Resumen de Functions, reciprocity and the obstruction to divisors on curves

Martin Bright, Samir Siksek

• 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 is a prime of k then denote the completion of k at by k, and let X X x k. 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 Pic(Xv) Gal(l/k) such that the image of Pic(X) in 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.