Bhargava, Manjul; Ho, Wei Coregular spaces and genus one curves. (English) Zbl 1342.14074 Camb. J. Math. 4, No. 1, 1-119 (2016). A geometric counting technique due to the first author [Ann. Math. (2) 172, No. 3, 1559–1591 (2010; Zbl 1220.11139)] recently enabled him with A. Shankar [Ann. Math. (2) 181, No. 2, 587–621 (2015; Zbl 1317.11038); Ann. Math. (2) 181, No. 1, 191–242 (2015; Zbl 1307.11071)] to obtain remarkable results on the average size of the \(2\)- and \(3\)- Selmer groups of the family of all elliptic curves over \({\mathbb{Q}}\), as well as upper bounds for the average rank of their Mordell-Weil group. These results are based on \(3\)-, \(4\)- and \(5\)-descent computations due to Cassels, Cremona, Cremona-Fisher-Stoll and Fisher, analogous to the original \(2\)-descent of B. J. Birch and H. P. F. Swinnerton-Dyer [J. Reine Angew. Math. 212, 7–25 (1963; Zbl 0118.27601)].A key ingredient in these descent computations is the orbit space parametrization for the moduli space of genus one curves equipped with degree \(2\) line bundle, in terms of the orbit of the action of \({\mathrm{GL}}_2\) on \({\mathrm{Sym}}^4(2)\) (space of binary quartic forms) for Birch and Swinnerton-Dyer, line bundles of degree \(3\), \(4\), \(5\) for subsequent works. The purpose of the paper under review is to develop further such correspondences.The original program of D. J. Wright and A. Yukie [Invent. Math. 110, No. 2, 283–314 (1992; Zbl 0803.12004)], of using prehomogeneous representations (meaning with only one open orbit over \({\mathbb{C}}\)) to determine densities of arithmetic objects, has been completed by the first author [Ann. Math. (2) 162, No. 2, 1031–1063 (2005; Zbl 1159.11045); Ann. Math. (2) 172, No. 3, 1559–1591 (2010; Zbl 1220.11139)]. Here, the authors show that the orbits of most coregular representations (i.e. representations in which the ring of polynomial invariants is freely generated) that are not prehomogeneous parametrize data involving genus one curves together with line bundles, vector bundles, points on their Jacobian curves. A number of explicit examples are given. Reviewer: Michel Waldschmidt (Paris) Cited in 3 ReviewsCited in 18 Documents MSC: 14H60 Vector bundles on curves and their moduli 11E12 Quadratic forms over global rings and fields 11E20 General ternary and quaternary quadratic forms; forms of more than two variables 11E76 Forms of degree higher than two 11G05 Elliptic curves over global fields 11R45 Density theorems 14H52 Elliptic curves Keywords:binary forms; ternary forms; symmetrization; Rubik cube; line bundles; vector bundles; Jordan algebra; Hermitian hypercubes; moduli space; Selmer group; Mordell Weil group; prehomogeneous vector spaces; genus one curves; coregular spaces Citations:Zbl 1220.11139; Zbl 1317.11038; Zbl 1307.11071; Zbl 0118.27601; Zbl 0803.12004; Zbl 1159.11045 PDFBibTeX XMLCite \textit{M. Bhargava} and \textit{W. Ho}, Camb. J. Math. 4, No. 1, 1--119 (2016; Zbl 1342.14074) Full Text: DOI arXiv