Abstract
We study n-point configurations in \({\mathbb{P}^1(\mathbb{F}_q)}\) modulo projective equivalence. For n = 4 and 5, a complete classification is given, along with the numbers of such configurations with a given symmetry group. Using Polya’s coloring theorem, we investigate the behavior of the numbers C(n, q) of classes of n-configurations resp. C spec(n, q) of classes with nontrivial symmetry group. Both are described by rational polynomials in q which depend on q modulo \({\lambda(n) = {\rm lcm} \{m \in \mathbb{N} | m \leq n\}}\) .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aigner M.: A Course in Enumeration. GTM 238. Springer, Berlin (2007)
Bohn, M.: Punktkonfigurationen über endlichen Körpern. Bachelor’s Thesis, Saarbrücken (2012)
Bombieri E., Husemöller D.: Classification and embeddings of surfaces. In: Proceedings of Symposium in Pure Mathematics 29, Algebraic Geometry Arcata 1974. AMS, Providence, pp. 329–420 (1975)
Gekeler E.-U.: Automorphe Formen über \({\mathbb{F}_q(T)}\) mit kleinem Führer. Abh. Math. Sem. Univ. Hamburg 55, 111–146 (1985)
Harris M.: Algebraic Geometry. GTM 133. Springer, Berlin (1992)
Hartshorne R.: Algebraic Geometry. GTM 52. Springer, Berlin (1977)
Lang S.: Introduction to Modular Forms. Grundlehren der math. Wiss. 222. Springer, Berlin (1976)
Mumford, D.: Geometric Invariant Theory. Ergeb. d. Mathematik 34, Springer, Berlin (1965)
Polya G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math. 68, 145–254 (1937)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gekeler, EU. Point configurations on the projective line over a finite field. manuscripta math. 144, 401–420 (2014). https://doi.org/10.1007/s00229-013-0651-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-013-0651-9