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,βk}. 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.
Similar content being viewed by others
References
Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves. vol. 1, Springer-Verlag, 1985
Childs, L.: Linearizing of n-ic forms and generalized Clifford algebras. Linear and Multilinear Algebra 5, 267–278 (1978)
DeMeyer, F., Ingraham, E.: Separable Algebras over Commutative Rings. Lecture Notes in Math. 181, Springer-Verlag, 1970
Gabber, O.: Some Theorems on Azumaya Algebras: Groupe de Brauer. Lecture Notes in Math., 844, Springer, Berlin-New York, 1981
Grothendieck, A.: Le Groupe De Brauer I-III: Dix exposés sur la cohomologie des schémas. North-Holland, Amsterdam, 1968, pp. 46-189
Haile, D.: On the Clifford algebra of a binary cubic form. Amer. J. Math. 106, 1269–1280 (1984)
Haile, D.: When is the Clifford algebra of a binary cubic form split. J. Algebra 146, 514–520 (1992)
Hoobler, R.T.: Brauer groups of abelian schemes. Ann. Sci. École Norm. Sup. 5, 45–70 (1972)
Kulkarni, R.S.: On the Clifford algebra of a binary form. Trans. of the AMS 355, 3181–3208 (2003)
Lichtenbaum, S.: Duality theorems for curves over p-adic fields. Inventiones Math. 7, 120–136 (1969)
Manin, Y.I.: Le Groupe De Brauer-Grothendieck En Géométrie Diophantinne: Actes, Congrés intern. math. 1970, Tome 1, pp. 401–411
Milne, J.: Étale Cohomology. Princeton University Press, 1980
Milne, J.: Jacobian Varieties: Arithmetic Geometry. Springer-Verlag, 1986, pp. 167–212
Mumford, D.: Abelian Varieties. 2nd ed., Oxford University Press, 1970
Revoy, P.: Algèbres de Clifford et algèbres extérieures. J. Algebra 46, 268–277 (1977)
Roby, N.: Algèbres de Clifford des formes polynomes. C. R. Acad. Sci. Paris Sér. I Math. A-B 268, A484–A486 (1969)
Tate, J.: WC-groups over p-adic fields. Séminaire Bourbaki 156, (156-1)–(156-13) (1957)
Van den Bergh, M.: Linearization of binary and ternary forms. J. Algebra 109, 172–183 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 16H05, 16G99, 16K50, 14H50, 14H40, 14K30
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-003-0392-2