Higher composition laws. I: A new view on Gauss composition, and quadratic generalizations
type: Item
instance of: scholarly article
title: Higher composition laws. I: A new view on Gauss composition, and quadratic generalizations
zbMATH Open document ID: 1072.11078
author: Manjul Bhargava
published in: Annals of Mathematics. Second Series
publication date: 2004-12-29T00:00:00Z
full work available at URL: p03.xhtml
review text: In this article, a new method for explaining the composition of binary quadratic forms is presented. It works as follows: Let \(a, b, c, d, e, f, g, h\) be integers, and consider the cube \[ \begin{tikzcd} & e \ar[rr,dash] \ar[dd, dash] \ar[dl, dash] & & f \ar[dd, dash] \ar[dl, dash]\\ a \ar[rr, dash, crossing over] \ar[dd, dash] & & b & \\ & g \ar[rr, dash] \ar[dl, dash] & & h \ar[dl, dash]\\ c \ar[rr, dash] & & d \ar[uu, dash, crossing over] & \end{tikzcd} \] Such a cube can be sliced into pairs of matrices \((M_i,N_i)\) in three different ways, giving rise to three binary quadratic forms \(Q_i = Q_i^A\) defined by \(Q_i(x,y) = - \det(M_i x - N_i y)\). Taking the front and the back face of the cube, for example, we get \(M_1 = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) and \(N_1 = \begin{pmatrix} e & f \\ g & h \end{pmatrix}\), hence \(Q_1(x,y) = A_1x^2 + B_1 xy + C_1y^2\) for \(A_1 = bc-ad\), \(B_1 = ah+de-bg-cf\), and \(C_1 = fg-eh\). These three forms all have the same discriminant \(D = B_i^2 - 4A_iC_i\), and their classes \([Q_i]\) satisfy \([Q_1] + [Q_2] + [Q_3] = 0\) in the additively written class group of forms of discriminant \(D\), where \(0\) is the class of the principal form \(Q_0\) associated to the principal cube \[ \begin{tikzcd} & 1 \ar[rr,dash] \ar[dd, dash] \ar[dl, dash] & & \sigma \ar[dd, dash] \ar[dl, dash]\\ 0 \ar[rr, dash, crossing over] \ar[dd, dash] & & 1 & \\ & \sigma \ar[rr, dash] \ar[dl, dash] & & m+\sigma \ar[dl, dash]\\ 1 \ar[rr, dash] & & \sigma \ar[uu, dash, crossing over] & \end{tikzcd} \] where the integers \(m\) and \(\sigma \in \{0, 1\}\) are defined by \(D = 4m+\sigma\). This definition of the composition of quadratic forms was first given (except for the visualization via cubes) by \textit{R. Dedekind} [``Über trilineare Formen und die Komposition der binären quadratischen Formen, J. Reine Angew. Math. 129, 1--34 (1905; JFM 36.0160.01], who credited \textit{Gauss} [Art. 235, Disquisitiones Arithmeticae] for the insight, a…
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 11E16
zbMATH DE Number: 2123893
MaRDI profile type: Publication
OpenAlex ID: W2004530105
Wikidata QID: Q56686832
DOI: 10.4007/ANNALS.2004.159.217
description: scientific article; zbMATH DE number 2123893