The finiteness of derivation-invariant prime ideals and the algebraic independence of the Eisenstein series
type: Item
instance of: scholarly article
title: The finiteness of derivation-invariant prime ideals and the algebraic independence of the Eisenstein series
zbMATH Open document ID: 1487.11072
author: Bo-Hae Im
author: Wonwoong Lee
published in: The Ramanujan Journal
publication date: 2021-12-03T00:00:00Z
review text: For \(m\ge 3\), the Hecke triangle group \(H(m)\) is the subgroup of the full modular group \({\mathrm{SL}}_2({\mathbb{Z}})\) generated by the two elements \(\begin{pmatrix}1&1\\ 0&1\end{pmatrix}\) and \(\begin{pmatrix}1&-1/(2\cos(\frac \pi m))\\ 2\cos(\frac \pi m)&0\end{pmatrix}\). The arithmetic triangle groups are \(H(3)={\mathrm{SL}}_2({\mathbb{Z}})\), \(H(4)=\Gamma_0^+(2)\) and \(H(6)=\Gamma_0^+(3)\). For \(k\ge 1\), the Eisenstein series \(E_{2k,m}\) for \(H(m)\) have been defined in [\textit{S. K. Ashok} et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 16, Paper 001, 26 p. (2020; Zbl 1444.34108)] and [\textit{C. F. Doran} et al., Commun. Number Theory Phys. 7, No. 4, 689--737 (2013; Zbl 1295.11050)]. The series \(E_{4,m}(\tau)\) and \(E_{6,m}(\tau)\) are modular forms for \(H(m)\), while \(E_{2,m}(\tau)\) is a quasi modular form for \(H(m)\). Let \(q= {\mathrm e}^{2\pi{\mathrm i}\tau}\) be a local parameter near \(\infty\) and \(\hat E_{2,m}(q)\), \(\hat E_{4,m}(q)\), \(\hat E_{6,m}(q)\) be the Fourier series for \(E_{2,m}(\tau)\), \(E_{4,m}(\tau)\) and \(E_{6,m}(\tau)\). The Ramanujan differential equations with respect to the derivation \(q{\mathrm{d}}/{\mathrm{d}}q\) for \(H(4)\) and \(H(6)\) are given in [Ramanujan J. 7, No. 4, 435--447 (2003; Zbl 1072.11052)], where \textit{W. Zudilin} proved the algebraic independence over \({\mathbb{C}}(q)\) of \(\hat E_{2,m}\), \(\hat E_{4,m}\) and \(\hat E_{6,m}\) for \(m=4\) and \(m=6\). \par The main result of the paper under review is that for \(m=3,4,6\) and for any point \(\alpha\) in the upper half plane \({\mathbb{H}}\), at least three among the four numbers \({\mathrm e}^{2\pi{\mathrm i}\alpha}\), \(E_{2,m}(\alpha)\), \(E_{4,m}(\alpha)\) and \(E_{6,m}(\alpha)\) are algebraically independent. For \(m=3\) this is a result of [\textit{Yu. V. Nesterenko}, Sb. Math. 187, No. 9, 1319--1348 (1996; Zbl 0898.11031); translation from Mat. Sb. 187, No. 9, 65--96 (1996)]. \par There is a further infinite Hecke g…
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J89
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 13N99
zbMATH DE Number: 7438405
zbMATH Keywords: differential ideal
zbMATH Keywords: transcendental
zbMATH Keywords: the Hecke group
zbMATH Keywords: modular form
zbMATH Keywords: quasi-modular form
zbMATH Keywords: Eisenstein series
MaRDI profile type: Publication
full work available at URL: s11139-021-00404-z
OpenAlex ID: W3133765709
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DOI: 10.1007/S11139-021-00404-Z
description: scientific article; zbMATH DE number 7438405