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Intrinsic Approach to Galois Theory of $q$-Difference Equations

About this Title

Lucia Di Vizio, Charlotte Hardouin and Anne Granier

Publication: Memoirs of the American Mathematical Society
Publication Year: 2022; Volume 279, Number 1376
ISBNs: 978-1-4704-5384-8 (print); 978-1-4704-7230-6 (online)
Published electronically: August 3, 2022
Keywords: Generic Galois group; intrinsic Galois group; $q$-difference equations; differential Tannakian categories; Kolchin differential groups; Grothendieck conjecture on $p$-curvatures; $D$-groupoid

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Table of Contents


  • Introduction

1. Introduction to $q$-difference equations

  • 1. Generalities on $q$-difference modules
  • 2. Formal classification of singularities

2. Triviality of $q$-difference equations with rational coefficients

  • 3. Rationality of solutions, when $q$ is an algebraic number
  • 4. Rationality of solutions when $q$ is transcendental
  • 5. A unified statement

3. Intrinsic Galois groups

  • 6. The intrinsic Galois group
  • 7. The parametrized intrinsic Galois group

4. Comparison with the non-linear theory

  • 8. Preface to Part 4. The Galois $D$-groupoid of a $q$-difference system, by Anne Granier
  • 9. Comparison of the parametrized intrinsic Galois group with the Galois $D$-groupoid


The Galois theory of difference equations has witnessed a major evolution in the last two decades. In the particular case of $q$-difference equations, authors have introduced several different Galois theories. In this memoir we consider an arithmetic approach to the Galois theory of $q$-difference equations and we use it to establish an arithmetical description of some of the Galois groups attached to $q$-difference systems.

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