The Clifford algebra of a cubic form

In this thesis we construct an algebra associated to a cubic curve C defined over a field F of characteristic not two or three. We prove 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 image of the induced function from the group of F-rational points on the Jacobian to the Brauer group of F is precisely the relative Brauer group of classes of central simple F-algebras split by the function field of C. For some special cases, we also prove that this induced function is a group homomorphism and the algebra is split if and only if the cubic curve C has an F-rational point.