DIS

Abstract

The local analytic classification of $q$-difference equations involves a $q$-analogue of Birkhoff-Malgrange-Sibuya theorem to the effect that the space of isoformal analytic classes is isomorphic to the $H^{1}$ of the so-called "Stokes sheaf", a sheaf of non abelian groups over the elliptic curve $E_{q} := \mathbf{C}^{*}/q^{\mathbf{Z}}$. On the other hand, the local analytic Galois theory of $q$-difference equations involves invariants of a new type, linked to "$q$-alien derivations". I hope to be able to use these new invariants to study the $H^{1}$ of some non abelian sheaves over $E_{q}$.