Mahler measure of polynomials defining genus 2 and 3 curves

TL;DR

This paper investigates Mahler measures of over 500 polynomial families defining genus 2 and 3 curves, revealing relations with special values of L-functions and discovering numerous identities through numerical analysis.

Contribution

It provides the first extensive numerical study linking Mahler measures of genus 2 and 3 curves to L-values and K_2 elements, with new identities and theoretical insights.

Findings

Discovered over 100 identities between Mahler measures of different polynomial families.

Established relations between Mahler measures and special L-values of elliptic and higher genus curves.

Identified explicit K_2 elements in genus 2 and 3 curves supporting Beilinson's conjecture.

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.