The criterion of Borel-Dwork-Polya-Bertrandias asserts that a power series with coefficients in a number field is a rational function, under analytic conditions which involve all places of the ground field. I will present a generalization of this criterion to algebraic curves of higher genus. The proof combines methods from diophantine approximation with the Hodge index theorem in Arakelov geometry. This is joint work with Jean-Benoît Bost.