Bhargava, Manjul Higher composition laws. IV: The parametrization of quintic rings. (English) Zbl 1173.11058 Ann. Math. (2) 167, No. 1, 53-94 (2008). In the first three articles in this series, the author worked out generalizations of Gauss composition for quadratic [part I: Ann. Math. (2) 159, No. 1, 217–250 (2004; Zbl 1072.11078)] and cubic rings [part II: Ann. Math. (2) 159, No. 2, 865–886 (2004; Zbl 1169.11044)], and applied these techniques for parametrizing quartic rings [part III: Ann. Math. (2) 159, No. 3, 1329–1360 (2004; Zbl 1169.11045)]. The aim of this article is providing a unified treatment of parametrizing rings of rank \(n\) for \(n = 2, 3, 4\) and, in particular, \(n = 5\).The parametrization of cubic rings [see part II] worked as follows: consider a cubic field \(K = \mathbb Q(\alpha)\); letting \(\alpha^{(i)}\) denote the conjugates of \(\alpha\), the map \(\delta(\alpha) = (\alpha^{(1)} - \alpha^{(2)}) (\alpha^{(2)} - \alpha^{(3)})(\alpha^{(3)} - \alpha^{(1)}) = \sqrt{\text{disc}\, \alpha}\) sends a generator \(\alpha\) of \(K\) to an element in the quadratic number field \(k = \mathbb Q(\sqrt{\text{disc}\, \alpha}\,)\), the quadratic resolvent field (actually, \(k = \mathbb Q\) if and only if \(K/\mathbb Q\) is abelian). By slightly modifying the function \(\delta\) we obtain a map \(\overline{\phi}: R \longrightarrow S\) from an order \(R\) in \(K\) to an order \(S\) in \(k\): we simply set \(\overline{\phi}(\alpha) = \frac12(\text{disc}\, \alpha + \sqrt{\text{disc}\, \alpha})\). Since \(\text{disc}\, (\alpha) = \text{disc}\, (\alpha + a)\) for any \(a \in \mathbb Z\), the map \( \overline{\phi}\) induces a map \(\phi: R/\mathbb Z \longrightarrow S/\mathbb Z\). Viewed as \(\mathbb Z\)-modules, \(\phi\) is a map from \(\mathbb Z^2\) to \(\mathbb Z\) and hence corresponds to a binary cubic form.For quartic rings \(R\) there is a cubic resolvent ring \(S\) and a map \(\phi: R/\mathbb Z \longrightarrow S/\mathbb Z\), which corresponds to a pair of ternary quadratic forms.The quintic case seems to behave different, since the object one needs to consider is not a map \(\phi: R/\mathbb Z \longrightarrow S/\mathbb Z\) from the quintic ring \(R\) to its sextic resolvent ring \(S\), but rather a map \(\phi: R/\mathbb Z \longrightarrow \wedge^2 (S/\mathbb Z)\), which may be viewed as a quadruple of alternating \(2\)-forms in five variables.The unified treatment of cubic, quartic and quintic rings employs the following strategy: to an order \(R\) in a number field of degree \(n\), the author attaches a set of \(n\) points in some projective space, which is well defined up to transformations in \(\mathrm{GL}_{n-1}(\mathbb Z)\). He then studies hypersurfaces defined over \(\mathbb Z\) with minimal degree passing through these \(n\) points, and finds that they correspond to functions from \(R\) to certain resolvent rings, which have degrees \(2\), \(3\) and \(6\) for \(n = 3\), \(n = 4\) and \(n = 5\). In the last case, pairs \((R,S)\) of quintic rings and their sextic resolvent rings \(S\) are in bijection with certain quadruples of quinary alternating \(2\)-forms. Maximal orders \(R\) have a unique resolvent ring \(R\) (up to isomorphism), and this gives a particularly simple parametrization of such rings. Reviewer: Franz Lemmermeyer (Jagstzell) Cited in 6 ReviewsCited in 30 Documents MSC: 11R29 Class numbers, class groups, discriminants 11R45 Density theorems 11R16 Cubic and quartic extensions Keywords:quintic rings; Pfaffians; discriminant; sextic resolvents; resolvent rings; composition of forms Citations:Zbl 1072.11078; Zbl 1169.11044; Zbl 1169.11045 × Cite Format Result Cite Review PDF Full Text: DOI Link