Elkies finiteness conjecture for exceptional points of bounded degree

Prove that, for every fixed integer d greater than or equal to1, only finitely many squarefree levels N have an exceptional point on X_0^*(N) defined over a number field of degree d, where an exceptional point is a rational or bounded-degree point that is neither cuspidal nor CM.

Background

The paper studies the star curves X_0*(N)=X_0(N)/W(N) for squarefree N. Their non-cuspidal points correspond to Q-curves and, more generally, to K-curves over number fields. Special points consist of cusps and CM points, while exceptional points are those that are not special.

The authors present Elkies's conjecture as a generalization of Serre's uniformity conjecture to discrete arithmetic subgroups of PGL_2+(Q). The conjecture predicts finiteness of levels supporting exceptional points of any fixed degree and motivates the computational search and geometric analysis carried out in the paper.

References

Elkies conjectures that for any fixed $d \geq 1$, there are finitely many $N$ such that $X_0*(N)$ has an exceptional point over a number field of degree $d$.

— Exceptional points on Atkin--Lehner quotients  (2609.00516 - Assaf et al., 1 Sep 2026) in Section 1, Introduction

In this section we give some evidence for a stronger boundedness conjecture.

\begin{conjecture}[Explicit boundedness of $Q$-curves] Let $N>0$ be a squarefree integer. If the genus of $X_0*(N)$ is at least 5, then the rational points are all special. \end{conjecture}

— Exceptional points on Atkin--Lehner quotients  (2609.00516 - Assaf et al., 1 Sep 2026) in Conjecture 2.1, Section 2, subsection "Exceptional points on Atkin--Lehner quotients of modular curves"