  In this paper we will establish functional equations for Mahler measures of
families of genus-one two-variable polynomials. These families were previously
studied by Beauville, and their Mahler measures were considered by Boyd,
Rodriguez-Villegas, Bertin, Zagier, and Stienstra. Our functional equations
allow us to prove identities between Mahler measures that were conjectured by
Boyd. As a corollary, we also establish some new transformations for
hypergeometric functions.
