On an algebra determined by a quartic curve of genus one

Let k be a field of characteristic not two. To each irreducible quartic f (x) over k we associate a certain algebra A(f), given by explicit generators and relations. We prove Af is an Azumaya algebra of rank four over its center and that its center is the coordinate ring of an affine piece of an elliptic curve, the Jacobian of the curve C: 2 = f (x). Its simple images are quaternion algebras and the resulting function from the group of k-rational points on the Jacobian to the Brauer group of k is a group homomorphism whose image is the relative Brauer group of central simple k-algebra split by the function field of C. We also show that the algebra Af is split if and only if C has a k-rational point. (c) 2006 Elsevier Inc. All rights reserved.