Mahler measure of some singular K3-surfaces

TL;DR

This paper investigates the Mahler measure of a family of Laurent polynomials defining K3-surfaces, establishing new formulas and extending prior research on their connection to L-functions.

Contribution

It introduces new formulas for Mahler measures of singular K3-surfaces, expanding the understanding of their arithmetic properties and relations to L-functions.

Findings

Derived new explicit formulas for Mahler measures.

Extended previous results on K3-surfaces with Picard number 20.

Connected Mahler measures to L-functions of K3-surfaces.

Abstract

We study the Mahler measure of the three-variable Laurent polynomial x + 1/x + y + 1/y + z + 1/z - k where k is a parameter. The zeros of this polynomial define (after desingularization) a family of K3-surfaces. In favorable cases, the K3-surface has Picard number 20, and the Mahler measure is related to its L-function. This was first studied by Marie-Jose Bertin. In this work, we prove several new formulas, extending the earlier work of Bertin.