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Mahler's Measure and the Dilogarithm (II) | | David W. Boyd
; Fernando Rodriguez-Villegas
; Nathan M. Dunfield
; | | Date: |
5 Aug 2003 | | Subject: | Number Theory; Geometric Topology MSC-class: 11C08, 11G10, 11G55 | math.NT math.GT | | 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 $F$ ($V_F := c_Fzeta_F(2)$ 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 $rin Q^*$. We give one such example in detail in the body of the paper and in the appendix. | | Source: | arXiv, math.NT/0308041 | | Services: | Forum | Review | PDF | Favorites | |
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