On the Zeta functions of supersingular isogeny graphs and modular curves

TL;DR

This paper establishes a formula connecting the Hasse--Weil zeta function of certain modular curves over finite fields to the Ihara zeta function of supersingular isogeny graphs, extending previous results to new cases.

Contribution

It provides a new explicit relation between zeta functions of modular curves and supersingular isogeny graphs, generalizing Sugiyama's earlier work.

Findings

Derived a formula linking Hasse--Weil and Ihara zeta functions.

Extended known results to modular curves with level structure.

Confirmed the relation for specific prime and level conditions.

Abstract

Let $p$ and $q$ be distinct prime numbers, with $q\equiv 1\pmod{12}$. 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.