The Mahler measure of a Calabi-Yau threefold and special L-values

TL;DR

This paper establishes a formula linking the Mahler measure of a specific Calabi-Yau threefold to special L-values and zeta values, revealing deep connections between algebraic geometry and number theory.

Contribution

It provides a new explicit formula for the Mahler measure of a Laurent polynomial defining a Calabi-Yau threefold, connecting it to special L-values and hypergeometric series.

Findings

Mahler measure expressed as a rational linear combination of L-value and zeta value

Derived a new formula for a 6F5-hypergeometric series at 1

Established a link between Calabi-Yau geometry and special values of L-functions

Abstract

The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi-Yau threefold. We show that its Mahler measure is a rational linear combination of a special L-value of the normalized newform in S_4(Gamma_0(8)) and a Riemann zeta value. This is equivalent to a new formula for a 6F5-hypergeometric series evaluated at 1.