Spectral Theory 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.