Finite field models of Raleigh-Akiyama polynomials for Hecke groups

Abstract

Following work of Raleigh and Akiyama (\cite{raleigh1962fourier, akiyama1992note}), in \cite{interpolating} we considered (among other objects) families of weight zero meromorphic modular forms $J_m$ for Hecke groups $G({\lambda }_m)$. We conjectured in \cite{interpolating} that, for a certain uniformizing variable $X_m$, the $J_m$ have Fourier expansions $J_m=1/X_m+{\sum }_{n=0}^{\infty }{A}_n(m)X_m^n$, where the $A_n(x)$ are polynomials in $\mathbb{Q}[x]$. The present article is concerned with models ${\mathcal{A}}_n[p](x)$ of the $A_n(x)$: polynomials representing self-maps of finite fields with characteristic $p$. The main content is a conjecture specifying ${\mathcal{A}}_n[p](x)$ up to a multiplicative constant for certain families of $n$ and $p$, based on numerical experiments.

arXiv commentAccepted for publication in the journal "Integers"