On the Zeta functions of supersingular isogeny graphs and modular curves

Abstract

Let $p$ and $q$ be distinct prime numbers, with $q\equiv 1(\mathrm{mod}12)$. Let $N$ be a positive integer that is coprime to $pq$. We prove a formula relating the Hasse--Weil zeta function of the modular curve $X_0(qN{)}_{{\mathbb{F}}_q}$ to the Ihara zeta function of the $p$-isogeny graphs of supersingular elliptic curves defined over $\overline{{\mathbb{F}}_q}$ equipped with a ${\Gamma }_0(N)$-level structure. When $N=1$, this recovers a result of Sugiyama.

minor changes, accepted for publication in Archiv der Mathematik