Improper Eisenstein series on Bruhat-Tits trees
type: Item
instance of: scholarly article
title: Improper Eisenstein series on Bruhat-Tits trees
zbMATH Open document ID: 0884.11025
DOI: 10.1007/BF02568000
author: Ernst-Ulrich Gekeler
published in: Manuscripta Mathematica
publication date: 1995-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/156060
review text: If \(\Delta(z)\) is the normalized cusp form of weight \(12\) for \(\text{SL}_{2}(\mathbb{Z})\), then the logarithmic derivative \(\Delta'(z)/\Delta(z)\), usually known as \(E_2(z)\), is almost a modular form of weight \(2\). \(E_2\) is invariant under translation by \(z\mapsto z+1\), but \(E_2(-1/z)-z^{2}E_2(z)\) is a simple non--zero function of \(z\) [see \textit{N. Koblitz}, Introduction to elliptic curves and modular forms (Springer, Berlin 1984; Zbl 0553.10019), pp. 108-115]. This paper constructs an analogue \(H\) of \(E_{2}(z)\) for the rational function field in one variable over a finite field, and establishes many of its properties (Fourier coefficients, values, etc.) Let \(k=\mathbb{F}_q(T)\). Automorphic forms for \(\text{GL}_{2}(\mathbb{A}_k)\) can be interpreted as functions on the Bruhat-Tits tree \(T\) of \(G=\text{GL}_2(k_\infty)\), where \(k_\infty\) is the completion at the place at infinity of \(k\). The author constructs \(H\) in these terms. The resulting function is invariant by the upper triangular Borel subgroup of \(G\) and transforms in a simple way under the action of the matrix \(\left(\begin{smallmatrix} 0 &1\\ 1 &0\end{smallmatrix}\right)\) The function \(H\) is also closely related to the automorphic Eisenstein series for \(\text{GL}_2(\mathbb{A}_k)\). Functions on the tree \(T\) are also closely related to the Drinfeld modular forms on the upper half plane over \(k_\infty\). These are rigid analytic functions which play a role in the theory of moduli of Drinfeld's elliptic modules similar to that of the classical modular forms in the moduli of elliptic curves. (See \textit{P. Deligne} and \textit{D. Husemoller}, Survey of Drinfeld modules, Contemp. Math. 67, 25-91 (1987; Zbl 0627.14026)). The author outlines how the function \(H\) can be interpreted as the logarithmic derivative of the Drinfeld modular form \(\Delta(z)\), but leaves the proofs for another paper. The function \(H\) is constructed using Fourier analysis on the tree \(…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11F12
zbMATH DE Number: 784035
zbMATH Keywords: automorphic forms
zbMATH Keywords: Eisenstein series
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: function fields
reviewed by: Jeremy Teitelbaum
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784035