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Higher composition laws. I: A new view on Gauss composition, and quadratic generalizations. (English) Zbl 1072.11078
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 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 Gauss [Art. 235, Disquisitiones Arithmeticae] for the insight, and later again by A. Speiser [H. Weber-Festschrift, 375–395 (1912; JFM 43.0277.14)], who gave a particularly clever proof of the relation \([Q_1] + [Q_2] + [Q_3] = 0\) [see L. E. Dickson, History of the Theory of Numbers. Vol II, p. 75–78].
Not all of Bhargava’s article, however, is already in Dedekind. Bhargava extends the action of \(\text{SL}_2(\mathbb Z)\) and the group law from binary quadratic forms to equivalence classes of projective cubes (these are cubes whose associated quadratic forms are primitive) with fixed discriminant \(D\) in the following way. An element \((\gamma, I, I)\), where \(I\) is the \(2 \times 2\)-identity matrix, and where \(\gamma = \begin{pmatrix} r & s \\ t & u \end{pmatrix}\) is in \(\text{SL}_2(\mathbb Z)\), acts on a cube by replacing \((M_1,N_1)\) with \((rM_1+sN_1, tM_1 + uN_1)\); similarly, the actions of \((I,\gamma,I)\) and \((I,I,\gamma)\) are defined. Since the action of the three factors in \(\Gamma = \text{SL}_2(\mathbb Z) \times \text{SL}_2(\mathbb Z) \times \text{SL}_2(\mathbb Z)\) commutes, this gives a well defined action of \(\Gamma\) on the set of projective cubes. Theorem 2 now shows that there is a unique group law on the set of \(\Gamma\)-equivalence classes \([A]\) of projective cubes \(A\) with discriminant \(D\) such that the class of the principal cube is the neutral element, and such that the maps \([A] \mapsto [Q_i^A]\) are group homomorphisms.
Next, Bhargava associates the binary cubic form \(px^3 + 3qx^2y + 3rxy^2 + sy^3\) to the triply symmetric cube
\[ \begin{tikzcd} & q \ar[rr,dash] \ar[dd, dash] \ar[dl, dash] & & r \ar[dd, dash] \ar[dl, dash]\\ p \ar[rr, dash, crossing over] \ar[dd, dash] & & q & \\ & r \ar[rr, dash] \ar[dl, dash] & & s \ar[dl, dash]\\ q \ar[rr, dash] & & r \ar[uu, dash, crossing over] & \end{tikzcd} \]
The quadratic forms associated to such a cube are all equal to the Hessian \(H(x,y) = (q^2-pr)x^2 + (ps-qr)xy + (r^2-qs)y^2\). The principal cube corresponds to the cubic form \(C_0 = 3x^2y + 3 \sigma xy^2 + (m+\sigma)y^3\), and there is a unique group law on projective binary cubics with discriminant \(D = 4m+\sigma\) with neutral element \([C_0]\) and such that the map sending the \(\text{SL}_2(\mathbb Z)\)-equivalence class of a cubic form to the \(\Gamma\)-equivalence class of its cube is a homomorphism. The natural epimorphism from the class group of cubic forms of discriminant \(D\) to the \(3\)-class group of the quadratic number field with discriminant \(D\), which was first studied by Eisenstein [J. Reine Angew. Math. 27, 75–79 (1844; ERAM 027.0780cj)], comes out as a corollary.
In a similar way, Bhargava now constructs composition laws on pairs of binary quadratic forms, pairs of quaternary alternating \(2\)-forms, and senary (sextic) alternating \(3\)-forms. These composition laws are then connected to the arithmetic of ideals in orders of quadratic number fields, and there are even relations to exceptional Lie groups with Dynkin diagrams \(D_4\), \(B_3\), \(D_5\), \(G_2\) and \(E_6\).
For Parts II and III, cf. ibid. 159, No. 2, 865–886 (2004; Zbl 1169.11044) and 159, No. 3, 1329–1360 (2004; Zbl 1169.11045).
In addition to Dedekind and Speiser, the possibility of defining composition of quadratic forms using “cubes” or \(2 \times 4\)-matrices of integers was also realized by A. Cayley [J. Reine Angew. Math. 39, 14–15 (1850)] (he was led to this insight by his theory of hyperdeterminants; cf. I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky [Discriminants, resultants, and multidimensional determinants (1994; Zbl 0827.14036)]), Riss [Sém. Théor. Nombres Bordeaux 1977/78, Exp. No. 18 (1978; Zbl 0391.00003)] and D. Shanks [Number Theory and Applications; Proc. Banff 1988, 179–204 (1989; Zbl 0691.10011)].

11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
11E76 Forms of degree higher than two
11E16 General binary quadratic forms
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