in dimension 3
• Mahler Measure for a Quiver Symphony
• Quantum holonomic link invariants derived from stated skein algebras
• Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition
• Mahler measure of $P_d$ polynomials
• An Introduction to A-polynomials and their Mahler Measures