Higher composition laws. III: The parametrization of quartic rings
It is well known that isomorphism classes of quadratic rings (rings that are free of rank 2 as \(\mathbb Z\)-modules) are parametrized by their discriminants, i.e., integers \(d\) congruent to 0 or 1 modulo 4. By a classical result due to \textit{F. Levi} [Leipz. Ber. 66, 26--37 (1914; JFM 45.0336.02)], the isomorphism classes of cubic rings are parametrized by \(\mathrm{GL}_2(\mathbb Z)\)-equivalence classes of integral binary cubic forms. For getting a parametrization of quartic rings, the author reformulates the cubic case as follows: there is a bijection between \(\mathrm{GL}_2(\mathbb Z)\)-equivalence classes of integral binary cubic forms and isomorphism classes of pairs \((R,S)\), where \(R\) is a cubic ring and \(S\) is the quadratic resolvent of \(R\). The main result of this article gives a bijection between isomorphism classes of pairs \((Q,R)\), where \(Q\) is a quartic ring and \(R\) a cubic resolvent of \(Q\), and \(\mathrm{GL}_3(\mathbb Z) \times \mathrm{GL}_2(\mathbb Z)\)-orbits on the space \((\mathrm{Sym}^2 \mathbb Z^3 \otimes \mathbb Z^2)^*\) of pairs of integral ternary quadratic forms. A similar result is obtained for the class of primitive quartic rings. In the final section, information on the splitting of primes in maximal orders is tied to properties of the ternary forms. These results were used for finding the density of discriminants of quartic rings and fields [Ann. Math. (2) 162, No. 2, 1031--1063 (2005; Zbl 1159.11045)].
scientific article; zbMATH DE number 2144178