Automorphe Formen über \({\mathbb{F}}_ q(T)\) mit kleinem Führer. (Automorphic forms over \({\mathbb{F}}_ q(T)\) with small conductor)
type: Item
instance of: scholarly article
title: Automorphe Formen über \({\mathbb{F}}_ q(T)\) mit kleinem Führer. (Automorphic forms over \({\mathbb{F}}_ q(T)\) with small conductor)
zbMATH Open document ID: 0564.10026
DOI: 10.1007/BF02941492
author: Ernst-Ulrich Gekeler
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1985-01-01T00:00:00Z
review text: By the work of \textit{H. Jacquet} and \textit{R. P. Langlands} [Automorphic forms on GL(2) (Lect. Notes Math. 114) (1975; Zbl 0236.12010)], the classical theory of modular forms on the complex upper halfplane may be formulated for arbitrary global fields. This generalization gives better insight into the nature of problems, and some questions, whose answers are far from being known in the classical context, may be handled for global function fields (for instance the conjecture of Taniyama and Weil on parametrization of elliptic curves). An interesting problem is that of splitting (and arithmetic nature of the components) of spaces of automorphic cusp forms under the action of the Hecke algebra. It is not known, for example, which way the space of cusp forms of weight k for the modular group SL(2,\({\mathbb{Z}})\) decomposes in general. In this paper, the problem is considered in the case of a rational function field \({\mathbb{F}}_ q(T)\) over a finite field \({\mathbb{F}}_ q\). Essentially, the most simple nontrivial case (conductor of degree 4) is handled, but some of the results (the algorithm for the computation of fundamental domains) are valid in greater generality. Main results are the computations of the matrices of Hecke operators, and of Fourier coefficients for cusp forms of Drinfeld type. They are independent of the splitting type of the conductor and the size q of the base field \({\mathbb{F}}_ q.\) Before, one has to determine fundamental domains of certain arithmetic subgroups of \(GL(2,{\mathbb{F}}_ q(T))\) on the Bruhat-Tits tree at the infinite place. Whereas the algorithm is developed in detail in {\S} 4, the resulting graphs in the considered cases are indicated without proofs in {\S} 5. Further, cusp forms of the above mentioned type are fully determined by their Fourier coefficients at places of degree 1. So, the splitting problem is reduced to the computation of eigenvectors of explicitly given matrices, and a great number of examples becomes…
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 3898980
zbMATH Keywords: Taniyama-Weil conjecture
zbMATH Keywords: splitting
zbMATH Keywords: spaces of automorphic cusp forms
zbMATH Keywords: Hecke algebra
zbMATH Keywords: rational function field
zbMATH Keywords: finite field
zbMATH Keywords: computation of fundamental domains
zbMATH Keywords: computations of the matrices of Hecke operators
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: cusp forms of Drinfield type
zbMATH Keywords: tables for cusp forms
zbMATH Keywords: elliptic curves
zbMATH Keywords: reduction behavior over \({\mathbb{F}}_ q(T)\)
MaRDI profile type: Publication
cites work: Modular functions of one variable. IV. Proceedings of the international summer school, University of Antwerp, RUCA, July 17 -- August 3, 1972
cites work: ELLIPTIC MODULES
cites work: Q3890820
cites work: Q3734575
cites work: Automorphic Forms on Adele Groups. (AM-83)
cites work: Halbeinfache Gruppenschemata über Dedekindringen
cites work: Chevalley groups over function fields and automorphic forms
cites work: Q3868003
cites work: Automorphic forms on GL (2)
cites work: Modular curves and the Eisenstein ideal
cites work: Elliptic Curves and Wild Ramification
cites work: A table of Hecke operators, II
cites work: Sur la théorie du corps de classes
cites work: Dirichlet series and automorphic forms
description: scientific article; zbMATH DE number 3898980