The Mahler measure of a genus 3 family
type: Item
instance of: scholarly article
title: The Mahler measure of a genus 3 family
zbMATH Open document ID: 1468.11222
author: Matilde Lalín
author: Gang Wu
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: In the paper under review, the authors show that the two polynomials \[ P_{k}(x,y):=y^{2}+(x^{6}+kx^{5}-x^{4}+2(1-k)x^{3}-x^{2}+kx+1)y+x^{6} \] and \[ Q_{k}(x,y):=xy^{2}+(kx-1)y-x^{2}+x, \] where \(k\in \mathbb{N}\) and \(k\geq 2\), have the same Mahler measure. The method of proof of this conjectural identity of [\textit{H. Liu} and \textit{H. Qin}, ``Data for Mahler measure of polynomials defining genus 2 and 3 curves'', \url{https://github. com/liuhangsnnu/mahler-measure-of-genus-2-and-3-curves}] is similar to that employed in [\textit{M. Lalín} and \textit{G. Wu}, Int. J. Number Theory 15, No. 5, 945--967 (2019; Zbl 1448.11191)], establishing identities between the regulators, but new strategies are used to simplify the regulator of the genus \(3\) curve \(P_{k}(x,y)=0\), when \(k\geq 3\), before comparing it to the regulator of the curve \(Q_{k}(x,y)=0\) of genus \(1\).
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 7344936
zbMATH Keywords: Mahler measure
zbMATH Keywords: special values of \(L\)-functions
zbMATH Keywords: genus 3 curves
zbMATH Keywords: elliptic curves
zbMATH Keywords: elliptic regulator
reviewed by: Toufik Zaïmi
MaRDI profile type: Publication
full work available at URL: s11139-019-00241-1
OpenAlex ID: W3034795274
cites work: Higher regulators and values of L-functions of curves
cites work: Higher Regulators, Algebraic 𝐾-Theory, and Zeta Functions of Elliptic Curves
cites work: Speculations Concerning the Range of Mahler's Measure
cites work: Mahler's Measure and Special Values of L-functions
cites work: On the Mahler measure of a family of genus 2 curves
cites work: On the Mahler measure of hyperelliptic families
cites work: Deligne periods of mixed motives, $K$-theory and the entropy of certain $\mathbb {Z}^n$-actions
cites work: Factorization of certain cyclotomic functions
cites work: Regulator proofs for Boyd’s identities on genus 2 curves
cites work: Elliptic dilogarithms and parallel lines
cites work: Q4252760
cites work: Q4431819
cites work: FromL-series of elliptic curves to Mahler measures
cites work: On Measures of Polynomials in Several Variables: Corrigendum
cites work: Q4952461
cites work: Regulator of modular units and Mahler measures
DOI: 10.1007/S11139-019-00241-1
describes a project that uses: GitHub
Recommended article: Mahler Measure of Families of Polynomials Defining Genus 2 and 3 Curves
Recommended article: On the Mahler measure of a family of genus 2 curves
Recommended article: An explicit formule for the Mahler measure of a family of 3-variable polynomials
Recommended article: The Mahler measure for \(K3\) hypersurfaces
Recommended article: On the Mahler measure of hyperelliptic families
Recommended article: Q3438252
Recommended article: The Mahler measure of a three-variable family and an application to the Boyd-Lawton formula
Recommended article: The parametrized family of metric Mahler measures
Recommended article: Mahler measure and elliptic curve \(L\)-functions at \(s = 3\)
Recommended article: Three-variable Mahler measures and special values of \(L\)-functions of modular forms
description: scientific article; zbMATH DE number 7344936