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).
