# A moduli interpretation of untwisted binary cubic forms

@inproceedings{Kulkarni2021AMI, title={A moduli interpretation of untwisted binary cubic forms}, author={Rajesh S. Kulkarni and Charlotte Ure}, year={2021} }

We give a moduli interpretation to the quotient of (nondegenerate) binary cubic forms with respect to the natural GL2-action on the variables. In particular, we show that these GL2 orbits are in bijection with pairs of j-invariant 0 elliptic curves together with 3-torsion Brauer classes that are invariant under complex multiplication. The binary cubic generic Clifford algebra plays a key role in the construction of this correspondence. Introduction The subject of homogeneous forms has…

## One Citation

### On the equivalence of binary cubic forms

- Mathematics, Computer Science
- 2022

. We consider the question of determining whether two binary cubic forms over an arbitrary ﬁeld K whose characteristic is not 2 or 3 are equivalent under the actions of either GL(2 ,K ) or SL(2 ,K ),…

## References

SHOWING 1-10 OF 47 REFERENCES

### The extension of the reduced Clifford algebra and its Brauer class

- Mathematics
- 2003

AbstractThe shape Clifford algebraCf of a binary form f of degree d is the k-algebra k{x,y}/I, where I is the ideal generated by {(αx+βy)d−f(α,β)∥α,βk}. Cf has a natural homomorphic image Af, called…

### Rings and ideals parameterized by binary n‐ic forms

- MathematicsJ. Lond. Math. Soc.
- 2011

This paper shows exactly what algebraic structures are parametrized by binary n-ic forms, and proves these parametrizations when any base scheme replaces the integers, and shows that the correspondences between forms and the algebraic data are functorial in the base scheme.

### A Note on Generic Clifford Algebras of Binary Cubic Forms

- MathematicsAlgebras and Representation Theory
- 2019

We study the representation theoretic results of the binary cubic generic Clifford algebra C $\mathcal C$ , which is an Artin-Schelter regular algebra of global dimension five. In particular, we show…

### On the Clifford algebra of a binary form

- Mathematics
- 2003

The Clifford algebra Cj of a binary form f of degree d is the k-algebra k{x,y}/I, where I is the ideal generated by {(ax + βy) d - f(α, β) | α, β E k}. C f has a natural homomorphic image A f that is…

### Arithmetic invariant theory

- Mathematics
- 2014

Let k be a field, let G be a reductive algebraic group over k, and let V be a linear representation of G. Geometric invariant theory involves the study of the k-algebra of G-invariant polynomials on…

### Linearizing of n-ic forms and generalized Clifford algebras

- Mathematics
- 1978

A homogeneous form fd(x1,…,xn) of degree d with coefficients in a field F has a finite linearization if for some m there are m×m matrices α1,…,αn with entries in F so that is the identity matrix.…

### Relative Brauer groups of genus 1 curves

- Mathematics
- 2007

In this paper we develop techniques for computing the relative Brauer group of curves, focusing particularly on the case where the genus is 1. We use these techniques to show that the relative Brauer…

### Crossed Products and Hereditary Orders

- MathematicsNagoya Mathematical Journal
- 1963

Let S be the integral closure of a discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and denote the Galois group of the quotient field extension by G. It has…

### NÉRON MODELS

- Mathematics
- 2011

§1.1. Motivation. The purpose of these notes is to explain the definition and basic properties of the Néron model A of an abelian variety A over a global or local field K. We also give some idea of…