Determinants of Mahler measures and special values of L-functions

2024/09/23 by Detchat Samart, Zhengyu Tao, Samart, Detchat +1 · 1 citation
Mathematics · #Analytic and geometric function theory #Mathematical Approximation and Integration #Meromorphic and Entire Functions #math.AG #math.KT #math.NT

paper · pdf · doi:10.48550/arxiv.2409.14714

openalex publication_date 2024/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider Mahler measures of two well-studied families of bivariate polynomials, namely P_t=x+x^-1+y+y^-1+√(t) and Q_t=x³+y³+1-√[3]txy, 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 F_t (resp. C_t), we prove that determinants of n× n matrices whose entries are Mahler measures corresponding to their Galois conjugates are non-zero rational multiples of L⁽ⁿ⁾(F_t,0) (resp. L⁽ⁿ⁾(C_t,0)).

Cited by

• Samart's conjecture n₄(81)=40M₇: the exact CM evaluation and the two obstructions. A status report

Related

• Mahler measures and L-values of elliptic curves over real quadratic fields
• The Mahler measure of exact polynomials in three variables
• CM points, class numbers, and the Mahler measures of x³+y³+1-kxy
• Mahler measures, K-theory and values of L-functions
• The Mahler measure of exact polynomials and special L-values of K3 surfaces