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 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.

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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