Higher composition laws. II: On cubic analogues of Gauss composition
type: Item
instance of: scholarly article
title: Higher composition laws. II: On cubic analogues of Gauss composition
zbMATH Open document ID: 1169.11044
author: Manjul Bhargava
published in: Annals of Mathematics. Second Series
publication date: 2004-10-21T00:00:00Z
review text: In the first article [Ann. Math. (2) 159, No. 1, 217--250 (2004; Zbl 1072.11078)] of this series, the author presented a new interpretation of Gauss composition on equivalence classes of binary quadratic forms, which led to several new composition laws on other spaces of forms. Replacing the \(2 \times 2 \times 2\)-cubes in the theory of binary quadratic forms by \(3 \times 3 \times 3\)-cubes, one ends up with a group law on ternary cubic forms in the following way: the cube \(C\) can be sliced in three different ways into triples \((L_i,M_i,N_i)\) of \(3 \times 3\)-matrices, and setting \(f_i(x,y,z) = - \det(L_i x + M_i y + N_i z)\) defines three ternary cubic forms attached to \(C\). Demanding that \([f_1] + [f_2] + [f_3] = [f]\) for some chosen ``principal form'' \(f\) and suitably defined equivalence classes with respect to the action of \(\text{GL}_3(\mathbb Z)^3\) gives a group law on classes of ternary cubic forms, which is, however, not presented in detail since the author is mainly interested in decomposable forms, i.e. norm forms attached to some cubic ring. To this end, he considers \(2 \times 3 \times 3\)-cubes, which are described as pairs \((A,B)\) of \(3 \times 3\)-matrices, or, equivalently, as elements of the tensor product \(\mathbb Z^2 \otimes \mathbb Z^3 \otimes \mathbb Z^3\). For studying the natural action of \(\text{GL}_2(\mathbb Z) \times \text{GL}_3(\mathbb Z) \times \text{GL}_3(\mathbb Z)\) on these pairs of matrices it is sufficient to consider the subgroup \(\Gamma = \text{GL}_2(\mathbb Z) \times \text{SL}_3(\mathbb Z) \times \text{SL}_3(\mathbb Z)\). The last two components leave the cubic form \(f(x,y) = \det (Ax - By)\) invariant, and the first component acts on \(f\) in the classical way. The unique polynomial invariant of the \(\Gamma\)-action on \((A,B)\) is \(\text{disc}(A,B) = \text{disc}(\det(Ax-By))\). A classical result due to \textit{F. Levi} [Leipz. Ber. 66, 26--37 (1914; JFM 45.0336.02)] and usually credited to \textit{B. N.…
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11E76
zbMATH DE Number: 2109181
zbMATH Keywords: Gauss composition
zbMATH Keywords: binary cubic forms
zbMATH Keywords: ternary quadratic forms
zbMATH Keywords: cubic rings
zbMATH Keywords: class groups
zbMATH Keywords: discriminant
zbMATH Keywords: parametrization of cubic rings
zbMATH Keywords: exceptional Lie groups
zbMATH Keywords: Dynkin diagrams
Wikidata QID: Q56686833
MaRDI profile type: Publication
full work available at URL: annals.2004.159.865
OpenAlex ID: W2165694723
DOI: 10.4007/ANNALS.2004.159.865
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description: scientific article; zbMATH DE number 2109181