Spectral Theory of Isogeny Graphs

TL;DR

This paper analyzes the spectral properties of isogeny graphs of supersingular elliptic curves, establishing bounds on eigenvalues and demonstrating their Ramanujan property, with implications for cryptography and modular forms.

Contribution

It provides an upper bound on the eigenvalues of adjacency matrices of supersingular elliptic curve isogeny graphs, proving they are Ramanujan graphs.

Findings

Eigenvalues are bounded above, confirming Ramanujan property.

Asymptotic distribution of eigenvalues studied.

Connections between graphs, modular forms, and automorphisms explored.

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.