The modular automorphisms of quotient modular curves

TL;DR

This paper determines the automorphism groups of quotient modular curves of level N, revealing new automorphisms of order 3 and providing the full automorphism group for certain large N cases.

Contribution

It explicitly computes the modular automorphism groups of quotient modular curves and uncovers unexpected automorphisms not defined over Q.

Findings

Identified automorphism groups for quotient modular curves of level N.

Discovered non-expected automorphisms of order 3.

Determined the full automorphism group of X_0^*(N^2) for large N.

Abstract

We obtain the modular automorphism group of any quotient modular curve of level $N$, with $4,9\nmid N$. In particular, we obtain some non-expected automorphisms of order 3 that appear for the quotient modular curves when the Atkin-Lehner involution $w_{25}$ belongs to the quotient modular group, such automorphisms are not necessarily defined over $\mathbb{Q}$. As a consequence of the results, we obtain the full automorphism group of the quotient modular curve $X_0^*(N^2)$, for sufficiently large $N$.