Higher composition laws I: A new view on Gauss composition, and quadratic generalizations

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Parametrization of ideal classes in rings associated to binary forms (2010, methodology): “d to binary n-ic forms for all n. This gives an explicit parametrization of class groups of a natural inﬁnite family of orders in rank nnumber ﬁelds. When n= 2,3, these are the results of Bhargava in [1], [2]. One can also study symmetric elements of Z2 ⊗Zn ⊗Zn, that is elements of Z2 ⊗ Sym2 Zn. We ”
Black Holes, Qubits and Octonions (2008, methodology): “28) The individual actions of the three SL(2)i all commute and, therefore, this provides a natural representation of [SL(2)]3 [197].”
Most hyperelliptic curves over Q have no rational points (2013, unknown): “n = 2 by Hardy and Williams [25] and more generally by Morales [29, 30]. A classiﬁcation of the orbits in the cases n= 2 and n= 3, in terms of ideal classes in quadratic and cubic rings, was given in [1] and [2], while a complete classiﬁcation for general nin terms of module classes of rings of rank”