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CM points, class numbers, and the Mahler measures of $x^3+y^3+1-kxy$

Zhengyu Tao, Xuejun Guo

Abstract

We study the Mahler measures of the polynomial family $Q_k(x,y) = x^3+y^3+1-kxy$ using the method previously developed by the authors. An algorithm is implemented to search for CM points with class numbers $\leqslant 3$, we employ these points to derive interesting formulas that link the Mahler measures of $Q_k(x,y)$ to $L$-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure $\tilde{n}(k)$ introduced by Samart recently. For $k=\sqrt[3]{729\pm405\sqrt{3}}$, we also prove an equality that expresses a $2\times 2$ determinant with entries the Mahler measures of $Q_k(x,y)$ as some multiple of the $L$-value of two isogenous elliptic curves over $\mathbb{Q}(\sqrt{3})$.

CM points, class numbers, and the Mahler measures of $x^3+y^3+1-kxy$

Abstract

We study the Mahler measures of the polynomial family using the method previously developed by the authors. An algorithm is implemented to search for CM points with class numbers , we employ these points to derive interesting formulas that link the Mahler measures of to -values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure introduced by Samart recently. For , we also prove an equality that expresses a determinant with entries the Mahler measures of as some multiple of the -value of two isogenous elliptic curves over .
Paper Structure (7 sections, 11 theorems, 123 equations, 2 figures, 2 tables)

This paper contains 7 sections, 11 theorems, 123 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. CM points and the algorithm
  3. The relations with cusp forms
  4. Proofs of the cases when h(tau)=1
  5. Proofs of the cases when h(tau)=2,3
  6. Linking to the L-values of elliptic curves
  7. Some remarks

Key Result

Theorem 1.1

Let $\mathcal{F}'\subset\mathcal{H}$ be the fundamental domain of the congruence subgroup $\Gamma_0(3)$ formed by the geodesic triangle of vertices $i\infty,0,(1+i/\sqrt{3})/2$ and its reflection along the imaginary axis. For any $\tau\in \mathcal{F}'$, if $\sqrt[3]{t(\tau)}\in \mathbb{C}-\mathcal{K where $\chi_{-3}(\cdot)=\left(\frac{-3}{\cdot}\right)$ and $\underset{m,n\in\mathbb{Z}}{\sum'}$ mea

Figures (2)

  • Figure 1: $\mathcal{F}'$ and its covering \ref{['coverofFprime']}.
  • Figure 2: The paths of $y^\pm_1(e^{i\theta}),y^\pm_2(e^{i\theta})$ and $y^\pm_3(e^{i\theta})$ on $\mathbb{C}$.

Theorems & Definitions (19)

  • Theorem 1.1: Rodriguez Villegas RV98
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1: Cox
  • Theorem 2.2
  • Proposition 3.1: CS17
  • Proposition 3.2: Sch74
  • proof : Proof of \ref{['iden4']}
  • ...and 9 more