Algebra & Number Theory

Graphs of Hecke operators

Oliver Lorscheid

Full-text: Open access


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

Export citation


  • J. K. Arason, R. Elman, and B. Jacob, “On indecomposable vector bundles”, Comm. Algebra 20:5 (1992), 1323–1351.
  • M. Atiyah, “On the Krull–Schmidt theorem with application to sheaves”, Bull. Soc. Math. France 84 (1956), 307–317.
  • N. Bourbaki, Éléments de mathématique, Algèbre commutative, Chapitre 7: Diviseurs, Actualités Scientifiques et Industrielles 1314, Hermann, Paris, 1965.
  • E. Frenkel, “Recent advances in the Langlands program”, Bull. Amer. Math. Soc. $($N.S.$)$ 41:2 (2004), 151–184.
  • D. Gaitsgory, “Informal introduction to geometric Langlands”, pp. 269–281 in An introduction to the Langlands program (Jerusalem, 2001), edited by J. Bernstein and S. Gelbart, Birkhäuser, Boston, MA, 2003.
  • E.-U. Gekeler, “Improper Eisenstein series on Bruhat–Tits trees”, Manuscripta Math. 86:3 (1995), 367–391.
  • E.-U. Gekeler, “On the Drinfeld discriminant function”, Compositio Math. 106:2 (1997), 181–202.
  • E.-U. Gekeler and U. Nonnengardt, “Fundamental domains of some arithmetic groups over function fields”, Internat. J. Math. 6:5 (1995), 689–708.
  • S. S. Gelbart, Automorphic forms on adèle groups, Annals of Mathematics Studies 83, Princeton University Press, 1975.
  • G. Harder and M. S. Narasimhan, “On the cohomology groups of moduli spaces of vector bundles on curves”, Math. Ann. 212 (1974/75), 215–248.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
  • M. Kneser, “Strong approximation”, pp. 187–196 in Algebraic Groups and Discontinuous Subgroups (Boulder, CO, 1965), edited by A. Borel and G. D. Mostow, Amer. Math. Soc., Providence, R.I., 1966.
  • S. Lang, “Algebraic groups over finite fields”, Amer. J. Math. 78 (1956), 555–563.
  • G. Laumon, Cohomology of Drinfeld modular varieties, II: Automorphic forms, trace formulas and Langlands correspondence, Cambridge Studies in Advanced Mathematics 56, Cambridge University Press, 1997.
  • O. Lorscheid, Toroidal automorphic forms for function fields, PhD thesis, University of Utrecht, 2008,
  • O. Lorscheid, “Toroidal automorphic forms for function fields”, preprint, 2010. To appear in Israel J. Math.
  • O. Lorscheid, “Automorphic forms for elliptic function fields”, Math. Z. 272:3-4 (2012), 885–911.
  • G. A. Margulis, “Cobounded subgroups in algebraic groups over local fields”, Funkcional. Anal. i Priložen. 11:2 (1977), 45–57, 95. In Russian; translated in Functional Anal. Appl., 11:2 (1977), 119–122.
  • C. C. Moore, “Group extensions of $p$-adic and adelic linear groups”, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 157–222.
  • C. J. Moreno, “Analytic proof of the strong multiplicity one theorem”, Amer. J. Math. 107:1 (1985), 163–206.
  • G. Prasad, “Strong approximation for semi-simple groups over function fields”, Ann. of Math. $(2)$ 105:3 (1977), 553–572.
  • M. van der Put and M. Reversat, “Automorphic forms and Drinfeld's reciprocity law”, pp. 188–223 in Drinfeld modules, modular schemes and applications (Alden-Biesen, 1996), edited by E.-U. Gekeler et al., World Sci. Publ., River Edge, NJ, 1997.
  • T. Schleich, Einige Bemerkungen zur Spektralzerlegung der Hecke-Algebra für die $\mathrm{PGL}\sb{2}$ über Funktionenkörpern, Bonner Mathematische Schriften 71, Mathematisches Institut, Universität Bonn, Bonn, 1974.
  • J.-P. Serre, Trees, Springer, Berlin, 2003.