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Descent on Picard groups using functions on curves

Published online by Cambridge University Press:  17 April 2009

Samir Siksek
Samir Siksek
Affiliation:
Department of Mathematics, Faculty of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Oman e-mail: [email protected]
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Abstract

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Let k be a perfect field, X a smooth curve over k, and denote by Xc the subset of closed points of X. We show that for any non-constant element f of the function field k (X) there exists a natural homomorphism Where

We explain how this generalises the usual results on descents on Jacobians and Picard groups of curves.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 1 , August 2002 , pp. 119 - 124
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Colliot-Thélène, J.-L. and Sansuc, J.-J., ‘La descente sur les variétés rationnelles, II’, Duke J. Math. 54 (1987), 375–492.CrossRefGoogle Scholar
[2]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics 52 (Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[3]Lang, S., Abelian varieties, Interscience Tracts in Pure and Applied Mathematics 7 (Interscience Publishers, New York, 1959).Google Scholar
[4]Lichtenbaum, S., ‘Duality theorems for curves over P-adic fields’, Invent. Math. 7 (1969), 120–136.CrossRefGoogle Scholar
[5]Poonen, B. and Schaefer, E.F., ‘Explicit descent for the Jacobians of cyclic covers of the projective line’, J. Reine Angew. Math. 488 (1997), 141–188.Google Scholar
[6]Schaefer, E.F., ‘Computing a Selmer group of a Jacobian using functions on the curve’, Math. Ann. 310 (1998), 447–471.CrossRefGoogle Scholar
[7]Serre, J.-P., Algebraic groups and class fields, Graduate Texts in Mathematics 117 (Springer-Verlag, New York, 1988).CrossRefGoogle Scholar
[8]Silverman, J.H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 (Springer–Verlag, Berlin, Heidelberg, New York, 1986).CrossRefGoogle Scholar
[9]Stichtenoth, H., Algebraic function fields and codes (Springer-Verlag, Berlin, Heidelberg, New York, 1993).Google Scholar