## Quadratic Chabauty for Atkin–Lehner quotients of modular curves of prime level and genus 4, 5, 6

### Volume 208 / 2023

Acta Arithmetica 208 (2023), 15-49
MSC: Primary 14G05; Secondary 11G30, 11G18.
DOI: 10.4064/aa220110-7-3
Published online: 5 June 2023

#### Abstract

We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus 4, 5, and 6. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus 5 and 6. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197, 211,223,269,271,359$}.