On an algebra associated to a ternary cubic curve

Abstract

In this paper we construct an algebra associated to a cubic curve C defined over a field F of characteristic not two or three. We prove that this algebra is an Azumaya algebra of rank nine. Its center is the affine coordinate ring of an elliptic curve, the Jacobian of the cubic curve C. The induced function from the group of F-rational points on the Jacobian into the Brauer group of F is a group homomorphism with image precisely the relative Brauer group of classes of central simple F-algebras split by the function field of C. We also prove that this algebra is split if and only if the cubic curve C has an F-rational point. These results generalize Haile's work on the Clifford algebra of a binary cubic form.