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The Ramanujan Journal

, Volume 47, Issue 2, pp 291–308 | Cite as

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

  • Pietro Mercuri
Article

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.

Keywords

Modular curve Modular parametrization 

Mathematics Subject Classification

Primary: 11G18 Secondary: 14H25 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Sapienza UniversityRomeItaly

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