Functional equations for Mahler measures of genus-one curves

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. Consider a family of two-variable polynomials whose coefficients depend on one parameter. Then the Mahler measure is a function of that parameter. Mat Rogers has discovered several examples for which this function satisfies functional equations. They all correspond to families of elliptic curves. We may deduce these functional equations from modularity properties or evaluations of elliptic regulators following works by Rodriguez-Villegas, Zagier, Deninger, etc.

(joint work with Mat Rogers)