Dynamical structure of AGM over finite fields with order congruent to 5 mod 8

TL;DR

This work analyzes the arithmetic-geometric mean (AGM) process over finite fields, with a focus on fields where $q\equiv 5\pmod{8}$. It develops advanceability and backtracking criteria, proving that indefinitely advanceable/backtrackable nodes form structured jellyfish components: bell-head cycles with Y-shaped tentacles/colons of length two. A contravariant reversal symmetry connects the advancement and backtracking graphs, and the total population of indefinitely advanceable nodes is expressed in terms of the Frobenius trace $a_q$ of the congruent-number elliptic curve $E: y^2=x^3-x$. The results culminate in explicit population formulas and structural descriptions (including a quarter-overlap property) that connect AGM dynamics on finite fields to elliptic curve arithmetic, revealing deep interplay between combinatorial graph structure and arithmetic geometry.

Abstract

Motivated by classical works of Gauss and Euler on the AGM, Ono and his collaborators investigated the union of AGM sequences over finite fields $\mathbb{F}_q$, where $q \equiv 3 \bmod 4$, which they refer to as swarms of jellyfish. A recent preprint extends some of their results to all finite fields with odd characteristic. For $q \equiv 5 \bmod 8$, we reveal finer details about the structure of the connected components, which turn out to be variants of jellyfish with longer and branched tentacles. Moreover, we determine the total population of these swarms in terms of the celebrated base "congruent number" elliptic curve.

Figures (6)

• Figure 1: The AGM over $\mathbb{F}_7$ consists of a single jellyfish.
• Figure 2: One connected component of $F_{{\mathbb F}_{29}}^{\mathrm{adv}_\infty}$, demonstrating the bell-head cycle and branched length-two tentacles, with the cyclic nodes and cyclic arrows colored blue
• Figure 4: A demonstration of the correspondences between members of $S_{K}^{\mathrm{adv}_{2}}$ and $S_{K}^{\mathrm{back}_{2}}$ using $(x, y) = \left(\frac{\mu^2}{1+k}, \frac{\pm 2\mu}{1+k}\right)$ and $(k, \mu) = \left(\frac{1-x^2}{1+x^2}, \frac{\pm y}{1+x^2}\right)$
• Figure 5: Skeletons of $F_{{\mathbb F}_{29}}^{\mathrm{adv}_\infty}$ and $F_{{\mathbb F}_{29}}^{\mathrm{back}_\infty}$ shown side-by-side
• Figure 6: One connected component each of $S_{{\mathbb F}_{29}}^{\mathrm{adv}_{\infty}}$ and $S_{{\mathbb F}_{29}}^{\mathrm{back}_{\infty}}$, showing contravariantly isomorphic structure

Theorems & Definitions (68)

• Example
• Theorem 1.1: MR4567422
• Theorem 1.2
• Example
• Theorem 1.3
• Remark
• Corollary 1.4
• Theorem 1.5
• Proposition 1.6
• Remark