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Arithmetic-geometric mean sequences over finite fields $\mathbb{F}_q$, where $q\equiv5\pmod{8}$

Natália Bátorová, Stevan Gajović

TL;DR

The work extends arithmetic-geometric mean sequences to finite fields with $q \ge 29$ and $q \equiv 5 \pmod{8}$, linking these sequences to a directed graph $\mathcal{J}_{\mathbb{F}_q}$ and establishing precise structural properties. It proves that every nontrivial component of the graph contains a cycle, provides a full description of the graph's components (including a nontrivial component that may consist solely of square vertices), and shows that for all such $q$ there exists at least one nontrivial component when $q \ge 29$, with explicit cycle lengths in the smallest case $q=29$. The results refine previous $q \equiv 3 \pmod{4}$ analyses, extend elementary number-theoretic methods to the $5 \pmod{8}$ setting, and set the stage for future extensions to $q \equiv 1 \pmod{8}$, contrasting with recent elliptic-curve-based approaches. Overall, the paper advances the understanding of AGM dynamics over finite fields and their associated graph-theoretic structures, contributing new existence and structural theorems and suggesting avenues for further elementary development.

Abstract

Arithmetic-geometric mean sequences were already studied over real and complex numbers, and recently, Michael J. Griffin, Ken Ono, Neelam Saikia and Wei-Lun Tsai considered them over finite fields $\mathbb{F}_q$ such that $q \equiv 3 \pmod 4$. In this paper, we extend the definition of arithmetic-geometric mean sequences over $\mathbb{F}_q$ such that $q \equiv 5 \pmod 8$. We explain the connection of these sequences with graphs and show the properties of the corresponding graphs in the case $q \equiv 5 \pmod 8$.

Arithmetic-geometric mean sequences over finite fields $\mathbb{F}_q$, where $q\equiv5\pmod{8}$

TL;DR

The work extends arithmetic-geometric mean sequences to finite fields with and , linking these sequences to a directed graph and establishing precise structural properties. It proves that every nontrivial component of the graph contains a cycle, provides a full description of the graph's components (including a nontrivial component that may consist solely of square vertices), and shows that for all such there exists at least one nontrivial component when , with explicit cycle lengths in the smallest case . The results refine previous analyses, extend elementary number-theoretic methods to the setting, and set the stage for future extensions to , contrasting with recent elliptic-curve-based approaches. Overall, the paper advances the understanding of AGM dynamics over finite fields and their associated graph-theoretic structures, contributing new existence and structural theorems and suggesting avenues for further elementary development.

Abstract

Arithmetic-geometric mean sequences were already studied over real and complex numbers, and recently, Michael J. Griffin, Ken Ono, Neelam Saikia and Wei-Lun Tsai considered them over finite fields such that . In this paper, we extend the definition of arithmetic-geometric mean sequences over such that . We explain the connection of these sequences with graphs and show the properties of the corresponding graphs in the case .
Paper Structure (8 sections, 19 theorems, 28 equations, 5 figures)

This paper contains 8 sections, 19 theorems, 28 equations, 5 figures.

Table of Contents

  1. Introduction.
  2. Main results
  3. Structure of the paper
  4. AGM over $\mathbb{R}$ and $\mathbb{C}$ and their applications.
  5. AGM as a directed graph over finite fields.
  6. AGM over $\mathbb{F}_q$, where ${q \equiv 3 \pmod 4}$.
  7. AGM over $\mathbb{F}_q$, where ${q\equiv~5~\pmod 8}$.
  8. Challenge for $q\equiv 1\pmod{8}$.

Key Result

Lemma 3.5

Let $(a,b)$ be a vertex of $\mathcal{J}_{\mathbb{F}_q}$. Then $(a,b)$ has a parent if and only if $\phi_q(a^2-b^2)=1$. Furthermore, if $(a,b)$ has a parent, there are exactly two parents, namely, $(a + S, a - S)$ and $(a-S,a+S)$, where $S^2 = a^2-b^2$.

Figures (5)

  • Figure 1: Components of $\mathcal{J}_{\mathbb{F}_{11}}$
  • Figure 2: A general nontrivial component of $\mathcal{J}_{\mathbb{F}_q}$ such that $q \equiv 5 \pmod 8$
  • Figure 3: Components of $\mathcal{J}_{\mathbb{F}_{29}}$
  • Figure :
  • Figure :

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • Theorem 3.7
  • proof
  • ...and 39 more