## Quadratic points on bielliptic modular curves

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Filip Najman and Borna Vukorepa
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## Abstract:

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus $2\leq g(X_0(n)) \leq 5$. Since all the hyperelliptic curves $X_0(n)$ are of genus $\leq 5$ and as a curve can have infinitely many quadratic points only if it is either of genus $\leq 1$, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves $X_0(n)$ naturally arises; this question has recently also been posed by Mazur.

We answer Mazur’s question completely and describe the quadratic points on all the bielliptic modular curves $X_0(n)$ for which this has not been done already. The values of $n$ that we deal with are $n=60$, $62$, $69$, $79$, $83$, $89$, $92$, $94$, $95$, $101$, $119$ and $131$; the curves $X_0(n)$ are of genus up to $11$. We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box’s relative symmetric Chabauty method and an application of a moduli description of $\mathbb {Q}$-curves of degree $d$ with an independent isogeny of degree $m$, which reduces the problem to finding the rational points on several quotients of modular curves.

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## Additional Information

**Filip Najman**- Affiliation: University of Zagreb, Bijenička Cesta 30, 10000 Zagreb, Croatia
- MR Author ID: 886852
- ORCID: 0000-0002-0994-0846
- Email: fnajman@math.hr
**Borna Vukorepa**- Affiliation: University of Zagreb, Bijenička Cesta 30, 10000 Zagreb, Croatia
- ORCID: 0000-0002-9560-9032
- Email: borna.vukorepa@math.hr
- Received by editor(s): December 21, 2021
- Received by editor(s) in revised form: April 13, 2022
- Published electronically: February 9, 2023
- Additional Notes: This work was supported by the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund – the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004) and by the Croatian Science Foundation under the project no. IP-2018-01-1313.
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp.
**92**(2023), 1791-1816 - MSC (2020): Primary 11G05, 14G05, 11G18
- DOI: https://doi.org/10.1090/mcom/3805
- MathSciNet review: 4570342