Rational points on \(X^ +_ 0(N)\) and quadratic \(\mathbb Q\)-curves.
type: Item
instance of: scholarly article
title: Rational points on \(X^ +_ 0(N)\) and quadratic \(\mathbb Q\)-curves.
zbMATH Open document ID: 1035.14008
author: Steven D. Galbraith
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2003-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93839
review text: Let \(N\) be an integer \(>1\) and \(X_0(N)\) be the modular curve whose non-cusp points classify the isomorphism classes of isogenies between elliptic curves \(\varphi:E\to E'\) of degree \(N\) with cyclic kernel. We denote by \(X^+_0(N)\) the quotient of \(X_0(N)\) by the Fricke involution \(W_N\). The curve \(X^+_0(N)\) has a model defined over \(\mathbb{Q}\). Rational points of \(X^+_0(N)\) which do not arise from elliptic curves with complex multiplication are called ``exceptional''. \textit{F. Momose} [J. Fac. Sci., Univ. Tokyo, Sect. 1 A Math. 33, 441--466 (1986; Zbl 0621.14018), J. Math. Soc. Japan 39, 269--286 (1987; Zbl 0623.14009)] has given interesting results on the \(\mathbb{Q}\)-rational points of \(X^+_0(N)\). The author of the paper under review gave a computational study of the case when \(N\) is a prime [\textit{S. D. Galbraith}, Exp. Math. 8, 311--318 (1999; Zbl 0960.14010)]. In this paper he continues his previous work studying some of the cases not covered by the results of Momose. Exceptional rational points are found in the cases \(N=91\), \(N=125\) and the \(j\)-invariants of the corresponding quadratic \(\mathbb{Q}\)-curves are obtained.
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 1943335
zbMATH Keywords: Heegner point
zbMATH Keywords: quadratic \(\mathbb{Q}\)-curves
zbMATH Keywords: \(j\)-invariants
zbMATH Keywords: modular curve
zbMATH Keywords: elliptic curves
zbMATH Keywords: exceptional rational points
reviewed by: Dimitros Poulakis
describes a project that uses: ecdata
MaRDI profile type: Publication
OpenAlex ID: W2083997496
cites work: Hecke operators on \(\Gamma_0(m)\)
cites work: Q4078161
cites work: Q4003857
cites work: Les Schémas de Modules de Courbes Elliptiques
cites work: Q4213383
cites work: Rational Points on XO+(p)
cites work: Constructing Isogenies between Elliptic Curves Over Finite Fields
cites work: Equations of hyperelliptic modular curves
cites work: Rational and elliptic parametrizations of \(\mathbb Q\)-curves
cites work: On the j -Invariants of the Quadratic Q -Curves
cites work: Arithmetic on elliptic curves with complex multiplication. With an appendix by B. Mazur
cites work: Q3347985
cites work: Q3335847
cites work: Table of quotient curves of modular curves \(X_ 0(N)\) with genus 2
cites work: \(\mathbb{Q}\)-curves over quadratic fields
cites work: On the Modular Curves X 0 (125), X 1 (25), X 1 (49)
cites work: Q4723876
cites work: Modular curves and the Eisenstein ideal
cites work: Q3758031
cites work: Rational points on the modular curves \(X^ +_ 0(N)\)
cites work: On normal forms of modular curves of genus 2
cites work: Q5181799
cites work: Hyperelliptic modular curves
cites work: Defining equations of modular curves \(X_ 0(N)\)
DOI: 10.5802/JTNB.354
description: scientific article; zbMATH DE number 1943335