Exceptional points on Atkin--Lehner quotients

Abstract

We study the rational points on the star curve $X_0^*(N) := X_0(N)/W(N)$, the quotient of the classical modular curve $X_0(N)$ by the full group of Atkin--Lehner involutions, for squarefree levels $N$. Rational points on $X_0^*(N)$ parameterize $\mathbb{Q}$-curves, i.e.\ elliptic curves $E/\overline{\mathbb{Q}}$ that are isogenous to all of their Galois conjugates. Elkies conjectures that $X_0^*(N)$ has only CM or cuspidal rational points for all large enough $N$. We call any other rational points "exceptional". In this article, we provide new examples of exceptional points in genus 3 and 4, and we give evidence that no exceptional points exist in genus $g \geq 5$. Moreover, we investigate the underlying geometric reasons that might "explain" why these exceptional points arise in the first place, in the vein of Ogg and Mazur. In particular, we propose geometric explanations for Galbraith's exceptional points on $X_0^*(137)$ and $X_0^*(311)$.