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October 1, 2026: Semi-stable models, local heights and quadratic Chabauty for $X_0(N)^*$

With Nikola Adžaga and Maarten Derickx, we developed methods to compute local heights for quadratic Chabauty on the Atkin–Lehner quotient $X_0(N)^*$ when $N$ is squarefree. This complements our work on the non-squarefree case.

Why this matters. Elkies’ conjecture asks whether, for large enough level $N$, all $\mathbf{Q}$-points on the modular curve quotient $X_0(N)^*$ are cusps or CM points. These points correspond to certain multiquadratic points on $X_0(N)$, so the conjecture tells us something about cyclic isogenies of $\mathbf{Q}$-curves, i.e., of elliptic curves that are isogenous to all their Galois conjugates.

Why squarefree levels are hard. Assuming the Birch–Swinnerton-Dyer conjecture, the root number forces the Mordell–Weil rank of the Jacobian of $X_0(N)^*$ to be at least its genus when $N$ is squarefree (see my blog entry from August 12, 2022). So neither classical Chabauty–Coleman (which needs rank smaller than genus) nor formal immersions (which need a rank-$0$ quotient) can be used to prove the conjecture. Only recently has progress been made, by quadratic Chabauty. However, this method requires knowledge of the local heights at the primes of bad reduction, and these are hard to compute in general.

Our results. For squarefree $N$, we explain a strategy for constructing a global $p$-adic height for which all the contributions at the primes of bad reduction vanish, so they need not be computed at all. Our first main result is that such a $p$-adic height exists for all squarefree levels $N > 714$.

The idea. We first give an explicit description of the minimal regular model of $X_0(N)^*$ at the primes of bad reduction and show that it is semi-stable. Each component of this model gives rise to a linear condition which ensures that the local height contribution at that component vanishes. Using embeddings of quadratic orders into quaternion algebras, we bound the number of these conditions; once the genus of $X_0(N)^*$ exceeds this bound by more than one, there is enough freedom to choose a suitable correspondence, the key input of quadratic Chabauty, which simplifies the computations.

Applications. As an application, we determine the rational points on several curves $X_0(N)^*$ for which this was not done before. The code is available on GitHub.