Université de Versailles-St Quentin, Laboratoire de Mathématiques, 45 avenue des États-Unis 78035 Versailles cedex, France
e-mail: divizio[at]math.cnrs.fr          Office: bâtiment Fermat, office 3305

with Charlotte Hardouin, with a preface to Part IV by Anne Granier. To appear in Memoirs of the American Methematical Society. 77 pages.

Abstract:

We give a complete answer to the analogue of Grothendieck conjecture on $$p$$-curvatures for $$q$$-difference equations defined over $$K(x)$$, where $$K$$ is any finitely generated extension of $$\mathbb Q$$ and $$q\in K$$ can be either a transcendental or an algebraic number. This generalizes the results in [DV02], proved under the assumption that $$K$$ is a number field and $$q$$ is an algebraic number. The results also hold for a field $$K$$ which is a finite extension of a purely transcendental extension $$k(q)$$ of a perfect field k. In Part 3, we consider two Galois groups attached to a $$q$$-difference module $$M$$ over $$K(x)$$:

1. the intrinsic Galois group $${\rm Gal}(M)$$, in the sense of [Kat82];
2. if char $$K=0$$, the intrinsic differential Galois group $${\rm Gal}^D(M)$$, which is a Kolchin differential algebraic group.

We deduce an arithmetic description of $${\rm Gal}(M)$$ (resp. $${\rm Gal}^D(M)$$). In Part 4, we show that the Galois $$D$$-groupoid [Gra09] of a nonlinear $$q$$-difference system generalizes $${\rm Gal}^D(M)$$.

ArXiv: 1002.4839. (This article contains the results of the unpublished preprints ArXiv: 1205.16941205.1692.)

HAL: hal-02147378