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Isogeny graphs with level structures arrising from the Verschiebung map

Antonio Lei, Katharina Müller

TL;DR

This work introduces isogeny graphs enhanced with level structures defined by kernels of iterates of the Verschiebung map, unifying ordinary and supersingular cases. By constructing $G_l^p(n,N)$ with vertices $(E,Q,P_n)$ where $P_n$ generates $ ext{ker}(V^n)$, the authors show that as $n$ grows the graphs form a $ obreak Z_p$-tower of coverings, generalizing previous level-structure frameworks. They then attach CM fields to connected components and classify the resulting volcanic structures according to how the prime $l$ splits in these CM fields, proving precise crater/volcano patterns in split, ramified, and inert cases. An extensive voltage-graph framework and a systematic extension to oriented supersingular curves underpin the main results, while an appendix addresses an inverse problem for tectonic craters. The results provide a p-adic-like tower structure for isogeny graphs with Verschiebung-level structures and a detailed volcanic classification of their components, with potential applications to isogeny-based cryptography and arithmetic geometry.

Abstract

We enhance an isogeny graph of elliptic curves by incorporating level structures defined by bases of the kernels of iterates of the Verschiebung map. We extend several previous results on isogeny graphs with level structures defined by geometric points to these graphs. Firstly, we prove that these graphs form $\mathbb{Z}_p$-towers of graph coverings as the power of the Verschiebung map varies. Secondly, we prove that the connected components of these graphs display a volcanic structure.

Isogeny graphs with level structures arrising from the Verschiebung map

TL;DR

This work introduces isogeny graphs enhanced with level structures defined by kernels of iterates of the Verschiebung map, unifying ordinary and supersingular cases. By constructing with vertices where generates , the authors show that as grows the graphs form a -tower of coverings, generalizing previous level-structure frameworks. They then attach CM fields to connected components and classify the resulting volcanic structures according to how the prime splits in these CM fields, proving precise crater/volcano patterns in split, ramified, and inert cases. An extensive voltage-graph framework and a systematic extension to oriented supersingular curves underpin the main results, while an appendix addresses an inverse problem for tectonic craters. The results provide a p-adic-like tower structure for isogeny graphs with Verschiebung-level structures and a detailed volcanic classification of their components, with potential applications to isogeny-based cryptography and arithmetic geometry.

Abstract

We enhance an isogeny graph of elliptic curves by incorporating level structures defined by bases of the kernels of iterates of the Verschiebung map. We extend several previous results on isogeny graphs with level structures defined by geometric points to these graphs. Firstly, we prove that these graphs form -towers of graph coverings as the power of the Verschiebung map varies. Secondly, we prove that the connected components of these graphs display a volcanic structure.
Paper Structure (12 sections, 20 theorems, 34 equations)

This paper contains 12 sections, 20 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Defining isogeny graphs with level structures
  4. Voltage assignments and derived graphs
  5. General results on voltage assignments and derived graphs
  6. Proof of Theorem \ref{['thmA']}
  7. Volcanic structures of isogeny graphs of oriented supersingular elliptic curves
  8. Definitions
  9. Preliminary lemmas - I
  10. Preliminary lemmas - II
  11. Proof of Theorem \ref{['thmB']}
  12. An inverse problem for isogeny graphs of oriented supersingular elliptic curves

Key Result

Theorem A

Let $E/\mathbb{F}_{p^k}$ be an elliptic curve representing a non-isolated vertex of $\mathbf{G}_l^p(0,1)$. Let $Q$ be a point on $E$ of order $N$. There is an integer $n_0$ such that for all $n\ge n_0$, we can find a connected component $X_n$ of $\mathbf{G}_l^p(n,N)$ that includes a vertex of the fo

Theorems & Definitions (60)

  • Theorem A
  • Remark 1.1
  • Theorem B: Theorems \ref{['thm:structure1']}, \ref{['thm:structure2']} and \ref{['thm:structure3']}
  • Remark 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 50 more