[Paper Review] Spectral Theory of Isogeny Graphs

Giulio Codogni, Guido Lido|arXiv (Cornell University)|Aug 26, 2023

Finite Group Theory ResearchMathematics3 citations

TL;DR

This paper establishes that isogeny graphs of supersingular elliptic curves with level structures possess the Ramanujan property, meaning their adjacency matrices have optimal spectral gaps. Using algebraic geometry and modular forms, the authors prove that eigenvalues are bounded within the optimal range, generalizing Mestre's method and confirming these graphs as optimal expanders, with implications for isogeny-based cryptography and modular form computations.

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.

Motivation & Objective

• To establish the Ramanujan property for isogeny graphs of supersingular elliptic curves with level structures, ensuring optimal spectral expansion.
• To generalize Mestre's 'Méthode des graphes' for computing modular forms via adjacency matrix eigenvectors.
• To analyze the asymptotic distribution of eigenvalues, number of connected components, automorphisms, and connections to modular forms.
• To confirm that isogeny graphs with level structures retain optimal expansion properties, extending results from classical isogeny graphs.
• To provide a spectral foundation for isogeny-based cryptography, particularly in zero-knowledge proof systems.

Proposed method

• Constructs isogeny graphs $ G(p, ho, H) $ with vertices as isomorphism classes of supersingular elliptic curves over $ ar{\mathbb{F}}_p $ with level $ H $-structures.
• Defines edges as degree $ \ell $ isogenies between such curves, modulo automorphisms of the target curve.
• Analyzes the adjacency matrix $ A = (a_{ij}) $, where $ a_{ij} $ counts isogenies from $ (E_j, \phi_j) $ to $ (E_i, \phi_i) $.
• Applies algebraic geometry techniques, including Picard groups and norm maps on curves with nodal singularities.
• Uses character groups of tori in Picard groups to model Hecke operators and study push-forwards of line bundles.
• Employs isomorphisms between character groups and kernels of maps $ \Sigma: \bigoplus \mathbb{Z} y_i^j \to \mathbb{Z}^r $ to relate geometry to spectral properties.

Experimental results

Research questions

• RQ1 Do isogeny graphs with level structures satisfy the Ramanujan property, i.e., do their adjacency matrices have eigenvalues bounded within the optimal spectral range?
• RQ2 How do the eigenvalues of the adjacency matrix distribute asymptotically as $ p \to \infty $?
• RQ3 What is the number of connected components in isogeny graphs with level structures, and how does it relate to class numbers or Galois actions?
• RQ4 How do automorphisms of curves and level structures affect the structure of the isogeny graph?
• RQ5 Can the spectral properties of these graphs be used to extend Mestre’s method for computing modular forms?

Key findings

• The adjacency matrices of isogeny graphs with level structures satisfy the Ramanujan property: all nontrivial eigenvalues lie within the optimal interval $ [-2\sqrt{\ell}, 2\sqrt{\ell}] $, matching the Ramanujan bound.
• The asymptotic distribution of eigenvalues follows the Kesten-McKay law, confirming the graphs as optimal expanders.
• The number of connected components in the graph corresponds to the class number of the associated quaternion algebra, generalizing classical results.
• Automorphism groups of vertices are controlled by the level structure: for example, Borel and full level structures reduce automorphisms, while trivial structures preserve full $ \mathrm{GL}_2 $-symmetry.
• The spectral theory provides a framework to generalize Mestre’s method for computing eigenforms via eigenvectors of adjacency matrices.
• Isogeny graphs with level structures inherit optimal expansion properties, as shown via spectral gap analysis and Hecke operator actions on line bundles.