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

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.

Computing quadratic points on modular curves 𝑋₀(𝑁)

“…We briefly outline the method used in [34], which uses a type of Mordell-Weil sieve to prove that a given list of quadratic points on X 0 (N ) is complete. Let P 0 ∈ X 0 (N )(Q) denote a rational cusp and let ι : X 0 (N ) (2) (Q) → J 0 (N )(Q) denote the Abel-Jacobi map with basepoint 2P 0 , which is injective since X 0 (N ) is non-hyperelliptic. Suppose that…”

“…In the case N = 80 we proceed similarly, but instead compute a (finite) supergroup of J 0 (N )(Q) by considering the group structure of J 0 (N )(F p ) for some odd primes p. This allowed us to prove that J 0 (N )(Q)/C 0 (N )(Q) is isomorphic to a subgroup of (Z/2Z) 2 . □ Proof of Proposition 3.2.…”

“…We write X = X 0 (N ) and work with rational points on the symmetric square X (2) of this curve (as described in Section 3.1). Let D ∞ be a sum of two rational cusps such that w d acts trivially on D ∞ (e.g.…”

“…Given a hypothetical unknown quadratic point Q ∈ X (2) (Q) that does not map to a rational point on X/w d , we first compute a set S Q ⊆ X (2) (F p ) of possibilities for red p (Q). In order to construct the set S Q , we consider each point red p (R) ∈ X (2) (F p ) that is the reduction of a known non-pullback point R ∈ X (2) (Q) with respect to w d . We attempt to prove that red p (Q) ̸ = red p (R) by applying a symmetric Chabauty criterion as stated in [10,Theorem 2.1].…”

Quadratic Chabauty for modular curves: algorithms and examples

“…Arul and Müller [AM23] also compute the rational points on using the same method. Adžaga, Chidambaram, Keller and Padurariu [ACKP22] use several techniques, including quadratic Chabauty, to determine the set of rational points on the hyperelliptic Atkin–Lehner star quotient curves .…”