A Note on the Rational Points of $X_0^+(N)$

Abstract

Let $C$ be the image of a canonical embedding $\phi$ of the Atkin--Lehner quotient $X_0^+(N)$ associated with the Fricke involution $w_N$. In this note we exhibit some relations among the rational points of $C$. For each $g=3$ (respectively the first $g=4$) curve $C$ we found that there are one or more lines (respectively planes) in $\PP^{g-1}$ whose intersection with $C$ consists entirely of rational Heegner points or the cusp point, where $N$ is prime. We also discuss an explanation of the first nonhyperelliptic exceptional rational point.