Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs

TL;DR

This work extends the classical Deuring correspondence to supersingular elliptic curves with level-$N$ structure by proving that the endomorphism ring $\mathcal{O}(E,G)$ is an Eichler order of level $N$ in the quaternion algebra $B_{p,\infty}^E$, and by establishing a contravariant equivalence between the category of level-$N$ SSEs and invertible left $\mathcal{O}(E,G)$-modules. It clarifies that the map from $|\mathcal{S}_N|$ to Eichler orders is surjective but not injective, describes the fiber structure via dualizing and Frobenius involutions, and analyzes the corresponding $\ell$-isogeny graphs, including connectedness results and explicit examples. A detailed counting framework relates $N$-isogenies to the two-sided ideal class group of level-$N$ Eichler orders, yielding approximate upper bounds and exact class-number formulas in terms of $\mathbb{Z}[\sqrt{-pN}]$ and its orders. The level-$N$ graphs and category equivalence have cryptographic relevance, offering structural insights into enhanced isogeny graphs and their potential resistance to certain attacks. Overall, the paper provides a rigorous, complete Deuring-type correspondence for level structures and a rich algebraic and graph-theoretic framework for level-$N$ SSEs.

Abstract

In this paper, we add the information of level structure to supersingular elliptic curves and study these objects with the motivation of isogeny-based cryptography. Supersingular elliptic curves with level structure map to Eichler orders in a quaternion algebra, just as supersingular elliptic curves map to maximal orders in a quaternion algebra via the classical Deuring correspondence. We study this map and the Eichler orders themselves. We also look at isogeny graphs of supersingular elliptic curves with level structure, and how they relate to graphs of Eichler orders.

Figures (6)

• Figure 4.1: Diagram to support Lemma \ref{['lem:FrobInvIsWellDef']} in proving $F_p$ is well-defined on equivalence classes of ${|\mathcal{S}_N|}$.
• Figure 4.2: Diagram to support Lemma \ref{['lem:DandFpCommute']} in proving that $F_p$ and $D_1$ commute on equivalence classes of ${|\mathcal{S}_N|}$.
• Figure 5.1: The counts of 3-isogenies between distinct $p$-power Frobenius conjugate supersingular elliptic curves over ${\overline{\mathbb{F}}_{p}}$. For $p\equiv 1\pmod{12}$, the estimated upper bound described in this section was never off by more than four: exactly accurate for $23.33\%$ of the data, over by two for $51.34\%$ of the data, and over by four for $25.32\%$ of the data. The lower bound from EHLMP_lisogconjcount is plotted in red.
• Figure 5.2: Illustrative examples for Proposition \ref{['prop:NIsogConj']}.
• Figure 7.1: Graph of $\mathcal{E}_{37,2}^3$, with groups labeled by the first term in the $x$-coordinate of a generating point.

Theorems & Definitions (75)

• Definition 1.1: Supersingular elliptic curve with level-$N$ structure, see Definition \ref{['def:SSECsWithLS']}
• Definition 1.2: Endomorphisms of $(E,G)$, see Definition \ref{['def:OO(E,G)']}
• Theorem 1.3: See Theorem \ref{['thm:O(E,G)_is_EO']}
• Theorem 1.4
• Theorem 1.5: Equivalence of Categories, see Theorem \ref{['thm:cats_equiv']}
• Theorem 2.1: Deuring Correspondence
• Theorem 2.2: Theorem 44 Kohel
• Proposition 2.3: Section 2DelGal01
• Definition 3.1
• Definition 3.2: $\mathcal{O}(E,G)$