Functional equations for Mahler measures of genus-one curves

Lalin, Matilde; Rogers, Mathew

doi:10.2140/ant.2007.1.87

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Functional equations for Mahler measures of genus-one curves

Matilde N. Lalin and Mathew D. Rogers

Vol. 1 (2007), No. 1, 87–117

DOI: 10.2140/ant.2007.1.87

Abstract

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.

Keywords

Mahler measure, L-functions, Bloch–Beilinson conjectures, Kronecker–Eisenstein series, elliptic regulator, hypergeometric identities, modular equations

Mathematical Subject Classification 2000

Primary: 11R09

Secondary: 11F66, 19F27, 33C05, 33C20

Milestones

Received: 9 February 2007

Accepted: 7 July 2007

Published: 1 February 2007

Authors

Matilde N. Lalin
Department of Mathematical and Statistical Sciences 632 Central Academic Building University of Alberta Edmonton, AB T6G 2G1 Canada
http://www.math.ubc.ca/~mlalin
Mathew D. Rogers
Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2 Canada
http://www.math.ubc.ca/~matrogers

Vol. 1, No. 1, 2007