Galois theory of q -difference equations

Marius van der Put[1]; Marc Reversat[2]

  • [1] Department of Mathematics, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands
  • [2] Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse cedex 9. 31062 Toulouse cedex 9, France,

Annales de la faculté des sciences de Toulouse Mathématiques (2007)

  • Volume: 16, Issue: 3, page 665-718
  • ISSN: 0240-2963

Abstract

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Choose q ∈ 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 . A systematic treatment of classification and moduli is developed. 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 : = ℂ * / q ℤ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification ( [ A t ] ) of the indecomposable vector bundles on the complex Tate curve. Linear q -difference equations are also studied in positive characteristic p in order to derive Atiyah’s results for elliptic curves for which the j -invariant is not algebraic over 𝔽 p .

How to cite

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van der Put, Marius, and Reversat, Marc. "Galois theory of $q$-difference equations." Annales de la faculté des sciences de Toulouse Mathématiques 16.3 (2007): 665-718. <http://eudml.org/doc/10067>.

@article{vanderPut2007,
abstract = {Choose $q\in \mathbb\{C\}$ with $0&lt;|q|&lt;1$. The main theme of this paper is the study of linear $q$-difference equations over the field $K$ of germs of meromorphic functions at $0$. A systematic treatment of classification and moduli is developed. It turns out that a difference module $M$ over $K$ induces in a functorial way a vector bundle $v(M)$ on the Tate curve $E_q\!:\,=\{\mathbb\{C\}\}^\{\ast \}/q^\{\mathbb\{Z\}\}$ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification $([\{ \textrm\{A\}t\}])$ of the indecomposable vector bundles on the complex Tate curve. Linear $q$-difference equations are also studied in positive characteristic $p$ in order to derive Atiyah’s results for elliptic curves for which the $j$-invariant is not algebraic over $\{\mathbb\{F\}\}_p$.},
affiliation = {Department of Mathematics, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands; Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse cedex 9. 31062 Toulouse cedex 9, France,},
author = {van der Put, Marius, Reversat, Marc},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
number = {3},
pages = {665-718},
publisher = {Université Paul Sabatier, Toulouse},
title = {Galois theory of $q$-difference equations},
url = {http://eudml.org/doc/10067},
volume = {16},
year = {2007},
}

TY - JOUR
AU - van der Put, Marius
AU - Reversat, Marc
TI - Galois theory of $q$-difference equations
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2007
PB - Université Paul Sabatier, Toulouse
VL - 16
IS - 3
SP - 665
EP - 718
AB - Choose $q\in \mathbb{C}$ with $0&lt;|q|&lt;1$. The main theme of this paper is the study of linear $q$-difference equations over the field $K$ of germs of meromorphic functions at $0$. A systematic treatment of classification and moduli is developed. It turns out that a difference module $M$ over $K$ induces in a functorial way a vector bundle $v(M)$ on the Tate curve $E_q\!:\,={\mathbb{C}}^{\ast }/q^{\mathbb{Z}}$ that was known for modules with ”integer slopes“, [Saul, 2]). As a corollary one rediscovers Atiyah’s classification $([{ \textrm{A}t}])$ of the indecomposable vector bundles on the complex Tate curve. Linear $q$-difference equations are also studied in positive characteristic $p$ in order to derive Atiyah’s results for elliptic curves for which the $j$-invariant is not algebraic over ${\mathbb{F}}_p$.
LA - eng
UR - http://eudml.org/doc/10067
ER -

References

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  1. Atiyah (M.F.).— Vector bundles on elliptic curves, Proc.London.Math. Soc., 7, 414-452 (1957). Zbl0084.17305MR131423
  2. Collected mathematical papers of George David Birkhoff, Volume 1. Dover publications, 1968. 
  3. Forster (O.), Riemannsche Fläche, Heidelberger Taschenbücher, Springer Verlag (1977). Zbl0381.30021MR447557
  4. Fresnel (J.), van der Put (M.), Rigid Analytic Geometry and its Applications, Progress in Math., 218, 2004. Zbl1096.14014
  5. van der Put (M.), Skew differential fields, differential and difference equations, Astérisque, 296, p. 191-207 (2004). Zbl1082.12005MR2136010
  6. van der Put (M.), Reversat (M.), Krichever modules for difference and differential equations, Astérisque, 296, 197-225 (2004). Zbl1086.12001MR2136011
  7. van der Put (M.), Singer (M.F.), Galois theory of difference equations, Lect. Notes in Math., vol 1666, Springer Verlag, 1997. Zbl0930.12006MR1480919
  8. van der Put (M.), Singer (M.F.), Galois theory of linear differential equations, Grundlehren der mathematische Wissenschaften 328, Springer Verlag, 2003. Zbl1036.12008MR1960772
  9. Ramis (J.-P.), Sauloy (J.), Zhang (C.), La variété des classes analytiques d’équations aux q -différences dans une classe formelle , C.R.Acad.Sci.Paris, Ser. I 338 (2004). Zbl1038.39011
  10. Sauloy (J.), Galois theory of fuchsian q -difference equations, Ann. Sci. Éc. Norm. Sup. 4 e série 36, no 6, p. 925-968 (2003). Zbl1053.39033MR2032530
  11. Sauloy (J.), Algebraic construction of the Stokes sheaf for irregular linear q -difference equations, Astérisque 296, p. 227-251 (2004). Zbl1075.39020MR2136012
  12. Sauloy (J.), La filtration canonique par les pentes d’un module aux q -différences et le gradué associé, Ann. Inst. Fourier 54, no. 1, p.181–210 (2004). Zbl1061.39013

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