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Mahler measures and $L$-values of elliptic curves over real quadratic fields

Zhengyu Tao, Xuejun Guo, Tao Wei

Abstract

A famous formula of Rodriguez Villegas shows that the Mahler measures $m(k)$ of $P_k(x,y)=x+1/x+y+1/y+k$ can be written as a Kronecker-Eisenstein series. We prove that the degree of $k$ in Villegas' formula can be bounded by the class numbers of CM points. This fact allows us to systematically derive $28$ new identities linking $m(k)$ to $L$-values of cusp forms. Guided by Beilinson's conjecture, we also prove $5$ formulas that express $L$-values of CM elliptic curves over real quadratic fields to some $2\times 2$ determinants of $m(k)$. This extends a recent work of Guo (the second author of this paper), Ji, Liu, and Qin, in which they dealt with the cases when $k=4\pm 4\sqrt{2}$.

Mahler measures and $L$-values of elliptic curves over real quadratic fields

Abstract

A famous formula of Rodriguez Villegas shows that the Mahler measures of can be written as a Kronecker-Eisenstein series. We prove that the degree of in Villegas' formula can be bounded by the class numbers of CM points. This fact allows us to systematically derive new identities linking to -values of cusp forms. Guided by Beilinson's conjecture, we also prove formulas that express -values of CM elliptic curves over real quadratic fields to some determinants of . This extends a recent work of Guo (the second author of this paper), Ji, Liu, and Qin, in which they dealt with the cases when .
Paper Structure (7 sections, 7 theorems, 148 equations, 2 figures, 2 tables)

This paper contains 7 sections, 7 theorems, 148 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The degree of $\lambda(2\tau)$
  3. CM points with $h(D_{\tau_0})h(D_{4\tau_0})\leqslant 4$
  4. Mahler Measures and $L$-values of modular forms
  5. Beilinson's Conjecture for curves over number fields
  6. Proofs of Theorem \ref{['mainresults1']}
  7. The other four cases

Key Result

Theorem 1.1

If $\tau$ lies in the region $\mathcal{F}'\subset\mathcal{H}$ formed by the geodesic triangle of vertices $i\infty,0,1/2$ and its reflection along the imaginary axis, then we have where $k=\frac{4}{\sqrt{\lambda(2\tau)}}$ and $\underset{m,n\in\mathbb{Z}}{{\sum}'}$ means $(0,0)$ is excluded from the summation.

Figures (2)

  • Figure 1: The region $\mathcal{F}'$ in Theorem \ref{['mahlermeasureaslambda']} and its covering \ref{['RIGHTCOSETS']}.
  • Figure 2: The path of $y^{E^\sigma}_1$ and $y^{E^\sigma}_2$ when $\theta\in[-\pi,\pi]$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Cox
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof : Proof of the cases $k=12\pm8\sqrt{2}$ in Theorem \ref{['mainresults1']}
  • Conjecture 5.1: Beilinson
  • ...and 5 more