Automorphisms of Cartan modular curves of prime and composite level

Dose, Valerio; Lido, Guido; Mercuri, Pietro

doi:10.2140/ant.2022.16.1423

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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

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.

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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