Vol. 7, No. 1, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Graphs of Hecke operators

Oliver Lorscheid

Vol. 7 (2013), No. 1, 19–61
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.

Keywords
curve over a finite field, vector bundles, automorphic forms, Hecke operator, Bruhat–Tits tree
Mathematical Subject Classification 2010
Primary: 11F41
Secondary: 05C75, 11G20, 14H60, 20C08
Milestones
Received: 11 April 2011
Revised: 25 January 2012
Accepted: 22 February 2012
Published: 28 March 2013
Authors
Oliver Lorscheid
Instituto Nacional de Matemática Pura e Aplicada
IMPA
Estrada Dona Castorina 110
22460-320 Rio de Janeiro, RJ
Brazil
http://w3.impa.br/~lorschei/