Title:
Rational points on hyperelliptic Atkin-Lehner quotients of modular
curves and their coverings
Abstract:
We complete the computation of all \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 \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.