Algebra & Number Theory

Graphs of Hecke operators

Oliver Lorscheid

Full-text: Open access

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
Received: 11 April 2011
Revised: 25 January 2012
Accepted: 22 February 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
Secondary: 05C75: Structural characterization of families of graphs 11G20: Curves over finite and local fields [See also 14H25] 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05] 20C08: Hecke algebras and their representations

Keywords
curve over a finite field vector bundles automorphic forms Hecke operator Bruhat–Tits tree

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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