Equations and Rational Points of the Modular Curves $X^+_0(p)$

TL;DR

This paper develops methods to explicitly compute equations and rational points on the modular curves $X_0^+(p)$, enhancing understanding of their structure and rational solutions for primes $p$.

Contribution

It provides explicit equations for the canonical models of $X_0^+(p)$ and algorithms to compute modular parametrizations and rational points.

Findings

Explicit equations for $X_0^+(p)$ were obtained.

Algorithms for modular parametrization were developed.

Rational points on $X_0^+(p)$ were determined up to large heights.

Abstract

Let $p$ be an odd prime number and let $X_0^+(p)$ be the quotient of the classical modular curve $X_0(p)$ by the action of the Atkin-Lehner operator $w_p$. In this paper we show how to compute explicit equations for the canonical model of $X_0^+(p)$. Then we show how to compute the modular parametrization, when it exists, from $X_0^+(p)$ to an isogeny factor $E$ of dimension 1 of its jacobian $J_0^+(p)$. Finally we show how use this map to determine the rational points on $X_0^+(p)$ up to a large fixed height.