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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On modular functions in characteristic $ p$

Author: Wen Ch’ing Winnie Li
Journal: Trans. Amer. Math. Soc. 246 (1978), 231-259
MSC: Primary 10D05
MathSciNet review: 515538
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Abstract: Let $ k\, = \,{{\textbf{F}}_q}\left( T \right)$ be a function field of one variable over a finite field $ {{\textbf{F}}_q}$. For a nonzero polynomial $ A\, \in \,{{\textbf{F}}_q}\left[ T \right]$ one can define the modular group $ \Gamma \left( A \right)$. In this paper, we continue a theme introduced by Weil, and study the $ \lambda $-harmonic modular functions for $ \Gamma \left( A \right)$. The main purpose of this paper is to give a natural definition of $ \lambda $-harmonic Eisenstein series for $ \Gamma \left( A \right)$ so that we obtain a decomposition theory of $ \lambda $-harmonic modular functions, analogous to the classical results of Hecke. That is, we prove

$\displaystyle {\text{Modular}}\,{\text{Function}}\,{\text{ = }}\,{\text{Eisenstein}}\,{\text{series}}\, \oplus \,{\text{Cusp Functions}}{\text{.}}$

Moreover, the dimension of the space generated by $ \lambda $-harmonic Eisenstein series for $ \Gamma \left( A \right)$ is equal to the number of cusps of $ \Gamma \left( A \right)$, and so is independent of $ \lambda $.

For the definition of $ \lambda $-harmonic Eisenstein series and the proof of decomposition theory, we consider two cases: (i) $ \lambda \, \ne \, \pm 2\sqrt q $ and (ii) $ \lambda \, = \, \pm 2\sqrt q $, separately. Case (i) is treated in the usual way. Case (ii), being a ``degenerate'' case, is more interesting and requires more complicated analysis.

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