Algebraic and differential generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier

Details Algebraic and differential generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier Algebraic and differential generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier

2010-02-25

arxiv.org/abs/1002.4839v4

Mathematics

Quantum Algebra

Minor changes. Revised version, after the referee suggestions

Scientific paper

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. If k is a number field and q is a parameter, one can either reduce the equation modulo a finite place of k or specialize the parameter q, or both. In particular for q=1, we obtain a differential equation defined over a number field or in positive characteristic. In \S II, we consider two Galois groups attached to a q-difference module M over K(x): the generic Galois group Gal(M), in the sense of [Kat82]; if char K=0, the generic 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 positive characteristic, we prove some devissage. There are many Galois theories for q-difference equations defined over fields such as C, the elliptic functions, or the differential closure of C. In \S III, we show that the Galois D-groupoid [Gra09] of a nonlinear q-difference system generalizes Gal^D(M). In \S IV we give some comparison results between the two generic Galois groups above and the other Galois groups for linear $q$-difference equations in the literature. We compare: the group introduced in [HS08] with the Gal^D(M) and hence with the Galois D-groupoid (cf [Mal09]); Gal(M) and Gal^D(M) to the generic Galois groups of the modules obtained by specialization of q or by reduction in positive characteristic. We relate the dimension of Gal^D(M) to the differential relations among the solutions of M.

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