Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6

TL;DR

This paper applies quadratic Chabauty to compute all rational points on certain modular curve quotients of genus 4 to 6, identifying only specific levels with exceptional points and confirming their absence in higher genus cases.

Contribution

It demonstrates the effectiveness of quadratic Chabauty in explicitly determining rational points on Atkin-Lehner quotients of modular curves of prime level and genus 4 to 6, with complete classifications.

Findings

Only levels 137 and 311 have exceptional rational points.

No exceptional rational points on curves of genus five and six.

Complete rational point determination for specified levels.

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 four, five, and six. 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 five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.