Functions, reciprocity and the obstruction to divisors on curves

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