The Mahler measure of exact polynomials in three variables

TL;DR

This paper establishes a connection between the Mahler measure of certain three-variable polynomials and special values of elliptic curve L-functions and dilogarithms, under specific conditions and conjectures.

Contribution

It generalizes previous results by relating Mahler measures to elliptic L-values and dilogarithms via K-theory constructions, extending to multiple new polynomial cases.

Findings

Mahler measure expressed in terms of elliptic L-values and dilogarithms

Simplification of dilogarithmic values to Dirichlet L-values in some cases

Application to conjectured Mahler measure identities by Boyd and Brunault

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 $K_4^{(3)}$ of a smooth projective curve over a number field. This generalizes a result of Lal\'in for the polynomial $z + (x+1)(y+1)$. We apply our method to several other Mahler measure identities conjectured by Boyd and Brunault.