Class invariants by Shimura's reciprocity law
type: Item
instance of: scholarly article
title: Class invariants by Shimura's reciprocity law
zbMATH Open document ID: 0957.11048
author: Alice Gee
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2001-04-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/226709
full work available at URL: jtnb11-1.html
review text: Let \(K\) be an imaginary quadratic number field of discriminant \(d<-4\) with ring of integers \({\mathcal O}=\mathbb{Z}[\theta]\) and minimal polynomial of \(\theta\), \(f(X)=X^2 +BX+C\). According to the theory of complex multiplication, the absolute invariant \(j({\mathcal O})=j(\theta)\) generates the Hilbert class field \(H\) of \(K\). For a modular function \(h\) of level \(N\), the value \(h(\theta)\) is called a class invariant whenever \(h(\theta)\) and \(j(\theta)\) generate the same field over \(K\). Class invariants are useful because their minimal polynomial has, in general much smaller coefficients than the minimal polynomial of \(j({\mathcal O})\). The author of the paper under review had the nice idea to use the Shimura reciprocity law in order to produce the following criterion: If \(h\) is a modular function of level \(N\) and \(\mathbb{Q}(j) \subseteq \mathbb{Q} (h)\), then \(h(\theta)\) is a class invariant if and only if the set of matrices \[ W_{N,\theta}= \left\{\left(\begin{matrix} t-Bs & -Cs\\ s & t\end{matrix} \right)\in GL_2 (\mathbb{Z}/N \mathbb{Z})\mid t,s\in \mathbb{Z}/N \mathbb{Z}\right\} \] acts trivially on \(h\). By using this result she gives, in an easy way results proven before by H. Weber, B. Birch or R. Schertz, concerning class invariants, of the Weber modular functions \(\gamma_2,\gamma_3,f,f_1,f_2\) as well as of Weber's resolvents of \(\omega_0\) and \(\omega_3\) of degree 5. After that she gives a modification of Shimura's reciprocity law which has been used to give the action of the Galois group \(\text{Gal}(H/K)\) on class invariants and proves a conjectural formula of \textit{F. Morain} [Algorithmic Number Theory, Lect. Notes Comput. Sci. 1423, 111-130 (1998; Zbl 0908.11061)] and another one of \textit{N. Yui} and \textit{D. Zagier} [Math. Comput. 66, 1645-1662 (1997; Zbl 0892.11022)] concerning conjugates of class invariants arising from some classical functions. An extensive list containing the polynomials for the cl…
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11F03
zbMATH DE Number: 1370104
zbMATH Keywords: minimal polynomial
zbMATH Keywords: Hilbert class field
zbMATH Keywords: class invariant
zbMATH Keywords: Shimura reciprocity law
zbMATH Keywords: Weber modular functions
reviewed by: Jannis A. Antoniadis
describes a project that uses: ECPP
MaRDI profile type: Publication
OpenAlex ID: W2147311270
cites work: Weber's class invariants
cites work: Q4723876
cites work: Q3840154
cites work: Die singulären Werte der Weberschen Funktionen f, f1, f2, 2, 3.
cites work: Q5625265
cites work: Q3213187
cites work: On the singular values of Weber modular functions
DOI: 10.5802/JTNB.238
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description: scientific article; zbMATH DE number 1370104