an:06587935
Zbl 1344.34096
Crespo, Teresa; Hajto, Zbigniew; van der Put, Marius
Real and \(p\)-adic Picard-Vessiot fields
EN
Math. Ann. 365, No. 1-2, 93-103 (2016).
00355583
2016
j
12H20 12H25 34M50 11E10 11R34
Picard-Vessiot field; differential module; field of constants; field extension
The authors consider differential modules \(M\) over real and \(p\)-adic differential fields \(K\) (with fields of constants \(k\) real closed or \(p\)-adic closed). A Picard-Vessiot field (PVF) \(L\) for \(M/K\) is a field extension of \(K\) with the same field of constants \(k\), where \(L\) is equipped with a differentiation extending the one of \(K\) and there exists an invertible \(d\times d\)-matrix \(F\) (\(d=\)dim\(M\)) with entries in \(L\) satisfying \(F'=AF\), as a field \(L\) being generated over \(K\) by the entries of \(F\). Using results of J.-P. Serre and P. Deligne, the authors obtain a purely algebraic proof of the existence and unicity of PVFs. They treat the inverse problem for real forms of a semisimple group and they give examples illustrating the relations between differential modules, PVFs and real forms of a linear algebraic group.
Vladimir P. Kostov (Nice)