Mahler measures of elliptic modular surfaces

TL;DR

This paper introduces a novel method connecting Mahler measures of three-variable polynomials defining elliptic modular surfaces to special L-values of modular forms, expanding the understanding of these mathematical objects.

Contribution

It develops a new approach using Deligne periods and an extension of the Rogers-Zudilin method to relate Mahler measures to L-values, advancing the study of elliptic modular surfaces.

Findings

Established a link between Mahler measures and L-values of modular forms.

Expressed Mahler measures as Deligne periods of elliptic modular surfaces.

Extended the Rogers-Zudilin method to Kuga-Sato varieties.

Abstract

In this article we develop a new method for relating Mahler measures of three-variable polynomials that define elliptic modular surfaces to L-values of modular forms. Using an idea of Deninger, we express the Mahler measure as a Deligne period of the surface and then apply the first author's extension of the Rogers-Zudilin method to Kuga-Sato varieties to arrive at an L-value.