Graphs of Hecke operators. (English) Zbl 1292.11061

Lorscheid, Oliver

Algebra Number Theory 7, No. 1, 19-61 (2013).

Summary: 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.

MSC:

11F41	Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11G20	Curves over finite and local fields
14G15	Finite ground fields in algebraic geometry

Keywords:

curve over a finite field; vector bundles; automorphic forms; Hecke operator; Bruhat-Tits tree