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Spectral Theory of Isogeny Graphs

Giulio Codogni, Guido Lido

TL;DR

This work develops a spectral theory for isogeny graphs built from supersingular elliptic curves with level structure. By relating the adjacency operator to Hecke operators on modular curves and employing Eichler–Shimura relations together with Deligne’s Weil conjectures, it derives sharp eigenvalue bounds, showing Ramanujan-type behavior for a broad class of graphs and revealing detailed component structure via Weil invariants. The analysis unifies graph-theoretic spectra with spaces of modular forms, clarifying how automorphisms of graphs correspond to modular-curve symmetries (Fricke, Atkin–Lehner, diamond) and enabling precise descriptions of eigenvalue distributions and mixing times. These results have implications for expander constructions, mixing in non-backtracking walks, and the security considerations of isogeny-based cryptographic protocols. Overall, the paper provides a rigorous bridge between algebraic geometry, modular forms, and spectral graph theory in the context of isogeny graphs.

Abstract

We consider finite graphs whose vertexes are supersingular elliptic curves, possibly with level structure, and edges are isogenies. They can be applied to the study of modular forms and to isogeny based cryptography. The main result of this paper is an upper bound on the modules of the eigenvalues of their adjacency matrices, which in particular implies that these graphs are Ramanujan. We also study the asymptotic distribution of the eigenvalues of the adjacency matrices, the number of connected components, the automorphisms of the graphs, and the connection between the graphs and modular forms.

Spectral Theory of Isogeny Graphs

TL;DR

This work develops a spectral theory for isogeny graphs built from supersingular elliptic curves with level structure. By relating the adjacency operator to Hecke operators on modular curves and employing Eichler–Shimura relations together with Deligne’s Weil conjectures, it derives sharp eigenvalue bounds, showing Ramanujan-type behavior for a broad class of graphs and revealing detailed component structure via Weil invariants. The analysis unifies graph-theoretic spectra with spaces of modular forms, clarifying how automorphisms of graphs correspond to modular-curve symmetries (Fricke, Atkin–Lehner, diamond) and enabling precise descriptions of eigenvalue distributions and mixing times. These results have implications for expander constructions, mixing in non-backtracking walks, and the security considerations of isogeny-based cryptographic protocols. Overall, the paper provides a rigorous bridge between algebraic geometry, modular forms, and spectral graph theory in the context of isogeny graphs.

Abstract

We consider finite graphs whose vertexes are supersingular elliptic curves, possibly with level structure, and edges are isogenies. They can be applied to the study of modular forms and to isogeny based cryptography. The main result of this paper is an upper bound on the modules of the eigenvalues of their adjacency matrices, which in particular implies that these graphs are Ramanujan. We also study the asymptotic distribution of the eigenvalues of the adjacency matrices, the number of connected components, the automorphisms of the graphs, and the connection between the graphs and modular forms.
Paper Structure (24 sections, 32 theorems, 91 equations, 2 figures)

This paper contains 24 sections, 32 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Definitions and Results
  3. Ramanujan graphs and expander sequences
  4. Non-backtracking random walks and mixing time
  5. Asymptotic distribution of the eigenvalues
  6. Relation with isogeny based cryptography
  7. Relation with other works
  8. First properties of isogeny graphs and reduction of Theorems \ref{['thm:main1']} and \ref{['thm:main2']} to Theorem \ref{['thm:spet1']}
  9. Automorphisms of isogeny graphs
  10. Hermitian form and diagonalization
  11. Weil pairing and reduction of Theorems \ref{['thm:main1']} and \ref{['thm:main2']} to Theorem \ref{['thm:spet1']}
  12. Isomorphism between Borel and Cartan level structure
  13. Preliminary results on modular curves
  14. Relation between modular curves and isogeny graphs and proof of Theorem \ref{['thm:spet1']}
  15. Relation with modular forms
  16. ...and 9 more sections

Key Result

Theorem 1.4

With the notation of Definition def:graph, if $H$ contains the scalar matrices and $\det(H)=(\mathbb Z/N\mathbb Z)^{\times}$, then the graph $G(p,\ell,H)$ is connected, its adjacency matrix $A$ is diagonalizable over $\mathbb R$, the eigenvalue $\ell+1$ has multiplicity one, and all the other eigenv where $|V|$ is the number of vertices of $G(p,\ell,H)$. In particular, all eigenvalues different fr

Figures (2)

  • Figure 1: This is an example of isogeny graph $G(p,\ell,H)$ with $p=23$, $\ell=3$ and $H = \langle \left( 5621 \right) ,\left( 1201 \right) ,\left( 7027 \right) ,\left( 5005 \right) ,\left( 2771 \right) ,\left( 1401 \right) ,\left( 1041 \right) \rangle$ the only index 8 subgroup of $\mathrm{GL}_2(\mathbb Z/8\mathbb Z)$. $\qquad$ Color indicates the elliptic curve, or equivalently the $j$-invariant. The level structure is given through a matrix: for each of the three elliptic curves we have chosen a (non-canonical) basis of the $8$-torsion, and the matrix gives the change of basis. For mere visual clarity, arrows going up are black, arrows going down are gray. $\qquad$ The graph $G(p,\ell,H)$ has two components, $G_1$ and $G_2$, which correspond via the Weil invariant to the two connected components $C_1$ and $C_2$ of the Cayley graph, depicted on the right. Since each $C_i$ has two vertices, each $G_i$ is bipartite, see Remark \ref{['rem:partition']}. At the bottom, the graph without level structure.
  • Figure 2: Numerical experiments on $\eta(p,\ell)$

Theorems & Definitions (54)

  • Definition 1.1: Level structure on elliptic curves
  • Remark 1.2
  • Definition 1.3: Supersingular isogeny graph
  • Theorem 1.4
  • Definition 1.5: Weil invariant of a level structure
  • Theorem 1.6
  • Remark 1.7: Multipartite graphs
  • Corollary 1.8
  • Corollary 1.9
  • Proposition 1.12
  • ...and 44 more