The extension of the reduced Clifford algebra and its Brauer class

The shape Clifford algebra C _f of a binary form f of degree d is the k-algebra k{x,y}/I, where I is the ideal generated by {(x+y) ^d –f(,),k}. C _f has a natural homomorphic image A _f , called the reduced Clifford algebra, which is a rank d ² Azumaya algebra over its center. The center is isomorphic to the coordinate ring of the complement of an explicit -divisor in Pic _C/k ^{d +g –1} , where C is the curve (w ^d –f(u,v)) and g is the genus of C (9]). We show that the Brauer class of A _f can be extended to a class in the Brauer group of Pic _C/k ^{d + g –1} . We also show that if d is odd, then the algebra A _f is split if and only if the principal homogeneous space Pic _C/k ¹ of the jacobian of C has a k-rational point. Mathematics Subject Classification (2000):16H05, 16G99, 16K50, 14H50, 14H40, 14K30