# zbMATH — the first resource for mathematics

Higher composition laws. I: A new view on Gauss composition, and quadratic generalizations. (English) Zbl 1072.11078
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)].

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