Let
be a curve
over
with
function field
.
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
of unramified
Hecke operators, which is of a similar vein to Serre’s theory of quotients of Bruhat–Tits trees.
To be precise,
is locally a quotient of a Bruhat–Tits tree and has finitely many components. An interpretation
of
in terms of
rank
bundles on
and methods from
reduction theory show that
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
.
We describe how one recovers unramified automorphic forms as functions on the graphs
. In the
exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on
functions on
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.
Keywords
curve over a finite field, vector bundles, automorphic
forms, Hecke operator, Bruhat–Tits tree
