On the Mahler measure of a family of genus 2 curves

TL;DR

This paper derives a general identity linking the Mahler measures of two polynomial families, one defining genus 2 hyperelliptic curves and the other elliptic curves, revealing deep connections between their geometric properties.

Contribution

It introduces a novel identity connecting Mahler measures of hyperelliptic and elliptic curves, expanding understanding of their interrelations.

Findings

Established a general identity between Mahler measures of hyperelliptic and elliptic curves.

Revealed deep connections between genus 2 and genus 1 curve Mahler measures.

Enhanced the theoretical framework relating polynomial families and their geometric properties.

Abstract

We establish a general identity between the Mahler measures $m(Q_k(x,y))$ and $m(P_k(x,y))$ of two polynomial families, where $Q_k(x,y)=0$ and $P_k(x,y)=0$ are generically hyperelliptic and elliptic curves, respectively.