The extension of the reduced Clifford algebra and its Brauer class

Abstract

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 {(αx+βy) d −f(α,β)∥α,β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 Θ -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.