Generalized lambda functions and modular function fields of principal congruence subgroups
type: Item
instance of: scholarly article
title: Generalized lambda functions and modular function fields of principal congruence subgroups
zbMATH Open document ID: 1307.11047
author: Noburo Ishii
published in: Tokyo Journal of Mathematics
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1406552442
review text: Given an element \(\tau\) of the complex upper half plane, let \(L_\tau\) be the lattice generated by 1 and \(\tau\), and let \(\wp (z; L_\tau)\) be the Weierstrass \(\wp\)-function associated to \(L_\tau\). If \(e_1\) \(e_2\) and \(e_3\) are 2-division points of the quotient group \(\mathbb C/ L_\tau\), then the elliptic modular lambda function \(\lambda\) is given by \[ \lambda (\tau) = \frac {\wp (e_1; L_\tau) - \wp (e_3; L_\tau)} {\wp (e_2; L_\tau) - \wp (e_3; L_\tau)} . \] In this paper the author proves that the lambda function \(\lambda\) and the modular invariant function \(j\) generate the modular function field with respect to the principal congruence subgroup \(\Gamma (N)\) of \(\mathrm{SL}(2, \mathbb Z)\) with \(N>1\). He also studies its values at at imaginary quadratic points.
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 6350033
zbMATH Keywords: modular function fields
zbMATH Keywords: lambda functions
zbMATH Keywords: Weierstrass \(\wp\)-function
MaRDI profile type: Publication
OpenAlex ID: W2087060671
cites work: Q3996190
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DOI: 10.3836/TJM/1406552442
description: scientific article; zbMATH DE number 6350033