Bounds for the torsion of elliptic curves over number fields
type: Item
instance of: scholarly article
title: Bounds for the torsion of elliptic curves over number fields
zbMATH Open document ID: 0936.11037
published in: Inventiones Mathematicae
publication date: 1996-05-29T00:00:00Z
review text: Let \(p\) be a prime number and \(X_0(p)\) be the modular curve associated to the group \(\Gamma_0(p)= \left\{\left( \begin{smallmatrix} a &b\\ c &d\end{smallmatrix} \right)\in \text{SL}_2(\mathbb{Z}):c\equiv 0\pmod p \right\}\). Let \(J_0(p)\) be the Jacobian variety of \(X_0(p)\). We put \(H= H_1(X_0(p), \mathbb{Z})\) and we denote by \(T\) the subring of \(\text{End} (J_0(p))\) generated by the Hecke operators \(T_r\) and the Atkin-Lehner involution. Let \({\mathcal T}_e\) be the kernel of the morphism of \(T\)-modules \(T\to H\otimes \mathbb{Q}\) defined by \(t\to te\), where \(e\) is the winding element. The image of the map \({\mathcal T}_e\times J_0(p)\to J_0(p)\) defined by \((t,x)\to tx\) generates an abelian subvariety of \(J_0(p)\) which we denote by \({\mathcal T}_e J_0(p)\). We define the winding quotient \(J_e\) as the quotient \(J_0(p)/{\mathcal T}_e J_0(p)\) which is an abelian variety defined over \(\mathbb{Q}\). The author proves that the group \(J_e(\mathbb{Q})\) is finite and the elements \(T_1e,\dots, T_d e\) are linearly independent in \(H\otimes \mathbb{Q}\) provided \(p> d^{3d^2}\). Finally, combining these two results he proves the following interesting theorem: Let \(E\) be an elliptic curve defined over a number field \(K\) of degree \(d>1\) over \(\mathbb{Q}\). If \(E(K)\) has a point of prime order \(p\), then \(p<d^{3d^2}\).
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 866098
zbMATH Keywords: Hecke algebra
zbMATH Keywords: Eisenstein element
zbMATH Keywords: modular curve
zbMATH Keywords: Jacobian variety
zbMATH Keywords: elliptic curve
author: Loïc Merel
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
full work available at URL: s002220050059
OpenAlex ID: W2082955605
Wikidata QID: Q55899081
DOI: 10.1007/S002220050059
description: scientific article; zbMATH DE number 866098