Isogeny Graphs: Eigenvalues and Cryptography

This lecture explores isogeny graphs of supersingular elliptic curves with level structure, studying the relationship between the adjacency matrix eigenvalues and a Frobenius on a cohomology group. The lecture shows that the graph is Ramanujan, leading to optimal mixing times for random walks. Applications to cryptography are also presented, based on joint work with Guido Maria Lido.