#### Vol. 16, No. 6, 2022

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Automorphisms of Cartan modular curves of prime and composite level

### Valerio Dose, Guido Lido and Pietro Mercuri

Vol. 16 (2022), No. 6, 1423–1461
##### Abstract

We study the automorphisms of modular curves associated to Cartan subgroups of ${\mathrm{GL}}_{2}$ 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 nonsplit curves of prime level $p\ge 13$: the curve has no nontrivial automorphisms, whereas the curve has exactly one nontrivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of ${X}_{0}^{\ast }\left(n\right):={X}_{0}\left(n\right)∕W$, where $W$ is the group generated by the Atkin–Lehner involutions of ${X}_{0}\left(n\right)$ and $n$ is a large enough square.

##### Keywords
modular curves, elliptic curves, complex multiplication, automorphisms
##### Mathematical Subject Classification
Primary: 11G05, 11G15, 11G18, 11G30, 14G35
##### Milestones
Received: 12 October 2020
Revised: 5 August 2021
Accepted: 5 October 2021
Published: 27 September 2022
##### Authors
 Valerio Dose Dipartimento di Ingegneria Informatica Automatica e Gestionale Università di Roma Sapienza Rome Italy Guido Lido Università di Roma Tor Vergata Rome Italy Pietro Mercuri Università di Roma Sapienza Rome Italy