The automorphism group of the modular curve $X_0^*(N)$ with square-free level

TL;DR

This paper determines the automorphism group of the modular curve $X_0^*(N)$, formed by quotienting $X_0(N)$ with Atkin-Lehner involutions, for all square-free levels $N$, advancing understanding of modular curve symmetries.

Contribution

It provides a complete characterization of the automorphism groups of $X_0^*(N)$ for all square-free $N$, a previously unresolved classification.

Findings

Automorphism groups explicitly computed for all square-free $N$

New insights into symmetries of modular curves

Extension of known results to a broader class of levels

Abstract

We determine the automorphism group of the modular curve $X_0^*(N)$, obtained as the quotient of the modular curve $X_0(N)$ by the group of its Atkin-Lehner involutions, for all square-free values of $N$.