Graphs of Hecke operators

Abstract

Let $X$ be a curve over ${\mathbb{F}}_q$ 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 ${\mathcal{G}}_x$ of unramified Hecke operators, which is of a similar vein to Serre’s theory of quotients of Bruhat–Tits trees. To be precise, ${\mathcal{G}}_x$ is locally a quotient of a Bruhat–Tits tree and has finitely many components. An interpretation of ${\mathcal{G}}_x$ in terms of rank $2$ bundles on $X$ and methods from reduction theory show that ${\mathcal{G}}_x$ 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 ${\mathcal{G}}_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on ${\mathcal{G}}_x$ 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.