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The extension of the reduced Clifford algebra and its Brauer class

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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 {(αxy)df(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 df(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.

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Correspondence to Rajesh S. Kulkarni.

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Mathematics Subject Classification (2000): 16H05, 16G99, 16K50, 14H50, 14H40, 14K30

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Kulkarni, R. The extension of the reduced Clifford algebra and its Brauer class. manuscripta math. 112, 297–311 (2003). https://doi.org/10.1007/s00229-003-0392-2

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