Equations and rational points of the modular curves $$X_0^+(p)$$ X 0 + ( p )

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 to use this map to determine the rational points on $$X_0^+(p)$$ up to a large fixed height.