Galbraith's exceptional-point conjecture for Atkin–Lehner quotients

For an integer $N>1$, let $w_N$ denote the Atkin–Lehner involution on the modular curve $X_0(N)$, and let

$$X^+_0(N)=X_0(N)/\langle w_N\rangle$$

be the Atkin–Lehner quotient. Write $g^+_N$ for the genus of $X^+_0(N)$. A non-cuspidal rational point on $X^+_0(N)$ is exceptional if the associated elliptic curves do not have complex multiplication.

Galbraith's conjecture. If $2\leq g^+_N\leq 5$, then $X^+_0(N)$ contains exceptional rational points if and only if

$$N\in\{73,91,103,125,137,191,311\}.$$

The conjecture predicts that exceptional rational points, equivalently certain quadratic $\mathbb{Q}$-curves without complex multiplication, occur only at these levels among the specified genera. The paper states that its computation of $X^+_0(125)(\mathbb{Q})$, together with work of several authors, completes the proof of Galbraith's conjecture.