Intrinsic approach to Galois theory of q-difference equations, with the preface to Part 4 "The Galois D-groupoid of a q-difference system'' by Anne Granier

TL;DR

This paper extends the understanding of Galois theory for q-difference equations over various fields, providing an arithmetic description of associated Galois groups and linking the Galois D-groupoid to these groups.

Contribution

It generalizes Grothendieck's p-curvature conjecture for q-difference equations over broader fields and relates Galois D-groupoids to differential Galois groups.

Findings

Complete answer to the p-curvature analogue for q-difference equations.

Arithmetic description of Galois groups attached to q-difference modules.

Galois D-groupoid generalizes the differential Galois group for nonlinear systems.

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).