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Higher composition laws. II: On cubic analogues of Gauss composition. (English) Zbl 1169.11044
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 F. Levi [Leipz. Ber. 66, 26–37 (1914; JFM 45.0336.02)] and usually credited to B. N. Delone and D. K. Faddeev [The theory of irrationalities of the third degree (Russian). Tr. Mat. Inst. Steklova 11, 340 p. (1940; Zbl 0061.09001; JFM 66.0120.03); Engl. Transl. Providence, R.I.: American Mathematical Society (1964; Zbl 0133.30202)] (Delone mentions Levi’s contribution in the preface of the book) states that there is a bijection between $$\text{GL}_2(\mathbb Z)$$-equivalence classes of integral binary cubic forms $$f$$ and isomorphism classes of cubic rings $$R$$ with the same discriminant.
Let $$R$$ be a cubic ring with quotient field $$K = R \otimes \mathbb Q$$. A pair $$(I,I')$$ of fractional $$R$$-ideals in $$K$$ is called balanced if $$II' \subseteq R$$ and $$N(I) N(I') = 1$$. Two such pairs $$(I_1,I_1')$$ and $$(I_2,I_2')$$ are called equivalent if there is a $$\kappa \in K^\times$$ with $$I_1 = \kappa I_2$$ and $$I_2' = \kappa I_1'$$. If $$R$$ is a Dedekind ring, the equivalence classes of balanced pairs are simply pairs of ideal classes inverse to each other. The author shows that there is a bijection between nondegenerate $$\Gamma$$-orbits on $$\mathbb Z^2 \otimes \mathbb Z^3 \otimes \mathbb Z^3$$ and the set of isomorphism classes of pairs $$(R,(I,I'))$$, where $$R$$ is a nondegenerate cubic ring and $$(I,I')$$ an equivalence class of balanced pairs of ideals in $$R$$.
Just as imposing a symmetry condition on integer cubes allows the extraction of information on the $$3$$-part of the class group of quadratic fields, it is possible to describe the $$2$$-class group of cubic rings by imposing a symmetry condition on $$2 \times 3 \times 3$$-cubes, which leads to pairs $$(A,B)$$ of symmetric $$3 \times 3$$-matrices, or equivalently, to pairs of ternary quadratic forms.
In the last section, the composition laws on binary cubic forms, pairs of ternary quadratic forms and pairs of senary alternating $$2$$-forms are connected with exceptional Lie groups.
For Parts III and IV, see Ann. Math. (2) 159, No. 3, 1329–1360 (2004; Zbl 1169.11045) and Ann. Math. (2) 167, No. 1, 53–94 (2008; Zbl 1173.11058).

MSC:
 11R11 Quadratic extensions 11R29 Class numbers, class groups, discriminants 11E76 Forms of degree higher than two
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