Rational Divisors in Rational Divisor Classes

Abstract

We discuss the situation where a curve $\mathcal{C}$, defined over a number field K, has a known K-rational divisor class of degree 1, and consider whether this class contains an actual K-rational divisor. When $\mathcal{C}$ has points everywhere locally, the local to global principle of the Brauer group gives the existence of such a divisor. In this situation, we give an alternative, more down to earth, approach, which indicates how to compute this divisor in certain situations. We also discuss examples where $\mathcal{C}$ does not have points everywhere locally, and where no such K-rational divisor is contained in the K-rational divisor class.