Zeta functions of abstract isogeny graphs and modular curves

Abstract

We introduce a ``non-orientable'' variation of Serre's definition of a graph, which we call an abstract isogeny graph. These objects capture the combinatorics of the graphs $G(p,\ell,H)$, the $\ell$-isogeny graphs of supersingular elliptic curves with $H$-level structure. In particular they allow for the study of non-backtracking walks, primes, and zeta functions. We prove an analogue of Ihara's determinant formula for the zeta function of an abstract isogeny graph. For $B_1(N) \subseteq H \subseteq B_0(N)$ and $p > 3$, we use this formula to relate the Ihara zeta function of $G(p,\ell,H)$ to the Hasse-Weil zeta functions of the modular curves $X_{H, {\mathbb{F}_{\ell}}}$ and $X_{H \times B_0(p), \mathbb{F}_{\ell}}$. As applications, we give an explicit formula relating point counts on $X_0(pN)_{\mathbb{F}_{\ell}}$ and $X_0(N)_{\mathbb{F}_{\ell}}$ to cycle counts in $G(p,\ell,B_0(N))$ and prove that the number of non-backtracking cycles of length $r$ in $G(p,\ell,B_0(N))$ is asymptotic to $\ell^r$.