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    Chen, Imin and Siksek, Samir 2009. Perfect powers expressible as sums of two cubes. Journal of Algebra, Vol. 322, Issue. 3, p. 638.


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  • Bulletin of the Australian Mathematical Society, Volume 66, Issue 1
  • August 2002, pp. 119-124

Descent on Picard groups using functions on curves

  • Samir Siksek (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972700020736
  • Published online: 01 April 2009
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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[1]J.-L. Colliot-Thélène and J.-J. Sansuc , ‘La descente sur les variétés rationnelles, II’, Duke J. Math. 54 (1987), 375492.

[4]S. Lichtenbaum , ‘Duality theorems for curves over P-adic fields’, Invent. Math. 7 (1969), 120136.

[6]E.F. Schaefer , ‘Computing a Selmer group of a Jacobian using functions on the curve’, Math. Ann. 310 (1998), 447471.

[7]J.-P. Serre , Algebraic groups and class fields, Graduate Texts in Mathematics 117 (Springer-Verlag, New York, 1988).

[8]J.H. Silverman , The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 (Springer–Verlag, Berlin, Heidelberg, New York, 1986).

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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