Finite field models of Raleigh-Akiyama polynomials for Hecke groups

Barry Brent

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arXiv · 2206.00642

Finite field models of Raleigh-Akiyama polynomials for Hecke groups

Barry Brent

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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.

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Barry Brent. 2022-06-01. Finite field models of Raleigh-Akiyama polynomials for Hecke groups. https://arxiv.org/abs/2206.00642

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