Shuji Morikawa, Hiroshi Umemura (2009)
Annales de l’institut Fourier
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We apply the General Galois Theory of difference equations introduced in the first part to concrete examples. The General Galois Theory allows us to define a discrete dynamical system being infinitesimally solvable, which is a finer notion than being integrable. We determine all the infinitesimally solvable discrete dynamical systems on the compact Riemann surfaces.
Shuji Morikawa (2009)
Annales de l’institut Fourier
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We know well difference Picard-Vessiot theory, Galois theory of linear difference equations. We propose a general Galois theory of difference equations that generalizes Picard-Vessiot theory. For every difference field extension of characteristic , we attach its Galois group, which is a group of coordinate transformation.
Pierre Dèbes, Nour Ghazi (2012)
Annales de l’institut Fourier
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Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a -adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over . The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.
Cornelius Greither, Henri Johnston (2012)
Annales de l’institut Fourier
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We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower forces the tower to be split in a very strong sense.
Moshe Jarden (2006)
Journal de Théorie des Nombres de Bordeaux
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We prove that if is a number field and is a Galois extension of which is not algebraically closed, then is not PAC over .