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 $ζ_F(2)$ of zeta functions of number fields. Specifically, we define a class $\A$ of polynomials $A$ with the property that $πm(A)$ 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 $πm(A)$ as a linear combination with unspecified rational coefficients of $V_F$ for certain number fields $F$ ($V_F := c_Fζ_F(2)$ 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 $πm(A)= r V_F$ 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.