The Stokes Phenomenon and Some Applications

Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlev\'e equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices, is useful for some quantum differential equations and for confluent generalized hypergeometric equations

Galois theory of q-difference equations

Choose $q\in {\mathbb C}$ with 0<|q|<1. The main theme of this paper is the study of linear q-difference equations over the field K of germs of meromorphic functions at 0. It turns out that a difference module M over K induces in a functorial way a vector bundle v(M) on the Tate curve $E_q:={\mathbb C}^*/q^{\mathbb Z}$. As a corollary one rediscovers Atiyah's classification of the indecomposable vector bundles on the complex Tate curve. Linear q-difference equations are also studied in positive characteristic in order to derive Atiyah's results for elliptic curves for which the j-invariant is not algebraic over ${\mathbb F}_p$. A universal difference ring and a universal formal difference Galois group are introduced. Part of the difference Galois group has an interpretation as `Stokes matrices', the above moduli space is the algebraic tool to compute it. It is possible to provide the vector bundle v(M) on E_q, corresponding to a difference module M over K, with a connection $\nabla_M$. If M is regular singular, then $\nabla_M$ is essentially determined by the absense of singularities and `unit circle monodromy'. More precisely, the monodromy of the connection $(v(M),\nabla_M)$ coincides with the action of two topological generators of the universal regular singular difference Galois group. For irregular difference modules, $\nabla_M$ will have singularities and there are various Tannakian choices for $M\mapsto (v(M),\nabla_M)$. Explicit computations are difficult, especially for the case of non integer slopes.Comment: Corrected versio

Mumford curves and Mumford groups in positive characteristic

A Mumford group is a discontinuous subgroup $\Gamma$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $\Gamma$ is a curve of genus 0. As abstract group $\Gamma$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to our paper "Discontinuous subgroups of PGL(2,K)" published in Journ. of Alg. (2004). Part of it clarifies, corrects and extends work of G.~Cornelissen, F.~Kato and K.~Kontogeorgis.Comment: 62 page

Stratified Bundles on Curves and Differential Galois Groups in Positive Characteristic

Stratifications and iterative differential equations are analogues in positive characteristic of complex linear differential equations. There are few explicit examples of stratifications. The main goal of this paper is to construct stratifications on projective or affine curves in positive characteristic and to determine the possibilities for their differential Galois groups. For the related "differential Abhyankar conjecture" we present partial answers, supplementing the literature. The tools for the construction of regular singular stratifications and the study of their differential Galois groups are $p$-adic methods and rigid analytic methods using Mumford curves and Mumford groups. These constructions produce many stratifications and differential Galois groups. In particular, some information on the tame fundamental groups of affine curves is obtained