Elkies' conjecture on rational points of Atkin–Lehner quotients

Let $N$ be a positive integer, let $X_0(N)$ be the modular curve, and let $X_0(N)^*$ denote its quotient by the full group of Atkin–Lehner involutions. The notation $X_0(N)^*({\mathbb Q})$ denotes its rational points; among these, cusps are the points corresponding to degenerate elliptic-curve data, and CM points correspond to elliptic curves with complex multiplication.

Elkies' conjecture. For all integers $N\gg0$, the points in

$$X_0(N)^*({\mathbb Q})$$

are only CM points and cusps.

This conjecture predicts that, for sufficiently large level, rational points on the Atkin–Lehner quotient arise only from these two exceptional types. The paper establishes integrality results for the $j$-invariants of non-cuspidal rational points and classifies rational points in the finite cases of genus at most $5$, providing progress toward the conjecture; its general case remains open.