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Functions, reciprocity and the obstruction to divisors on curves
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Bright, M. (Martin) and Siksek, Samir (2008) Functions, reciprocity and the obstruction to divisors on curves. Journal of the London Mathematical Society, Vol.77 (Pt.3). pp. 789-807. doi:10.1112/jlms/jdn007 ISSN 0024-6107.
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Official URL: http://dx.doi.org/10.1112/jlms/jdn007
Abstract
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 v is a prime of k then denote the completion of k at v by k(v), and let X-v := X x k(v). In this paper, we introduce an abelian extension Ilk, depending on f in a natural way, which we call the class field of k belonging to f. We give an explicit homomorphism Pi Pic(X-v) -> Gal(l/k) such that the image of Pic(X) in Pi Pic(X-v) is in the kernel of this map. We explain how this can often obstruct the existence of k-rational divisors of certain degrees.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Divisor theory, Curves, Algebraic, Functions, Reciprocity theorems | ||||
Journal or Publication Title: | Journal of the London Mathematical Society | ||||
Publisher: | Cambridge University Press | ||||
ISSN: | 0024-6107 | ||||
Official Date: | June 2008 | ||||
Dates: |
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Volume: | Vol.77 | ||||
Number: | Pt.3 | ||||
Number of Pages: | 19 | ||||
Page Range: | pp. 789-807 | ||||
DOI: | 10.1112/jlms/jdn007 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC), Seventh Framework Programme (European Commission) (FP7) |
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