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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 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 . 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 . 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
. If M is regular singular, then is essentially determined
by the absense of singularities and `unit circle monodromy'. More precisely,
the monodromy of the connection coincides with the action of
two topological generators of the universal regular singular difference Galois
group. For irregular difference modules, will have singularities and
there are various Tannakian choices for . 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 of PGL(2,K), where K
denotes a non archimedean valued field, such that the quotient by is a
curve of genus 0. As abstract group 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
Periodic behaviors
This paper studies behaviors that are defined on a torus, or equivalently,
behaviors defined in spaces of periodic functions, and establishes their basic
properties analogous to classical results of Malgrange, Palamodov, Oberst et
al. for behaviors on R^n. These properties - in particular the Nullstellensatz
describing the Willems closure - are closely related to integral and rational
points on affine algebraic varieties.Comment: 13 page
Effective descent for differential operators
A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential
operator over a suitable differential field , which has an isotypical
decomposition over the algebraic closure of , is a tensor product
of an absolutely irreducible operator over and an
irreducible operator over having a finite differential Galois group.
Using the existence of the tensor decomposition , an algorithm is
given in \cite{C-W}, which computes an absolutely irreducible factor of
over a finite extension of . Here, an algorithmic approach to finding
and is given, based on the knowledge of . This involves a subtle descent
problem for differential operators which can be solved for explicit
differential fields which are -fields.Comment: 21 page
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