Automorphisms of Cartan modular curves of prime and composite level

TL;DR

This paper investigates the automorphisms of Cartan modular curves, establishing conditions under which all automorphisms are geometric and determining automorphism groups for specific prime and composite levels.

Contribution

It proves that for large enough levels, all automorphisms are induced by the covering of the upper half-plane, and computes automorphism groups for certain modular curves.

Findings

Non-split curves of prime level p ≥ 13 have limited automorphisms.

The curve X_ns^+(p) has no non-trivial automorphisms.

The curve X_ns(p) has exactly one non-trivial automorphism.

Abstract

We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for non-split curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one non-trivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^*(n):=X_0(n)/W$, where $W$ is the group generated by the Atkin-Lehner involutions of $X_0(n)$ and $n$ is a large enough square.