Skip to content

DIS

Seminar

Galois theory of q-difference in the roots of unity

Hardouin, C (Heidelberg)
Friday 15 May 2009, 09:30-10:30

Satellite

Abstract

For $q \in \mathbb{C}*$ non equal to $1$, we denote by $\sigma_q$ the automorphism of $\mathbb{C}(z)$ given by $\sigma_q(f)(z)=f(qz)$. As q goes to $1$, a q-difference equation w.r.t. $\sigma_q$ goes to a differential equation. The theory related to this fact is also called \textit{confluence} and one part of its study is the behaviour of the related Galois groups during this process. Therefore it seems interresting to have a good Galois theory of q-difference equation for q equal to a root of unity. Because of the increasing size of the constant field at these points, such construction has been avoided for a long time. Recently P.Hendricks has proposed a solution to this problems but his Galois groups were defined over very transcendant fields. We propose here a new approach based, in a certain sense, on a q-deformation of the work of B.H. Matzat and Marius van der Put for Differential Galois theory in positive characteristic. We consider also a family of \textit{iterative difference operator} instead of considering, just one difference operator, and by this way we stop the increasing of the constant field and succeed to set up a Picard-Vessiot Theory for q-difference equations where q is a root of unity and relate it to a Tannakian approach.

Video

The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.

Back to top ∧