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AGM aquariums and elliptic curves over arbitrary finite fields

June Kayath, Connor Lane, Ben Neifeld, Tianyu Ni, Hui Xue

Abstract

In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields $\mathbb{F}_q$, and study the resulting AGM graph with points $(a,b) \in \mathbb{F}_q \times \mathbb{F}_q$ and directed edges between points $(a,b)$, $(\frac{a+b}{2},\sqrt{ab})$ and $(a,b)$, $(\frac{a+b}{2},-\sqrt{ab})$. The points in this graph are naturally associated to elliptic curves over $\mathbb{F}_q$ in Legendre normal form, with the AGM function defining a 2-isogeny between the associated curves. We use this correspondence to prove several results on the structure, size, and multiplicity of the connected components in the AGM graph.

AGM aquariums and elliptic curves over arbitrary finite fields

Abstract

In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields , and study the resulting AGM graph with points and directed edges between points , and , . The points in this graph are naturally associated to elliptic curves over in Legendre normal form, with the AGM function defining a 2-isogeny between the associated curves. We use this correspondence to prove several results on the structure, size, and multiplicity of the connected components in the AGM graph.

Paper Structure

This paper contains 7 sections, 44 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pairs That Define Isomorphic Elliptic Curves
  4. Taxonomy
  5. Ideal Classes and Jellyfish
  6. Counting Components
  7. Conclusions

Key Result

Theorem 1

Let $q = p^r$ be an odd prime power, and let $P_0$ be a point on a jellyfish head $H$, associated to an elliptic curve $E_\lambda$. Let $|H|$ be the number of vertices in $H$, and let $M_H$ be the number of scalar multiples of the jellyfish head component in the AGM graph. Then if $\mathcal{O} := \t

Figures (5)

  • Figure 1: The aquarium $A(\mathbb{F}_{11})$. Here the lighter blue points and edges denote the "dead end" branches of the square root, so that the black part of the graph is the jellyfish swarm for $\mathbb{F}_{11}$.
  • Figure 2: Two views of a jellyfish with 48 total points and a strongly connected component of size 6 in the center, highlighted in purple. This jellyfish arises in the graph $A(\mathbb{F}_{125})$.
  • Figure 3: A turtle in $A(\mathbb{F}_{9})$ of size $16$. Here the duplicate points are identified, and we note that the component can be naturally drawn on a torus.
  • Figure 4: A large acyclic component of size 504, arising in $A(\mathbb{F}_{113})$, and a closeup of the same component. The edges are directed upwards; edges highlighted in blue are going directly from the middle level to the top level.
  • Figure 5: A "tangled" jellyfish of size 112 in $A(\mathbb{F}_{29})$, with two disjoint cycles of size 7. The cycles are highlighted in blue and red respectively.

Theorems & Definitions (103)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1: Generalization of hypergeometryagm, Theorem 1.3
  • Theorem 2
  • Theorem 3
  • Lemma 2.1: Classical $2$-descent
  • Lemma 2.2
  • proof
  • ...and 93 more