Galois theory of q-difference equations
| dc.creator | van der Put, Marius | |
| dc.creator | Reversat, Marc | |
| dc.date | 2005-07-05 | |
| dc.date | 2006-01-26 | |
| dc.date.accessioned | 2026-07-07T06:42:35Z | |
| dc.date.available | 2026-07-07T06:42:35Z | |
| dc.description | 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. | |
| dc.description | Corrected version | |
| dc.identifier | https://arxiv.org/abs/math/0507098 | |
| dc.identifier | http://arxiv.org/abs/math/0507098 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/102093 | |
| dc.subject | Quantum Algebra | |
| dc.title | Galois theory of q-difference equations | |
| dc.type | text |