Automorphisms of Cartan modular curves of prime and composite level
| Autor: | Dose, Valerio, Lido, Guido, Mercuri, Pietro |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Alg. Number Th. 16 (2022) 1423-1461 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/ant.2022.16.1423 |
| Popis: | We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ 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 non-split curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one non-trivial 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. Comment: 36 pages, 4 tables. Some proofs rely on MAGMA scripts available at https://github.com/guidoshore/automorphisms_of_Cartan_modular_curves |
| Databáze: | arXiv |
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