Correspondences on hyperbolic curves Original Research Article

This paper concerns correspondences on hyperbolic curves, which are analogous to isogenies of abelian varieties. The first main result states that given a fixed hyperbolic curve in characteristic zero and a fixed “type” (g, r) (where 2g − 2 + r ≥ 1), there are only finitely many hyperbolic curves of type (g, r) that are isogenous to the given curve. The second main result states if 2g − 2 + r ≥ 3, then the only curves isogenous to a general hyperbolic curve of type (g, r) are the curves that arise as its coverings. Finally, we discuss the meaning of these results relative to the analogy with abelian varieties, especially in light of a certain result of Royden on automorphisms of Teichmüller space.