Zeta functions of isogeny graphs and modular curves

Isogeny graphs of supersingular elliptic curves have broad application, from the study and computation of modular forms to post-quantum cryptography. This is in part because the family of q-isogeny graphs of supersingular elliptic curves in characteristic p (with prime p varying, for a fixed prime q) is Ramanujan. One tool for studying a graph is its Ihara zeta function, defined as an Euler product over the primes of the graph. Defining the zeta function formally requires a graph in the sense of Serre and Bass, i.e. a directed graph equipped with a fixed-point free involution on the edge set. In general, isogeny graphs fail to be graphs in this sense. In this talk, I will discuss joint work with Lau, Orvis, Scullard, and Zobernig in which we introduce abstract isogeny graphs along with their zeta functions; these graphs capture the combinatorial structure of supersingular isogeny graphs (with level structure) . I will survey some of our results, including an analogue of Ihara’s determinant formula, showing in particular that the zeta function is rational. We use this formula and the Eichler-Shimura relation to give a formula relating the zeta function of a q-isogeny graph with level-H structure (for certain H, including B0(N) and B1(N)) to the Hasse-Weil zeta functions of two associated modular curves over the finite field Fq, generalizing results of Hashimoto, Sugiyama, and Lei-Muller.