CM relations in fibered powers of elliptic families

Let \(E_\lambda\) be the Legendre family of elliptic curves. Given \(n\) linearly independent points \(P_1,\dots , P_n \in E_\lambda\left(\overline{\mathbb{Q}(\lambda)}\right)\) we prove that there are at most finitely many complex numbers \(\lambda_0\) such that \(E_{\lambda_0} \) has complex multi...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2018-07
Main Author: Barroero, Fabrizio
Format: Article
Language:English
Subjects:
Complex numbers
Curves
Multiplication
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Let \(E_\lambda\) be the Legendre family of elliptic curves. Given \(n\) linearly independent points \(P_1,\dots , P_n \in E_\lambda\left(\overline{\mathbb{Q}(\lambda)}\right)\) we prove that there are at most finitely many complex numbers \(\lambda_0\) such that \(E_{\lambda_0} \) has complex multiplication and \(P_1(\lambda_0), \dots ,P_n(\lambda_0)\) are dependent over \(End(E_{\lambda_0})\). This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber-Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over \(\overline{\mathbb{Q}}\).
ISSN:2331-8422
DOI:10.48550/arxiv.1611.01955