Vol. 1, No. 1, 2007

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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

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.

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
Received: 9 February 2007
Accepted: 7 July 2007
Published: 1 February 2007
Matilde N. Lalin
Department of Mathematical and Statistical Sciences
632 Central Academic Building
University of Alberta
Edmonton, AB T6G 2G1
Mathew D. Rogers
Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2