Heegner points on modular curves
type: Item
instance of: scholarly article
zbMATH Open document ID: 0997.11044
publication date: 2002-11-17T00:00:00Z
full work available at URL: https://eudml.org/doc/42413
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 1693176
zbMATH Keywords: Heegner triplet
zbMATH Keywords: modular curve
zbMATH Keywords: integral quadratic forms
zbMATH Keywords: number of representations
zbMATH Keywords: genera of binary quadratic forms
MaRDI profile type: Publication
author: A. Arenas
author: Pilar Bayer
title: Heegner points on modular curves
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales (España)
review text: Let \({\mathfrak O}\) be an order of a quadratic field of discriminant \(D\) and fix an integer \(N\geq 1\). We denote by \(\text{Pic}^+({\mathfrak O})\) the abelian group of invertible fractional \({\mathfrak O}\)-ideals modulo the principal ones defined by generators of positive norm. In this paper the authors define a Heegner triplet of type \((N,D)\) as a triplet \(({\mathfrak O},{\mathfrak n},[{\mathfrak a}])\), where \({\mathfrak n}\) is a primitive \({\mathfrak O}\)-ideal of norm \(N\) and \([{\mathfrak a}]\) an element of \(\text{Pic}^+({\mathfrak O})\), and prove that this concept generalizes the concept of Heegner point on the modular curve \(X_0(N)\). Let \({\mathfrak H}(N,D)\) be the set of integral quadratic forms of type \(aNX^2+ bXY+ cY^2\), with \(D= b^2-4Nac\), \(\gcd(aN,b,c)= \gcd(a,b,Nc)=1\). Furthermore, the authors prove that the group NEWLINE\[NEWLINE\Gamma_0(N)= \left\{ \left[ \begin{smallmatrix} \alpha&\beta\\ \gamma&\delta \end{smallmatrix} \right]\in \text{SL}(2,\mathbb{Z})\mid \gamma\equiv 0\pmod N\right\}NEWLINE\]NEWLINE operates on \({\mathfrak H}(N,D)\), the quotient \(H(N,D)= {\mathfrak H}(N,D)/ \Gamma_0(N)\) is a finite set and there is a bijection between \(H(N,D)\) and the set of pairs \(({\mathfrak n},[{\mathfrak a}])\). Moreover, when \(D<0\), they obtain a formula for the number of elements of \(H(N,D)\) in terms of the number of representations by genera of binary quadratic forms of discriminant \(D\).
reviewed by: Dimitros Poulakis
description: scientific article; zbMATH DE number 1693176