Arithmetic of the modular function \(j_{1,8}\)
type: Item
instance of: scholarly article
title: Arithmetic of the modular function \(j_{1,8}\)
zbMATH Open document ID: 0976.11024
DOI: 10.1023/A:1009857205327
author: Chang Heon Kim
author: Ja Kyung Koo
published in: The Ramanujan Journal
publication date: 2001-07-12T00:00:00Z
review text: The modular curve \(X_1(N)= \Gamma_1(N) \setminus \mathbb{H}^*\) is of genus zero, if and only if \(1\leq N\leq 10\) and \(N=12\) and so, for these values of \(N\) the corresponding function field \(\mathbb{C}(X_1(N))\) is a rational function field. The authors consider the case for \(N=8\), find a generator, \(j_{1,8}\), of this field by means of Jacobi theta series, show that the normalized generator \(N(j_{1,8})\), is an algebraic integer when evaluated at a point of \(\mathbb{H}\cap \mathbb{Q} (\sqrt{-d})\), apply this function to construct some class fields over an imaginary quadratic field and calculate the minimal polynomial of \(N(j_{1,8} (\alpha))\). The article is a continuation of their previous work [Acta Arith. 84, 129-143 (1998; Zbl 0907.11014)].
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11F22
zbMATH DE Number: 1563056
zbMATH Keywords: modular curve
zbMATH Keywords: rational function field
zbMATH Keywords: Jacobi theta series
zbMATH Keywords: normalized generator
zbMATH Keywords: class fields
zbMATH Keywords: minimal polynomial
reviewed by: Jannis A. Antoniadis
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1563056