Galois theory of fuchsian q-difference equations

Abstract

RésuméNous proposons une approche analytique de la théorie de Galois des systèmes aux q-différences linéaires singuliers réguliers. Nous combinons la dualité de Tannaka avec la méthode de classification de Birkhoffà l'aide de la matrice de connexion pour définir et décrire leurs groupes de Galois. Puis nous décrivons des sous-groupes fondamentaux qui donnent lieuà une correspondance de Riemann-Hilbert età un théorème de densité de type Schlesinger. AbstractWe propose an analytical approach to the Galois theory of singular regular linear q-difference systems. We use Tannaka duality along with Birkhoff's classification scheme with the connection matrix to define and describe their Galois groups. Then we describe fundamental subgroups that give rise to a Riemann-Hilbert correspondence and to a density theorem of Schlesinger's type.

Galois theory of q-difference equations

“…A more detailed and systematic treatment of classification and moduli is developed as a continuation of [vdP-S1] (Chapter 12), [vdP-R] and [vdP]. It turns out that a difference module M over K induces in a functorial way a vector bundle v(M ) on the Tate curve E q := C * /q Z (this is done here for all slopes, the case of integral slopes has been treated in [Sau1] and [Sau2]). As a corollary one rediscovers Atiyah's classification ( [At]) of the indecomposable vector bundles on the complex Tate curve.…”

“…If M is regular singular, then ∇ M is essentially determined by the absense of singularities and 'unit circle monodromy'. More precisely, the monodromy of the connection (v(M ), ∇ M ) coincides with the action of two topological generators of the universal regular singular difference Galois group ( Sau1]). For irregular difference modules, ∇ M will have singularities and there are various Tannakian choices for M → (v(M ), ∇ M ).…”

Birkhoff matrices, residues and rigidity for q-difference equations

“…This section follows the presentation of J. Sauloy in [13,14]. We also refer the reader to M. van der Put and M. Singer's book [9]; especially to section 12.3.…”

“…such that M (qz)A (0) = A (∞) M (z) where M denotes the sheaf over P 1 C of meromorphic functions (this is not exactly Birkhoff's original equivalence, but a modified version introduced by J. Sauloy in [14]). …”

On the definitions of difference Galois groups

“…In the local case and for certain restricted equations one does not necessarily need constants beyond those in C (see [9], [22], [23] as well as connections between the local and global cases. Another approach to q-difference equations is given by Sauloy in [26] and Ramis and Sauloy in [25] where a Galois group is produced using a combination of analytic and tannakian tools. The Galois groups discussed in these papers do not appear to act on rings or fields and, at present, it is not apparent how the techniques presented here can be used to compare these groups to other putative Galois groups.)…”