Equations and Rational Points of the Modular Curves X⁺₀(p)

2016/07/15 by Pietro Mercuri, Mercuri, Pietro · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1607.04558

arxiv created 2016/07/15 · openalex publication_date 2016/07/15 · arxiv updated 2016/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime number and let X₀⁺(p) be the quotient of the classical modular curve X₀(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 X₀⁺(p). Then we show how to compute the modular parametrization, when it exists, from X₀⁺(p) to an isogeny factor E of dimension 1 of its jacobian J₀⁺(p). Finally we show how use this map to determine the rational points on X₀⁺(p) up to a large fixed height.

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• Maximal curves over finite fields and a modular isogeny

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