The Mahler measure of x + 1/x + y + 1/y + 4 ± 42 and Beilinson’s conjecture

Abstract

In this note we study the Mahler measures of reciprocal polynomials [Formula: see text] for [Formula: see text]. We prove identities relating the Mahler measure of [Formula: see text] to special values of the [Formula: see text]-functions of weight 2 cusp forms of level 64. We also express the Beilinson regulator which is a [Formula: see text] determinant of Mahler measures as special value of [Formula: see text]-function of the elliptic curve defined by [Formula: see text].