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 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 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 Gal(M), in the sense of [Kat82]; (2) if char K=0, the intrinsic differential Galois group Gal^D(M), which is a Kolchin differential algebraic group. We deduce an arithmetic description of Gal(M) (resp. Gal^D(M)). In Part 4, we show that the Galois D-groupoid [Gra09] of a nonlinear q-difference system generalizes Gal^D(M).

ArXiv:1002.4839. (This article also contains the results of the unpublished preprints ArXiv: 1205.1694, 1205.1692.)

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