Exercising in complex Mahler measures: diamonds are not forever

TL;DR

This paper proves a relation between Mahler measures of hyperelliptic and elliptic families using existing diamond-free methods, highlighting the complex analysis involved in confirming such identities.

Contribution

It demonstrates how to prove Mahler measure relations with established techniques, extending the understanding of these mathematical objects.

Findings

Confirmed a specific Mahler measure relation

Showed the effectiveness of diamond-free methodology

Illustrated the complexity of proving Mahler measure identities

Abstract

Recently, Hang Liu and Hourong Qin came up with a numerical observation about the relation between the Mahler measures of one hyperelliptic and two elliptic families. The discoverers foresee a proof of the identities "by extending ideas in" two papers of Matilde Lal\'{\i}n and Gang Wu, the ideas based on a theorem of Spencer Bloch and explicit diamond-operation calculations on the underlying curves. We prove the relation using the already available diamond-free methodology. While finding such relations for the Mahler measures remains an art, proving them afterwards is mere complex (analysis) exercising.