Rational points on $X_0(N)^*$ and local heights in quadratic ChabautyMPIM

by Nikola Adžaga (University of Zagreb/z. Z. MPIM)

Wednesday Nov 4, 2026, 2:30 PM → 3:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

Rational points on the full Atkin--Lehner quotient $X_0(N)^*$ correspond to $\mathbb{Q}$-curves, and Elkies conjectured that $X_0(N)^*$ has exceptional (non-cuspidal, non-CM) rational points for only finitely many levels $N$. Quadratic Chabauty has become the main tool for provably determining the rational points on these curves. After briefly surveying joint work on the hyperelliptic curves $X_0(N)^*$ and on $X_0^+(p)$ of genus 4--6, I will focus on recent joint work with Maarten Derickx and Timo Keller. That work removes the main obstacle in these computations, namely the determination of local heights at primes of bad reduction. For squarefree $N$, we describe the dual graphs of semi-stable regular models of $X_0(N)^*$. Their long cycles impose linear conditions on the correspondence, and we bound the number of these conditions using embeddings of quadratic orders into quaternion algebras. As a result, for all but finitely many squarefree $N$, we can choose a correspondence for which the local heights at all primes of bad reduction vanish. As an application, we determine the rational points on several new curves $X_0(N)^*$.