Much work has gone into matrix representations of Galois groups, but there is a whole new class of naturally occurring representations that have as yet gone almost unnoticed. In fact, it is well-known in various areas of mathematics that the main sources of totally disconnected groups are matrix groups over local fields AND automorphism groups of locally finite trees [12]. It is perhaps surpris...

This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these modules with iterative connection (and also the modules with integrable iterative connection) form a Tannakian category, assuming some nice properties for the und...

We study the inverse problem in difference Galois theory of linear differential equations over difference-differential field C(x) with derivation ddx and endomorphism f(x)?f(x+1). Our main result is that every algebraic group, considered as a occurs group some equation C(x).

A Galois correspondence for finitely generated field extensions k/h is presented in the case characteristic h = p ^ 0. A field extension k/h is Galois if it is modular and h is separably algebraically closed in k. Galois groups are the direct limit of groups of higher derivations having rank a power of p. Galois groups are characterized in terms of abelian iterative generating sets in a manner ...

We study the middle convolution of local systems on the punctured affine line in the singular and the étale case. We give a motivic interpretation of the middle convolution which yields information on the occurring determinants. Finally, we use these methods to realize special linear groups regularly as Galois groups over Q(t). Introduction If K is a field, then we set GK := Gal(K/K), where K d...

This work presents a difference geometric approach to the strongly normal Galois theory of difference equations. In this approach, a system of ordinary difference equations is encoded in a difference extension, and the Galois groups are group schemes of finite type over the constants. The Galois groups need neither be linear nor reduced. The main result is a characterization of the extensions t...