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Finite field models of Raleigh-Akiyama polynomials for Hecke groups

Barry Brent

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 $G(λ_m)$. 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.

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 for Hecke groups . We conjectured in \cite{interpolating} that, for a certain uniformizing variable , the have Fourier expansions , where the are polynomials in . The present article is concerned with models of the : polynomials representing self-maps of finite fields with characteristic . The main content is a conjecture specifying up to a multiplicative constant for certain families of and , based on numerical experiments.
Paper Structure (11 sections, 2 theorems, 15 equations)

This paper contains 11 sections, 2 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Why Hecke groups?
  3. Existence of the interpolating polynomials.
  4. Hecke's theory of modular forms.
  5. The existence argument.
  6. Finite field models.
  7. How do finite field models help?
  8. A sketch of the models.
  9. Triangle functions.
  10. Polynomial interpolation of Fourier coefficients of modular functions for Hecke groups.
  11. Finite-field models of interpolating polynomials.

Key Result

Proposition 1

Let $\mathcal{K} = \{J_3, , J_4,...\}$ and $\overline{\mathcal{K}} = \{j_3, j_4,....\}$ Then there exist polynomials $Q_{\mathcal{K},k,n}(x)$ and $Q_{\overline{\mathcal{K}},k,n}(x)$ in $\mathbb{Q}[x]$ such that and In other words, $A_{\mathcal{K},k,m}(n) = Q_{\mathcal{K},k,n}(m)$ and $A_{\overline{\mathcal{K}},k,m}(n) = Q_{\overline{\mathcal{K}},k,n}(m)$ for $k = 1, 2, ..., m = 3, 4, ...$, and $

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 1
  • Proposition 2
  • ...and 9 more