Characterization of Modular Correspondences by Geometric Properties

In this paper, we will show how certain Hecke correspondences on modular curves may be characterized by their geometrical properties. We introduce the notion of a cuspidal correspondence and of an almost unramified correspondence (Definition 5) and prove (Theorem 1) that an irreducible almost unramified cuspidal correspondence on a modular curve is a modular correspondence. By considering the bidegree and the invariance properties of the correspondence we are able to some extent to identify the correspondences which arise (cf. Theorem 2 of §4). In §5, we give some simple criteria which sometimes make it easier to show that a correspondence is cuspidal. It would be very useful to have similar criteria for a correspondence to be almost unramified. We illustrate the theory with nontrivial examples on the curves X(5) and X(7). (1980) AMS Classification Numbers: 10D05, 10D07, 10D12, 14C30, 14E10, 14H35, 14H45, 20H05, 20H10, 22E40