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Improper Eisenstein series on Bruhat-Tits trees. (English) Zbl 0884.11025
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 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 P. Deligne and 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 \(T\). In the course of the construction the author carefully explains the theory of Fourier analysis on \(T\), and the exposition in the paper is a useful “tree–based” complement to the paper “On the analogue of the modular group in characteristic \(p\)”, by A. Weil [in Functional analysis and related fields, Proc. Conf. in Honor of M. Stone, Chicago 1968, 211-223 (1970; Zbl 0226.10031)], see also his Collected Works.

MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
11F12 Automorphic forms, one variable
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References:
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