Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distribution of cycles in supersingular $\ell$-isogeny graphs

Eli Orvis

Abstract

Recent work by Arpin, Chen, Lauter, Scheidler, Stange, and Tran counted the number of cycles of length $r$ in supersingular $\ell$-isogeny graphs. In this paper, we extend this work to count the number of cycles that occur along the spine. We provide formulas for both the number of such cycles, and the average number as $p \to \infty$, with $\ell$ and $r$ fixed. In particular, we show that when $r$ is not a power of $2$, cycles of length $r$ are disproportionately likely to occur along the spine. We provide experimental evidence that this result holds in the case that $r$ is a power of $2$ as well.

Distribution of cycles in supersingular $\ell$-isogeny graphs

Abstract

Recent work by Arpin, Chen, Lauter, Scheidler, Stange, and Tran counted the number of cycles of length in supersingular -isogeny graphs. In this paper, we extend this work to count the number of cycles that occur along the spine. We provide formulas for both the number of such cycles, and the average number as , with and fixed. In particular, we show that when is not a power of , cycles of length are disproportionately likely to occur along the spine. We provide experimental evidence that this result holds in the case that is a power of as well.
Paper Structure (13 sections, 35 theorems, 25 equations, 2 figures, 1 table)

This paper contains 13 sections, 35 theorems, 25 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Roadmap
  3. Acknowledgments
  4. Preliminaries
  5. Background on Supersingular Elliptic Curves
  6. Background on Quadratic Orders in $B_{p, \infty}$
  7. Setup
  8. $\mathbb F_p$-vertices on oriented isogeny cycles and the even class number case
  9. Limiting Distribution of cycles on and off the spine
  10. Explicit examples
  11. Expected values for $n_s$
  12. Remarks on computational verification
  13. Cycles of even length

Key Result

Theorem 1

Let $\ell$ and $r$ be fixed, with $r$ odd. Then there exists a modulus $M$ depending on $\ell$ and $r$ such that for $p \gg 0$, $n_t$ and $n_s$ depend only on $p$ modulo $M$.

Figures (2)

  • Figure 1: Average numbers of $3$-cycles along the spine in the $3$-isogeny and $5$-isogeny graphs
  • Figure 2: Average numbers of $4$-cycles (resp. $6$-cycles) along the spine in the $3$-isogeny (resp. $2$-isogeny) graph

Theorems & Definitions (70)

  • Theorem : Theorem \ref{['EventualDistributionThm']}
  • Corollary : Corollary \ref{['EventualDistributionCor']}
  • Theorem : Theorem \ref{['AverageNsThm']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 60 more