The q-analogue of the wild fundamental group and the inverse problem of the Galois theory of q-difference equations

TL;DR

This paper fully characterizes the wild fundamental group of linear q-difference equations, advancing the understanding of their Galois groups and solving the inverse Galois problem in this context.

Contribution

It provides a complete description of the wild fundamental group for q-difference equations and applies this to solve the inverse Galois problem, extending previous theoretical results.

Findings

Complete description of the wild fundamental group

Application to the inverse Galois problem

Enhanced results on the direct problem

Abstract

In previous papers, we defined $q$-analogues of alien derivations for linear analytic $q$-difference equations with integral slopes and proved a density theorem (in the Galois group) and a freeness theorem. In this paper, we completely describe the wild fundamental group and apply this result to the inverse problem in $q$-difference Galois theory. The new version contains an appendix on pronilpotent completion and the main result on the direct problem is made more precise. (Submitted for publication)