Modular symbols for \(F_ q(T)\)
type: Item
instance of: scholarly article
title: Modular symbols for \(F_ q(T)\)
zbMATH Open document ID: 0777.11021
DOI: 10.1215/S0012-7094-92-06811-6
author: Jeremy Teitelbaum
published in: Duke Mathematical Journal
publication date: 1993-12-12T00:00:00Z
review text: Let \(A\) be the polynomial ring \(\mathbb{F}_ q[T]\) with quotient field \(k=\mathbb{F}_ q(T)\) over a finite field \(\mathbb{F}_ q\). H. Hasse, E. Witt, A. Weil, and others established deep and far-reaching analogies between the respective arithmetics of \(A\) and of \(\mathbb{Q}\). In 1973, V. G. Drinfeld constructed two kinds of analogues for the classical modular setting (modular groups, modular curves, modular forms): The first one, dealing with functions on the ``Drinfeld upper half- plane'' leads to (rigid analytic) Drinfeld modular forms and curves. The second one uses the Bruhat-Tits tree \({\mathfrak T}\) of \(\text{PGL}_ 2(K_ \infty)\) as a substitute for the complex upper half-plane, and leads to automorphic forms on a certain adelic group. Both of these are related (and are simultaneously related to the classification of elliptic curves over \(k\) à la Shimura-Taniyama-Weil) by work of M. Reversat and the reviewer. Let \(\text{GL}_ 2(A)\) be the ``modular group'', which acts on \({\mathfrak T}\), and \(\Gamma=\Gamma_ 0(N)\) a ``Hecke congruence subgroup'' \((0\neq N\in A)\). As has become apparent in the last years, the ``right'' analogue of the \(\mathbb{Z}\)-module of cusp forms of weight two with integral Fourier coefficients for a Hecke congruence subgroup of \(\text{SL}_ 2(\mathbb{Z})\) is the \(\mathbb{Z}\)-module \(C_{\text{har}}(\Gamma_ 0(N),\mathbb{Z})\) of harmonic \(\Gamma_ 0(N)\)-invariant cuspidal functions on the set of edges of \({\mathfrak T}\). Some questions (mainly about Hecke correspondence on \(\Gamma\setminus{\mathfrak T}\) and the associated modular curve \(X_ \Gamma\)) are technically easier to handle after dualizing \(C_{\text{har}}\), which leads to the definition of modular symbols given in the paper. It follows the one given by Manin in the classical case. Essentially, a modular symbol is some functional on \(C_{\text{har}}\) induced by ``path integration''. The contents of the paper are as follows: --- definition and elemen…
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11T55
zbMATH DE Number: 179215
zbMATH Keywords: function fields
zbMATH Keywords: \(\mathbb{Z}\)-module of harmonic cuspidal functions
zbMATH Keywords: elliptic curves
zbMATH Keywords: modular symbol
zbMATH Keywords: generators
zbMATH Keywords: relations
reviewed by: Ernst-Ulrich Gekeler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 179215