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

TL;DR

The paper develops a CM-elliptic-curve based framework to relate the Mahler measures of two bivariate polynomial families, $P_t$ and $Q_t$, to special values of $L$-functions of cusp forms when their zero loci define CM curves. It provides explicit formulas expressing $ ext{m}(P_t)$ and $ ext{m}(Q_t)$ in terms of $L$-values of cusp forms at CM points and proves determinant identities for totally real base fields of degrees $2$ and $4$, linking determinants of Mahler measures across Galois conjugates to $L^{(n)}$-values of the corresponding CM elliptic curves. An algorithm classifies CM elliptic curves in these families for degrees up to $4$, and the results are supported by detailed computations, PSLQ-based identifications, and extensive data from LMFDB. The significance lies in providing concrete, verifiable links between Mahler measures and special $L$-values in a CM setting, complemented by explicit examples and a framework that may inform Beilinson-type regulator viewpoints and potential generalizations. The work also discusses limitations when the base field is not totally real and outlines conjectures for non-CM cases, indicating directions for future exploration of Mahler measures and $L$-values beyond CM.

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. $L^{(n)}(\mathcal{C}_t,0)$).

Figures (2)

• Figure 1: The closed region $\mathcal{K}_Q$ ($\omega=e^{\frac{2\pi i}{3}}$)
• Figure 2: Fundamental domains of $\Gamma_0(4)$ and $\Gamma_0(3)$

Theorems & Definitions (35)

• Theorem 2.1: BZ20Deninger97
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 3.1: RV99
• Remark 3.3
• Proposition 3.4