Equations and rational points of the modular curves X_0^+(p)

Abstract

Let p be an odd prime number and let X+0 (p) be the quotient of the classical modular curve X0(p) by the action of the Atkin-Lehner operator wp. 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.