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Bielliptic modular curves $ X_0^*(N)$ with square-free levels

Authors: Francesc Bars and Josep González Rovira
Journal: Math. Comp. 88 (2019), 2939-2957
MSC (2010): Primary 11G18, 11G30; Secondary 14G05, 14H37
Published electronically: April 9, 2019
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Abstract: Let $ N\geq 1$ be a square-free integer such that the modular curve $ X_0^*(N)$ has genus $ \geq 2$. We prove that $ X_0^*(N)$ is bielliptic exactly for $ 19$ values of $ N$, and we determine the automorphism group of these bielliptic curves. In particular, we obtain the first examples of nontrivial $ \operatorname {Aut}(X_0^*(N))$ when the genus of $ X_0^*(N)$ is $ \geq 3$. Moreover, we prove that the set of all quadratic points over $ \mathbb{Q}$ for the modular curve $ X_0^*(N)$ with genus $ \geq 2$ and $ N$ square-free is not finite exactly for $ 51$ values of $ N$.

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Additional Information

Francesc Bars
Affiliation: Departament Matemàtiques, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia; and BGSMath (at CMR), Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Catalonia

Josep González Rovira
Affiliation: Departament de Matemàtiques, Universitat Politècnica de Catalunya EPSEVG, Avinguda Víctor Balaguer 1, 08800 Vilanova i la Geltrú, Catalonia

Received by editor(s): June 6, 2018
Received by editor(s) in revised form: August 1, 2018, October 23, 2018, and December 31, 2018
Published electronically: April 9, 2019
Additional Notes: The first author was supported by MTM2016-75980-P and MDM-2014-0445
The second author was partially supported by DGI grant MTM2015-66180-R
Article copyright: © Copyright 2019 American Mathematical Society