Bielliptic modular curves $X_0^*(N)$ with square-free levels
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- by Francesc Bars and Josep González Rovira;
- Math. Comp. 88 (2019), 2939-2957
- DOI: https://doi.org/10.1090/mcom/3424
- 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 $N$.References
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Bibliographic 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
- MR Author ID: 647724
- ORCID: 0000-0003-4779-3995
- Email: francesc@mat.uab.cat
- Josep González Rovira
- Affiliation: Departament de Matemàtiques, Universitat Politècnica de Catalunya EPSEVG, Avinguda Víctor Balaguer 1, 08800 Vilanova i la Geltrú, Catalonia
- MR Author ID: 319937
- Email: josep.gonzalez@upc.edu
- 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 - © Copyright 2019 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 2939-2957
- MSC (2010): Primary 11G18, 11G30; Secondary 14G05, 14H37
- DOI: https://doi.org/10.1090/mcom/3424
- MathSciNet review: 3985482