The Mahler measure of exact polynomials in three variables

2023/10/10 by Trieu, Thu Ha
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2310.06563

Abstract

We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve L-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In some cases, these dilogarithmic values simplify to Dirichlet L-values. The proof involves a construction of an element in K₄⁽³⁾ of a smooth projective curve over a number field. This generalizes a result of Lalín for the polynomial z + (x+1)(y+1). We apply our method to several other Mahler measure identities conjectured by Boyd and Brunault.

Related

• The Mahler measure of exact polynomials and special L-values of K3 surfaces
• Mahler measures, K-theory and values of L-functions
• Mahler's Measure and the Dilogarithm (II)
• The Mahler measure of a family of polynomials with arbitrarily many variables
• Mahler's measure and elliptic curves with potential complex multiplication