Higher composition laws. III: The parametrization of quartic rings
type: Item
instance of: scholarly article
title: Higher composition laws. III: The parametrization of quartic rings
zbMATH Open document ID: 1169.11045
author: Manjul Bhargava
published in: Annals of Mathematics. Second Series
publication date: 2005-03-11T00:00:00Z
review text: It is well known that isomorphism classes of quadratic rings (rings that are free of rank 2 as \(\mathbb Z\)-modules) are parametrized by their discriminants, i.e., integers \(d\) congruent to 0 or 1 modulo 4. By a classical result due to \textit{F. Levi} [Leipz. Ber. 66, 26--37 (1914; JFM 45.0336.02)], the isomorphism classes of cubic rings are parametrized by \(\mathrm{GL}_2(\mathbb Z)\)-equivalence classes of integral binary cubic forms. For getting a parametrization of quartic rings, the author reformulates the cubic case as follows: there is a bijection between \(\mathrm{GL}_2(\mathbb Z)\)-equivalence classes of integral binary cubic forms and isomorphism classes of pairs \((R,S)\), where \(R\) is a cubic ring and \(S\) is the quadratic resolvent of \(R\). The main result of this article gives a bijection between isomorphism classes of pairs \((Q,R)\), where \(Q\) is a quartic ring and \(R\) a cubic resolvent of \(Q\), and \(\mathrm{GL}_3(\mathbb Z) \times \mathrm{GL}_2(\mathbb Z)\)-orbits on the space \((\mathrm{Sym}^2 \mathbb Z^3 \otimes \mathbb Z^2)^*\) of pairs of integral ternary quadratic forms. A similar result is obtained for the class of primitive quartic rings. In the final section, information on the splitting of primes in maximal orders is tied to properties of the ternary forms. These results were used for finding the density of discriminants of quartic rings and fields [Ann. Math. (2) 162, No. 2, 1031--1063 (2005; Zbl 1159.11045)].
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E20
zbMATH DE Number: 2144178
zbMATH Keywords: quartic rings
zbMATH Keywords: resolvents
zbMATH Keywords: ternary quadratic forms
zbMATH Keywords: maximal orders
zbMATH Keywords: splitting of primes
zbMATH Keywords: binary cubic forms
zbMATH Keywords: discriminants
MaRDI profile type: Publication
full work available at URL: annals.2004.159.1329
OpenAlex ID: W2055916423
Wikidata QID: Q56686834
DOI: 10.4007/ANNALS.2004.159.1329
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description: scientific article; zbMATH DE number 2144178