Rings and ideals parameterized by binary n‐ic forms

@article{Wood2011RingsAI,
  title={Rings and ideals parameterized by binary n‐ic forms},
  author={Melanie Matchett Wood},
  journal={Journal of the London Mathematical Society},
  year={2011},
  volume={83}
}
  • M. Wood
  • Published 30 July 2010
  • Mathematics
  • Journal of the London Mathematical Society
The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied composition of binary quadratic forms, which we now understand via Dedekind's association of ideal classes of quadratic rings to integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parameterized by equivalence classes of integral binary cubic forms. Birch, Merriman, Nakagawa, del Corso, Dvornicich, and Simon have all… 
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The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic
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In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could
Parametrization of ideal classes in rings associated to binary forms
We give a parametrization of the ideal classes of rings associated to integral binary forms by classes of tensors in $\mathbb Z^2\tensor \mathbb Z^n\tensor \mathbb Z^n$. This generalizes Bhargava's
Finiteness Theorems for Binary Forms with Given Discriminant
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Parametrizing quartic algebras over an arbitrary base
We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the
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In our first article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that
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