On modular functions in characteristic $p$

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 \[ {\text {Modular}} {\text {Function}} {\text { = }} {\text {Eisenstein}} {\text {series}} \oplus {\text {Cusp Functions}}{\text {.}}\] 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.