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arXiv · 2203.05541

Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings

Nikola Adžaga·Shiva Chidambaram·Timo Keller·Oana Padurariu
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Abstract

We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this to compute the $\mathbb{Q}$-rational points on all of their modular coverings.

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Nikola Adžaga, Shiva Chidambaram, Timo Keller, Oana Padurariu. 2022-03-10. Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings. https://arxiv.org/abs/2203.05541

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