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

authored by

Nikola Adzaga, Shiva Chidambaram, Timo Keller, Oana Padurariu

Abstract

We complete the computation of all Q-rational points on all the 64 maximal Atkin-Lehner quotients X(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 (from a forthcoming preprint of the fourth-named author) 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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

University of Zagreb
Massachusetts Institute of Technology
University of Bayreuth
Boston University (BU)

Type

Article

Journal

Research in Number Theory

Volume

8

Publication date

12.10.2022

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Electronic version(s)

https://doi.org/10.1007/s40993-022-00388-9 (Access: Open)