with Charlotte Hardouin, with a preface to Part IV by Anne Granier. Memoirs of the American Methematical Society, 2022, vol. 279, n. 1376. 
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 ${\rm Gal}^D(M)$.
DOI: 10.1090/memo/1376
ArXiv: 1002.4839. (This article contains the results of the unpublished preprints ArXiv: 1205.1694, 1205.1692.)
HAL: hal-02147378