Mahler Measure of Families of Polynomials Defining Genus 2 and 3 Curves

Abstract

In this article, we study the Mahler measures of more than 500 families of reciprocal polynomials defining genus 2 and genus 3 curves. We numerically find relations between the Mahler measures of these polynomials with special values of L-functions. We also numerically discover more than 100 identities between Mahler measures involving different families of polynomials defining genus 2 and genus 3 curves. Furthermore, we study the Mahler measures of several families of nonreciprocal polynomials defining genus 2 curves and numerically find relations between the Mahler measures of these families and special values of L-functions of elliptic curves. We also find identities between the Mahler measures of these nonreciprocal families and tempered polynomials defining genus 1 curves. We will explain these relations by considering the pushforward and pullback of certain elements in K 2 of curves defined by these polynomials and applying Beilinson’s conjecture on K 2 of curves. We show that there are two and three explicit linearly independent elements in K 2 of certain families of genus 2 and genus 3 curves, respectively.