Title: Mahler measures and regulators

Abstract:

The Mahler measure of an n-variable polynomial P is the integral of log|P| over the n-dimensional unit torus T^n with the Haar measure. For one-variable polynomials, this is a natural quantity that appears in different problems such as Lehmer's question. While the algebraic nature of the values of the Mahler measure for one-variable polynomials with integral coefficients is well understood, the knowledge of the several-variable case is reduced to a collection of examples, some of which are related to special values of L-functions and/or polylogarithms. This phenomenon may be explained in terms of evaluations of regulators. In turn, regulators also allow us to prove new formulas for Mahler measures involving genus-one curves.