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Galois theory of q -difference equations

Marius van der Put;  Marc Reversat — 2007

Annales de la faculté des sciences de Toulouse Mathématiques

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...

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