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Descent on Picard groups using functions on curves

  • Samir Siksek (a1)

Abstract

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.

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References

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[1]Colliot-Thélène, J.-L. and Sansuc, J.-J., ‘La descente sur les variétés rationnelles, II’, Duke J. Math. 54 (1987), 375492.
[2]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics 52 (Springer-Verlag, Berlin, Heidelberg, New York, 1977).
[3]Lang, S., Abelian varieties, Interscience Tracts in Pure and Applied Mathematics 7 (Interscience Publishers, New York, 1959).
[4]Lichtenbaum, S., ‘Duality theorems for curves over P-adic fields’, Invent. Math. 7 (1969), 120136.
[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), 141188.
[6]Schaefer, E.F., ‘Computing a Selmer group of a Jacobian using functions on the curve’, Math. Ann. 310 (1998), 447471.
[7]Serre, J.-P., Algebraic groups and class fields, Graduate Texts in Mathematics 117 (Springer-Verlag, New York, 1988).
[8]Silverman, J.H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 (Springer–Verlag, Berlin, Heidelberg, New York, 1986).
[9]Stichtenoth, H., Algebraic function fields and codes (Springer-Verlag, Berlin, Heidelberg, New York, 1993).
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Descent on Picard groups using functions on curves

  • Samir Siksek (a1)

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