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Journal of the London Mathematical Society Advance Access originally published online on March 21, 2008
Journal of the London Mathematical Society 2008 77(3):789-807; doi:10.1112/jlms/jdn007
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© 2008 London Mathematical Society

Functions, reciprocity and the obstruction to divisors on curves

Martin Bright

Department of Mathematics
University of Bristol
University Walk
Bristol
BS8 1TW
United Kingdom
M.Bright@bristol.ac.uk

Samir Siksek

Mathematics Institute
University of Warwick
Coventry
CV4 7AL
United Kingdom

Let k be a number field, X a smooth curve over k, and f a non-constant element of the function field k(X). If {nu} is a prime of k then denote the completion of k at {nu} by k{nu}, and let X{nu} colone X x k{nu}. In this paper, we introduce an abelian extension l/k, depending on f in a natural way, which we call the class field of k belonging to f. We give an explicit homomorphism prod Pic(Xv) -> Gal(l/k) such that the image of Pic(X) in prod Pic(Xv) is in the kernel of this map. We explain how this can often obstruct the existence of k-rational divisors of certain degrees.

2000 Mathematics Subject Classification 11G30 primary, 11G25 secondary.

The second author's research is supported by an Engineering and Physical Science Research Council (UK) grant, and by a Marie-Curie International Reintegration Grant.

Received December 7, 2006; revised September 19, 2007; published online March 21, 2008.


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