Let Y be a curve in the moduli space of principally polarized abelian varieties of a given dimension. An isogeny correspondence on Y is by definition an (irreducible) curve Z $\subset $ Y $\times $ Y such that for any point $(y', y'')$ of Z the abelian varieties corresponding to $y'$ and $y''$ are isogenous. There are plenty of curves Y which carry infinitely many isogeny correspondences ; the union of all these Y's is dense in the complex topology of the moduli space. However, we prove that for “most” curves Y there exist only finitely many isogeny correspondences. Here “most curves” mean “all curves belonging to a dense open subset of the space of all curves in the moduli space”, where the space of curves is given a suitable topology called the Kolchin topology, defined using algebraic differential equations