On the Mahler measure of hyperelliptic families

TL;DR

This paper proves a surprising coincidence in Mahler measures for certain genus 2 curves and relates these measures to elliptic curves, revealing new connections in algebraic geometry.

Contribution

It establishes the equivalence of Mahler measures for specific genus 2 families and elliptic curves, advancing understanding of Mahler measure relationships.

Findings

Proves Boyd's coincidence for genus 2 families

Relates Mahler measures to elliptic curve polynomials

Enhances knowledge of Mahler measure interrelations

Abstract

We prove Boyd's "unexpected coincidence" of the Mahler measures for two families of two-variate polynomials defining curves of genus 2. We further equate the same measures to the Mahler measures of polynomials $y^3-y+x^3-x+kxy$ whose zero loci define elliptic curves for $k\ne0,\pm3$.