Isogeny and Ramanujan Graphs

I (Jack) know nothing about this. At all. But it will need to be talked about.

  • Isogeny graph (general definition, degree, duals...).

  • Starting vertex (Bröker's algorithm).

  • Isogeny Volcanos: Sutherland might be a good source https://arxiv.org/abs/1208.5370.

  • Supersingular isogeny graphs

    • Size, everything is defined over GF(p^2). (*as long as the degree divides (p+1)^2 or (p-1)^2).

    • Random walks are probably the best motivation to define Ramanujanness, and are directly applicable to cryptography. A (perhaps too large) source is Hoory-Linial-Wigderson.

    • Consequence from Ramanujan + random walk convergence: O(log p) diameter.