On some algebras associated to genus one curves

TL;DR

This paper extends the construction of non-commutative algebras associated with genus one curves, providing explicit realizations of isomorphisms between the Weil-Chatelet and Brauer groups of elliptic curves.

Contribution

It generalizes previous work to pairs of quadratic forms in four variables and conjectures a broader framework for genus one curves of any degree.

Findings

Extended algebraic constructions to new classes of genus one curves.

Provided explicit realizations of key isomorphisms between algebraic groups.

Conjectured a universal generalization for all genus one curves.

Abstract

Haile, Han and Kuo have studied certain non-commutative algebras associated to a binary quartic or ternary cubic form. We extend their construction to pairs of quadratic forms in four variables, and conjecture a further generalisation to genus one curves of arbitrary degree. These constructions give an explicit realisation of an isomorphism relating the Weil-Chatelet and Brauer groups of an elliptic curve.