Papers › Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings
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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We complete the computation of all ℚ-rational points on all the $64$ maximal Atkin-Lehner quotients X₀(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 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 ℚ-rational points on all of their modular coverings.
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