Density of positive rank fibers in elliptic fibrations
type: Item
instance of: scholarly article
title: Density of positive rank fibers in elliptic fibrations
zbMATH Open document ID: 1126.11031
author: Ritabrata Munshi
published in: Journal of Number Theory
publication date: 2007-06-06T00:00:00Z
review text: A one-parameter elliptic fibration over a field \(K\) is a familly of elliptic curves \(E_r: Y^2=X^3+A(r)X+B(r)\) where \(A\) and \(B\) are polynomials with coefficients in \(K\). A conjecture of Mazur states that, for \(K={\mathbb Q}\), the set \(\{r\in{\mathbb Q}\,| \, \text{rank}(E_r({\mathbb Q})>0\}\) is either dense in \({\mathbb R}\) or finite. The conjecture is known to be true under the parity conjecture. Also, there are unconditional results of Rohrlich and Kuwata-Wang proving the conjecture for the quadratic twists of a fixed elliptic curve by a quadratic or a cubic polynomial. In the paper under review the Mazur conjecture is settled affirmatively for all general quadratic and cubic filtrations, i.e., whenever \(\max(\deg A,\deg B)\leq 3\). By the method of proof, the same result holds when \({\mathbb Q}\) is replaced by any real number field.
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 5161947
zbMATH Keywords: ellipric pencil
zbMATH Keywords: rational points
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.10.019
OpenAlex ID: W2001809585
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DOI: 10.1016/J.JNT.2006.10.019
description: scientific article; zbMATH DE number 5161947