Vol. 16, No. 6, 2022

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 8, 1777–2003
Issue 7, 1547–1776
Issue 6, 1327–1546
Issue 5, 1025–1326
Issue 4, 777–1024
Issue 3, 521–775
Issue 2, 231–519
Issue 1, 1–230

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
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
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 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 X0(n) := X0(n)W, where W is the group generated by the Atkin–Lehner involutions of X0(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