Automorphisms of hyperelliptic modular curves $X_0(N)$ in positive characteristic

TL;DR

This paper investigates the automorphism groups of hyperelliptic modular curves $X_0(N)$ when reduced modulo primes not dividing N, focusing on their structure in positive characteristic.

Contribution

It provides new insights into the automorphism groups of $X_0(N)$ over algebraic closures of finite fields, especially in the hyperelliptic case, extending understanding in positive characteristic.

Findings

Automorphism groups vary with prime p and level N.

Identification of conditions under which automorphisms extend to reductions.

Classification of automorphism groups for certain hyperelliptic $X_0(N)$.

Abstract

We study the automorphism groups of the reduction $X_0(N) \times \bar{\mathbb{F}}_p$ of a modular curve $X_0(N)$ over primes $ p\nmid N$.