Determinants of Mahler measures and special values of $L$-functions

TL;DR

This paper explores Mahler measures of specific bivariate polynomials linked to CM elliptic curves, deriving formulas in terms of $L$-values, classifying special parameters, and relating determinants of Mahler measures to derivatives of $L$-functions.

Contribution

It provides explicit formulas for Mahler measures of two polynomial families in terms of $L$-values of cusp forms and classifies CM elliptic curves over number fields of degree up to 4.

Findings

Mahler measures expressed via $L$-values of cusp forms.

Classification of parameters $t$ for CM elliptic curves over degree ≤ 4 fields.

Determinants of Mahler measure matrices relate to derivatives of $L$-functions.

Abstract

We consider Mahler measures of two well-studied families of bivariate polynomials, namely $P_t=x+x^{-1}+y+y^{-1}+\sqrt{t}$ and $Q_t=x^3+y^3+1-\sqrt[3]{t}xy$, where $t$ is a complex parameter. In the cases when the zero loci of these polynomials define CM elliptic curves over number fields, we derive general formulas for their Mahler measures in terms of $L$-values of cusp forms. For each family, we also classify all possible values of $t$ in number fields of degree not exceeding $4$ for which the corresponding elliptic curves have complex multiplication. Finally, for all such values of $t$ in totally real number fields of degree $n=2$ and $n=4$, corresponding to elliptic curves $\mathcal{F}_t$ (resp. $\mathcal{C}_t$), we prove that determinants of $n\times n$ matrices whose entries are Mahler measures corresponding to their Galois conjugates are non-zero rational multiples of $L^{(n)}(\mathcal{F}_t,0)$ (resp. $L^{(n)}(\mathcal{C}_t,0)$).