Automorphisms of Cartan modular curves of prime and composite level

Dose, Valerio; Lido, Guido; Mercuri, Pietro

doi:10.2140/ant.2022.16.1423

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Automorphisms of Cartan modular curves of prime and composite level

Valerio Dose, Guido Lido and Pietro Mercuri

Vol. 16 (2022), No. 6, 1423–1461

DOI: 10.2140/ant.2022.16.1423

Abstract

We study the automorphisms of modular curves associated to Cartan subgroups of ${GL}_{2}$ 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 $p \geq 13$ : the curve $X_{ns}^{+} (p)$ has no nontrivial automorphisms, whereas the curve $X_{ns} (p)$ has exactly one nontrivial 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.

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

Vol. 16, No. 6, 2022