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 \(K(x)\):

- the intrinsic Galois group \({\rm Gal}(M)\), in the sense of [Kat82];
- 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 (This article contains the results of the unpublished preprints ArXiv: 1205.1694, 1205.1692.)

HAL: hal-02147378