Mahler measure and volumes in hyperbolic space

The Mahler measure of the polynomials t(x^m-1) y - (x^n-1) \in \dC[x,y] is essentially the sum of volumes of a certain collection of ideal hyperbolic polyhedra in \HH^3, which can be determined a priori as a function on the parameter $t$. We obtain a formula that generalizes some previous formulas given by Cassaigne and Maillot \cite{M} and Vandervelde \cite{V}. These examples seem to be related to the ones studied by Boyd \cite{B1}, \cite{B2} and Boyd and Rodriguez Villegas \cite{BRV2} for some cases of the $A$-polynomial of one-cusped manifolds.Comment: 25 pages, 11 figure

On certain combination of colored multizeta values

We reduce the sum $\sum_{k=1}^\infty \sum_{j=0}^{k-1} \frac{(-1)^{j+k+1}}{(2j+1)^mk}$ in terms of special values of the Dirichlet L-series in the character of conductor 4. This sum is a combination of colored zeta values

An algebraic integration for Mahler measure

There are many examples of several-variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger, and later Rodriguez-Villegas, and Maillot. While Rodriguez-Villegas made this relationship explicit for the two variable case, it is our goal to understand the three variable case and shed some light on the examples with more variables.Comment: 3 figure

Higher Mahler measures and zeta functions

We consider a generalization of the Mahler measure of a multivariable polynomial $P$ as the integral of $\log^k|P|$ in the unit torus, as opposed to the classical definition with the integral of $\log|P|$. A zeta Mahler measure, involving the integral of $|P|^s$, is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet $L$-functions, and polylogarithms

Mahler measure under variations of the base group

We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of Mahler measures over finite groups, for example, in the classical case of the free abelian group or the infinite dihedral group, and others.Comment: 21 page

Biased statistics for traces of cyclic p-fold covers over finite fields

In this paper, we discuss in more detail some of the results on the statistics of the trace of the Frobenius endomorphism associated to cyclic p-fold covers of the projective line that were presented in [1]. We also show new findings regarding statistics associated to such curves where we fix the number of zeros in some of the factors of the equation in the affine model