Mahler measures as linear combinations of $L$-values of multiple modular forms

Abstract

We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$-values of at most one newform and/or at most one quadratic character. In this paper, we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$-values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$-values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of $L$-functions.

24 pages

References in corpus (2)

• Regulator of modular units and Mahler measures
• Three-variable Mahler measures and special values of modular and Dirichlet $L$-series

Cited by in corpus (3)

• Mahler measure of a non-reciprocal family of elliptic curves
• The elliptic trilogarithm and Mahler measures of $K3$ surfaces
• The Mahler measure of a Calabi-Yau threefold and special L-values