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2022 Automorphisms of Cartan modular curves of prime and composite level
Valerio Dose, Guido Lido, Pietro Mercuri
Algebra Number Theory 16(6): 1423-1461 (2022). DOI: 10.2140/ant.2022.16.1423

Abstract

We study the automorphisms of modular curves associated to Cartan subgroups of GL2 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 p13: the curve Xns+(p) has no nontrivial automorphisms, whereas the curve Xns(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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Valerio Dose. Guido Lido. Pietro Mercuri. "Automorphisms of Cartan modular curves of prime and composite level." Algebra Number Theory 16 (6) 1423 - 1461, 2022. https://doi.org/10.2140/ant.2022.16.1423

Information

Received: 12 October 2020; Revised: 5 August 2021; Accepted: 5 October 2021; Published: 2022
First available in Project Euclid: 2 October 2022

Digital Object Identifier: 10.2140/ant.2022.16.1423

Subjects:
Primary: 11G05 , 11G15 , 11G18 , 11G30 , 14G35

Keywords: automorphisms , Complex Multiplication , Elliptic curves , modular curves

Rights: Copyright © 2022 Mathematical Sciences Publishers

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Vol.16 • No. 6 • 2022
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