Graphs of Hecke operators

Let X be a curve over F-q with function field F. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results. We develop a structure theory for certain graphs g(x) of unramified Hecke operators, which is of a similar vein to Serre's theory of quotients of Bruhat-Tits trees. To be precise, g(x) is locally a quotient of a Bruhat-Tits tree and has finitely many components. An interpretation of g(x) in terms of rank 2 bundles on X and methods from reduction theory show that g(x) is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetic of F. We describe how one recovers unramified automorphic forms as functions on the graphs g(x). In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on g(x) leads to a finite dimensionality result. In particular, we reobtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms. In an appendix, we calculate a variety of examples of graphs over rational function fields.