Mahler's Measure and the Dilogarithm (II)

TL;DR

This paper explores the relationship between Mahler measures of certain polynomials, dilogarithm values, and zeta functions of number fields, revealing new formulas and connections with hyperbolic geometry.

Contribution

It introduces a class of polynomials with Mahler measures linked to dilogarithm values and zeta functions, including polynomials from hyperbolic geometry, and provides explicit formulas and examples.

Findings

Mahler measures relate to dilogarithm values at algebraic points.

Explicit formulas connect Mahler measures to zeta function values of number fields.

Hyperbolic geometry aids in proving identities involving Mahler measures.

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.