Higher composition laws and applications

Bhargava, Manjul

doi:10.4171/022-2/13

Cited by 27 publications

(70 citation statements)

References 24 publications

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“…The technique used to construct this map is similar to that originally used by Delone and Fadeev, with several additional complications. In [2], Bhargava also conjectures the existence of several other orbit-ring maps over the integers, refining those given in [32] over fields.…”

Section: §1 Introductionmentioning

confidence: 99%

“…In [15], integral orbits in one of the smaller examples identified in [32] were considered. Bhargava [2] considered integral orbits of several important cases. One of the cases Bhargava considered was the space of pairs of ternary quadratic forms.…”

Section: §1 Introductionmentioning

confidence: 99%

“…The prehomogeneous representation studied in this paper is discriminantal and so its fundamental relatively invariant polynomial is also its discriminant. This kind of terminology was used in [15] and [2] instead of the more usual terminology of relatively invariant polynomials.…”

Section: §1 Introductionmentioning

confidence: 99%

“…It is also natural to attempt to use invariant-theoretic methods to construct orbit-ring maps and this method was used in Section 2 of [31] to construct an orbit-ring map over a field with parameter space the space of trivectors on sevendimensional affine space, parameterizing Cayley algebras over the field. In [2], Bhargava constructed orbit-ring maps for several cases and identified additional structures on the ring side, necessary to render the maps bijective. He also applied his results, in the spirit of Davenport-Heilbronn, to obtain a density theorem for symmetric totally real quartic number fields, with a more explicit constant than that previously announced by Cohen, Diaz y Diaz and Olivier [5] for all quartic number fields, thus confirming a conjecture of the second author [33].…”

Section: §1 Introductionmentioning

confidence: 99%

“…This case is one of the cases considered in [32] and orbits over fields correspond to quartic extensions of the ground field. In [2], Bhargava constructed a bijective map from the set of integral orbits of this prehomogeneous vector space to the set of triples (R, C, f ) where R is a quartic ring, C is its resolvent ring (this notion of the resolvent ring is due to him) and f is a certain quadratic map from R to C. If one only considers R, this gives a map from the set of integral orbits of pairs of ternary quadratic forms to the set of quartic rings, thus giving an analogue of Delone and Fadeev's map. He also showed that this map is surjective and determined the orders of the fibers.…”

Section: §1 Introductionmentioning

confidence: 99%

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A construction of quintic rings

Kable¹

2004

Nagoya Mathematical Journal

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Abstract. We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant. §1. IntroductionTo set the stage, we shall first remind the reader of the work of Delone and Fadeev. In [9], these authors began with an integral binary cubic form f and showed how to construct from it a based cubic ring R f over Z. (Here, as below, an n-tic ring over Z is a commutative Z-algebra that is free of rank n as a Z-module. A ring is based if it is equipped with a distinguished basis containing 1.) They showed that the isomorphism class of R f as a ring depends only upon the GL(2, Z)-equivalence class of f and that the map [f ] → [R f ] from the set of equivalence classes of non-singular forms to the set of isomorphism classes of separable cubic rings is bijective. Moreover, they showed that their map is discriminant-preserving, where the discriminant of f is understood in the sense of invariant theory and that of R f in the sense of ring theory. It is also (probably more) well known that Gauss constructed a correspondence between the set of equivalence

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Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

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A construction of quintic rings

Kable¹

2004

Nagoya Mathematical Journal

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Local-global principles for representations of quadratic forms

Ellenberg

Venkatesh

2007

Invent. math.

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Counting and effective rigidity in algebra and geometry

et al. 2018

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The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.

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Higher composition laws and applications

Cited by 27 publications

References 24 publications

A construction of quintic rings

A construction of quintic rings

Local-global principles for representations of quadratic forms

Counting and effective rigidity in algebra and geometry

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