Supersimplicity and quadratic extensions
type: Item
instance of: scholarly article
title: Supersimplicity and quadratic extensions
zbMATH Open document ID: 1163.03019
published in: Archive for Mathematical Logic
publication date: 2009-03-17T00:00:00Z
review text: The paper deals with the algebraic characterization of supersimple fields. Pillay conjectured that such a field \(K\) is perfect, bounded and pseudo algebraically closed, and with Poizat proved both perfection and boundedness. Thus it remains to prove or disprove pseudo algebraic closedness, in other words that every absolutely irreducible plane curve defined over \(K\) has a \(K\)-rational point. The paper shows that if \(K\) is a supersimple field admitting exactly one extension of degree 2 (up to isomorphism), then every elliptic curve \(E\) defined over \(K\) has an \(s\)-generic \(K\)-rational point, i.e., a point \(P\) in \(E(K)\) such that the \(SU\)-rank of \(P\) over \(F\) equals the \(SU\)-rank of \(K\), where \(F\) is some small set of parameters over which \(E\) is defined.
reviewed by: Carlo Toffalori
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 5529379
zbMATH Keywords: Supersimple field
zbMATH Keywords: Elliptic curve
zbMATH Keywords: Generic point
author: Amador Martin-Pizarro
author: F. O. Wagner
MaRDI profile type: Publication
full work available at URL: s00153-008-0115-1
OpenAlex ID: W1996609381
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DOI: 10.1007/S00153-008-0115-1
description: scientific article; zbMATH DE number 5529379