Automorphe Formen über \({\mathbb{F}}_ q(T)\) mit kleinem Führer. (Automorphic forms over \({\mathbb{F}}_ q(T)\) with small conductor)
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 available. In {\S} 10, tables for cusp forms of the indicated type and \(q\leq 16\) are given, corresponding roughly speaking with the tables in [Modular functions of one variable. IV (Lect. Notes Math. 476) (1975; Zbl 0315.14014)]. In particular, by results of Deligne, Drinfeld, and Jacquet/Langlands (Taniyama/Weil's conjecture is true in the function field case), these tables give a complete classification of elliptic curves with a certain reduction behavior over \({\mathbb{F}}_ q(T)\) (q\(\leq 16)\). In some special cases, equations for such curves are given in {\S} 9. A special feature, arising from the existence of nontrivial automorphisms, is treated in {\S} 8.
computations of the matrices of Hecke operators
reduction behavior over \({\mathbb{F}}_ q(T)\)
scientific article; zbMATH DE number 3898980
scientific article