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

Abstract

Abstract We complete the computation of all

$$\mathbb {Q}$$

Q

-rational points on all the 64 maximal Atkin-Lehner quotients

$$X_0(N)^*$$

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 (from a forthcoming preprint of the fourth-named author) combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all

$$\mathbb {Q}$$

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

$$\mathbb {Q}$$

Q

-rational points on all of their modular coverings.