A construction of everywhere good \(\mathbb{Q}\)-curves with \(p\)-isogeny
type: Item
instance of: scholarly article
title: A construction of everywhere good \(\mathbb{Q}\)-curves with \(p\)-isogeny
zbMATH Open document ID: 0922.14021
author: Atsuki Umegaki
published in: Tokyo Journal of Mathematics
publication date: 1999-01-24T00:00:00Z
review text: Let \(E\) be an elliptic curve defined over the algebraic closure \(\overline{\mathbb Q}\) of the field of rational numbers \(\mathbb Q\). We call \(E\) a \(\mathbb Q\)-curve if \(E\) and its Galois conjugate \(E^{\sigma}\) are isogenous over \(\overline{\mathbb Q}\) for any \(\sigma\in\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\). It is conjectured by Ribet that \(\mathbb Q\)-curves are modular in the sense that each should be a factor over \(\overline{\mathbb Q}\) of the jacobian of the modular curve \(X_1(N)\) for some \(N\). In this paper, a \(\mathbb Q\)-curve \(E\) with the following properties is considered: For a real quadratic field \(K\) and a rational prime \(p\), (1) \(E\) is defined over \(K\), (2) \(E\) has everywhere good reduction over \(K\), and (3) \(E\) has an isogeny to its conjugate \(E^{\sigma}\) of degree \(p\). A method of constructing \(\mathbb Q\)-curves satisfying the above three properties is described. Then several examples of such \(\mathbb Q\)-curves are presented, and their modularity is established.
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14K02
zbMATH DE Number: 1207093
zbMATH Keywords: \({\mathbb Q}\)-curves
zbMATH Keywords: isogeny
zbMATH Keywords: good reduction
zbMATH Keywords: elliptic curve
zbMATH Keywords: jacobian of the modular curve
zbMATH Keywords: modularity of \({\mathbb Q}\)-curves
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270041995
OpenAlex ID: W2040396278
DOI: 10.3836/TJM/1270041995
description: scientific article; zbMATH DE number 1207093