Nous \'etendons aux courbes de genre arbitraire le th\'eor\`eme de rationalit\'e de Cantor, lui-m\^eme une extension de th\'eor\`emes de Borel, P\'olya, Dwork, Bertrandias et Robinson. La d\'emonstration s'effectue en deux \'etapes. La premi\`ere est un crit\`ere d'alg\'ebricit\'e, d\'emontr\'e par une m\'ethode d'approximation diophantienne. La seconde repose sur le th\'eor\`eme de l'indice de Hodge en th\'eorie d'Arakelov. -- We extend to algebraic curves of arbitrary genus the rationality theorem of Cantor, itself an extension of theorems of Borel, P\'olya, Dwork, Bertrandias and Robinson. The proof runs in two steps. The first step is an algebraicity criterion, which is proved using a method of diophantine approximation. The second step relies on the Hodge index theorem in Arakelov geometry.
Comment: in French language