Functional equations for Mahler measures of genus-one curves

By Matilde Lalín, Mathew Rogers. 2006

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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. MSC: 11R09, 11F66, 11G40, 19F27 1 History and introduction The goal of this paper is to establish identities between the logarithmic Mahler measures of polynomials with zero varieties corresponding to genus-one curves. Recall that the logarithmic Mahler measure (which we shall henceforth simply refer to as the Mahler measure) of an n-variable Laurent polynomial P (x1, x2, . . . , xn) is defined by m (P (x1, . . . , xn)) = ∫ 1

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

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