A
familiar object in isogeny based cryptography is the graph whose
vertices are supersingular elliptic curves and whose edges are isogenies
of fixed degree l. It is immediate to prove that from each vertex there
are exactly l+1 outgoing edges, while it is less obvious that such a
graph is connected and that it has the Ramanujan property, a property
about the spectrum of the adjacency matrix that implies that random
walks very soon visit all vertices with the same probability. In our
talk we look at a generalization of these graphs, namely graphs whose
vertices are pairs (E,T), where E is a supersingular elliptic curve and T
is some information on the n-torsion of E (e.g. a basis, a point, a
subgroup) for fixed n. It is easy to notice that the secret keys in SIDH
are exactly walks in such a generalized isogeny graph, and the public
keys are vertices. These graphs can be multipartite, implying that the
Ramanujan property is not always satisfied. By studying modular curves
over mixed characteristics we relate isogeny graphs to geometric and
cohomological objects, which allows us to prove the appropriate
modification of the Ramanujan property. The properties of these
generalized isogeny graphs are useful for a statistical zero knowledge
proof of knowledge of an isogeny, which is part of a work with Basso,
Connolly, De Feo, Fouotsa, Morrison, Panny, Patranabis and Wesolowski to
generate elliptic curves with unknown endomorphisms.