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Special Values of Mahler's measure
by
David W. Boyd
University of British Columbia
The logarithmic Mahler measure m(P) of a polynomial P(x, y) is the average of log|P(x, y)| over the unit 2-torus (the product of two unit circles). In the early 1980's, Smyth showed that m(1 + x + y) has the interesting value L'(-1, \chi-3), where L(s, \chi-3) is the Dirichlet L-function for the odd character of conductor 3. Since then, special values of m(P) have been shown to be connected with various interesting objects, such as elliptic curves. We will describe some recent work that studies m(P) when P=AM is the ``A-polynomial'' of a hyperbolic 3-manifold M. We show that \pi m(AM) can often be computed as a sum of dilogarithms of algebraic integers, and is related to a generalization of the Borel regulator of the manifold M. In some simple cases, \pi m(AM) is simply the volume of M but more generally, it is a sum of terms each of which can be interpreted as a ``pseudo-volume'' of M in a certain well-defined sense.
Date received: March 9, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadx-66.