Characterization of modular correspondences by geometric properties

Abstract

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{1). 0. Introduction. In this paper, we will show how certain Hecke correspondences on modular curves may be characterized by their geometric properties. The study of the equations defining modular correspondences was initiated by Adolf Hurwitz, Felix Klein and Ernst Wilhelm Fiedler in the last century. In [H], Adolf Hurwitz obtained a general coincidence formula for correspondences on an algebraic curve and considered from a general point of view the number of equations needed to define a correspondence . In Ch. VI, 6 §6 of [K-F], Felix Klein obtained some explicit equations for modular correspondences and gave criteria for a correspondence to arise as a SchnittsystemCorrespondenz. Since then, this aspect of the theory of modular correspondences has been largely neglected in favor of more powerful and general analytic and number theoretic methods which do not involve explicit equations. Our methods are essentially different from those of these classical authors. In the first section, we discuss correspondences in the category of covering spaces. While the results of §1 are certainly well known, it is convenient to include them for easy reference. In §2, we consider the different actions of the group PSL2(Z/7VZ) on the modular curve X(N) of level N. In §3, we introduce the notion of a cuspidal