Higher composition laws III: The parametrization of quartic rings

By Manjul Bhargava. 2004

171 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fe94aedb4f035700374d4c2ffe8f4e95c333afa1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Manjul Bhargava
2. Melanie Matchett Wood
3. A. Shankar
4. B. Gross
5. Bjorn Poonen
6. Alexei Oblomkov
7. Zhiwei Yun
8. Kiran S. Kedlaya

In Geometry and Topology

1. Manjul Bhargava
2. Melanie Matchett Wood
3. A. Shankar
4. Bjorn Poonen
5. B. Gross
6. Alexei Oblomkov
7. Zhiwei Yun
8. Xiaoheng Wang

In Mathematical Physics

1. Melanie Matchett Wood
2. A. Shankar
3. Manjul Bhargava
4. B. Gross
5. Andr'as M'esz'aros
6. Kiran S. Kedlaya
7. Bjorn Poonen
8. Alexei Oblomkov

Citations in the corpus

• Parametrization of ideal classes in rings associated to binary forms (2010, background): “ary quartic forms parametrize quartic rings with monogenic cubic resolvents. The cubic resolvent of of a quartic ring is an integral model of the classical cubic resolvent field, and was introduced in [3] to parametrize quartic rings. Thus the orders associated to binary forms are interesting example”
• Higher composition laws IV: The parametrization of quintic rings (2008, background): “Then (A, B) is precisely the pair of ternary quadratic forms corresponding to R (and S) in the parametrization of quartic rings laid down in [4].”
• Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces (2015, unknown): “When R is a PID, maximal rank n rings over R have unique resolvent rings ([31, § 15], [3, Corollary 5], and [4, Corollary 3]).”
• Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010, background): “15 It is proven in [3] that the orbit space (GL2(Z) × SL3(Z))\WZ corresponds to isomorphism classes of pairs (Q,R), where Q is a quartic ring and R is a cubic resolvent ring of Q.”
• On ℓ-torsion in class groups of number fields (2016, unknown): “The parametrization of quartic rings with their cubic resolvents due to Bhargava [Bha04] (see also [BBP10, Theorem 4.1]) gives an injection from the set of isomorphism classes of maximal quartic orders to the set of GL2(Z)×SL3(Z) classes of pairs of ternary quadratic forms with integral coefficients”