The Theory of Irrationalities of the Third Degree

Decomposable form equations

“…For that reason, I(X1, ... , Xn-l) is called the index form of the integral basis { ao, ... , t'.l'.n-d in question, and the equations of the type (8) are called index form equation8. For cubic number fields, index form equations were earlier extensively studied by Nagell [47], Delone and Faddeev [10] and others. For further references, see [28].…”

“…Results over function fields will be discussed in Mason's paper in this volume. For the methods which have been used, related results, further applications and references we refer to [62], [10], [7], [4], [56], [28], [33), [67], [44], [58], [16].…”

Error estimates for the Davenport-Heilbronn theorems

“…Theorem 2.1 (Delone-Faddeev [13]). The isomorphism classes of cubic orders are in bijection with the classes of irreducible integral binary cubic forms modulo GL(2, Z).…”

Higher composition laws IV: The parametrization of quintic rings

“…When n = 3, one finds that cubic rings are parametrized by integer equivalence classes of binary cubic forms. Specifically, there is a natural bijection between the GL 2 (Z)-orbits on the space of binary cubic forms, and the set of isomorphism classes of pairs (R, S), where R is a cubic ring and S is a quadratic resolvent of R. We are thus able to recover, from a geometric viewpoint, the celebrated result of Delone-Faddeev [11] and Gan-Gross-Savin [12] parametrizing cubic rings (as reformulated in [4]). …”