## Algebra & Number Theory

### Graphs of Hecke operators

Oliver Lorscheid

#### Abstract

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

#### Article information

Source
Algebra Number Theory, Volume 7, Number 1 (2013), 19-61.

Dates
Revised: 25 January 2012
Accepted: 22 February 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729928

Digital Object Identifier
doi:10.2140/ant.2013.7.19

Mathematical Reviews number (MathSciNet)
MR3037889

Zentralblatt MATH identifier
1292.11061

#### Citation

Lorscheid, Oliver. Graphs of Hecke operators. Algebra Number Theory 7 (2013), no. 1, 19--61. doi:10.2140/ant.2013.7.19. https://projecteuclid.org/euclid.ant/1513729928

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