Differential Galois realization of double covers
type: Item
instance of: scholarly article
title: Differential Galois realization of double covers
zbMATH Open document ID: 1017.12005
author: Teresa Crespo
author: Zbigniew Hajto
publication date: 2002-09-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116000
review text: Let \(G\) be one of the alternating groups \(A_4\), \(A_5\), or the symmetric group \(S_4\), and let \(2G\) be one of its double covers. Let \(k\) be a differential field of characteristic \(0\) with algebraically closed field of constants \(\mathcal C\). The authors are interested in exhibiting \(2G\) as the (polynomial or differential) Galois group of a field extension \(\widetilde{K} \supset k\). More precisely, they want to produce explicit second order differential equations over \(k\) whose differential Galois group is \(2G\). If there is an extension \(\widetilde{K}\), it contains a subextension \(K \supset k\) with Galois group \(G\). Thus they phrase their main result as follows: with \(k\), \(G\), \(2G\) as above, suppose that \(K \supset k\) is the splitting field of a polynomial \(P(x) \in k[X]\) with Galois group \(G\). Then there is a quadratic form \(Q\) in three variables over \(k\) such that the Galois embedding problem \( 2G \to G = \text{Gal}(K/k)\) is solvable if and only if \(Q\) represents \(0\) (nontrivially) over \(k\). Then for each such representation there are explicitly computable elements \(A,B,C\) of \(k\) such that \(Y^{\prime \prime \prime}+AY^{\prime \prime}+BY^\prime+CY=0\) has \(K\) as a differential Galois extension, while \(Y^{\prime \prime}+(A/3)Y'+(1/4)(B-(2/9)A^2-(1/3)A')Y=0\) has differential Galois group \(2G\). For the case that \(k=\overline{\mathbb Q}(T)\), they exhibit the constants \(A,B,C\) (and hence the corresponding differential equations) starting from explicit choices for \(P(X)\).
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 12F12
zbMATH DE Number: 1794821
zbMATH Keywords: Picard-Vessiot extension
zbMATH Keywords: differential Galois group
zbMATH Keywords: alternating groups
zbMATH Keywords: symmetric group
zbMATH Keywords: double covers
zbMATH Keywords: differential field of characteristic 0
reviewed by: Andy R. Magid
MaRDI profile type: Publication
OpenAlex ID: W2325628558
cites work: Finite linear groups as differential Galois groups.
cites work: Primitive unimodular groups of degree 2 as differential Galois groups
cites work: Double covers as differential Galois groups
cites work: Q3255437
cites work: Q4321755
cites work: Q4263461
cites work: The Witt invariant of the form \(\text{Tr}(x^ 2)\)
cites work: Galois Cohomology
cites work: Q3197951
cites work: Galois groups of second and third order linear differential equations
DOI: 10.5802/AIF.1908
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1794821