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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)
DOI: https://doi.org/10.1090/memo/1376
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
Table of Contents
Chapters
- 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
Abstract
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.- Yves AndrĂ©, DiffĂ©rentielles non commutatives et thĂ©orie de Galois diffĂ©rentielle ou aux diffĂ©rences, Ann. Sci. Ăcole Norm. Sup. (4) 34 (2001), no. 5, 685â739 (French, with English and French summaries). MR 1862024, DOI 10.1016/S0012-9593(01)01074-6
- Yves AndrĂ©, Sur la conjecture des $p$-courbures de Grothendieck-Katz et un problĂšme de Dwork, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 55â112 (French, with French summary). MR 2023288
- S. A. Abramov and A. A. Ryabenko, Linear $q$-difference equations depending on a parameter, J. Symbolic Comput. 49 (2013), 65â77. MR 2997840, DOI 10.1016/j.jsc.2011.12.017
- Pascal Autissier, Points entiers sur les surfaces arithmĂ©tiques, J. Reine Angew. Math. 531 (2001), 201â235 (French). MR 1810122, DOI 10.1515/crll.2001.015
- Moulay Barkatou, Thomas Cluzeau, Jacques-Arthur Weil, and Lucia Di Vizio, Computing the Lie algebra of the differential Galois group of a linear differential system, Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2016, pp. 63â70. MR 3565698, DOI 10.1145/2930889.2930932
- Alin Bostan, Xavier Caruso, and Ăric Schost, A fast algorithm for computing the characteristic polynomial of the p-curvature, ISSAC 2014âProceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2014, pp. 59â66. MR 3239909, DOI 10.1145/2608628.2608650
- Alin Bostan, Xavier Caruso, and Ăric Schost, A fast algorithm for computing the $p$-curvature, ISSACâ15âProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2015, pp. 69â76. MR 3388284
- Alin Bostan, Xavier Caruso, and Ăric Schost, Computation of the similarity class of the $p$-curvature, Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2016, pp. 111â118. MR 3565704, DOI 10.1145/2930889.2930897
- N. Bourbaki, ĂlĂ©ments de mathĂ©matique. Fasc. XXX. AlgĂšbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, ActualitĂ©s Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1308, Hermann, Paris, 1964 (French). MR 194450
- Alin Bostan and Ăric Schost, Fast algorithms for differential equations in positive characteristic, ISSAC 2009âProceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2009, pp. 47â54. MR 2742690, DOI 10.1145/1576702.1576712
- F. Casorati, Il calcolo delle differenze finite interpretato ed accresciuto di nuovi teoremi a sussidio principalmente delle odierne ricerche basate sulla variabilitĂ complessa, Annali di Matematica Pura ed Applicata, Series 2 (1880) 10 (1880), no. 1, 10â45.
- P. J. Cassidy, Differential algebraic groups, Amer. J. Math. 94 (1972), 891â954. MR 360611, DOI 10.2307/2373764
- Richard M. Cohn, Difference algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1965. MR 205987
- Guy Casale and Julien Roques, Dynamics of rational symplectic mappings and difference Galois theory, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 103, 23. MR 2439539, DOI 10.1093/imrn/rnn103
- Shaoshi Chen and Michael F. Singer, Residues and telescopers for bivariate rational functions, Adv. in Appl. Math. 49 (2012), no. 2, 111â133. MR 2946428, DOI 10.1016/j.aam.2012.04.003
- Shaoshi Chen and Michael F. Singer, Residues and telescopers for bivariate rational functions, Adv. in Appl. Math. 49 (2012), no. 2, 111â133. MR 2946428, DOI 10.1016/j.aam.2012.04.003
- Michel Demazure and Pierre Gabriel, Groupes algĂ©briques. Tome I: GĂ©omĂ©trie algĂ©brique, gĂ©nĂ©ralitĂ©s, groupes commutatifs, Masson & Cie, Ăditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 302656
- Lucia Di Vizio, Arithmetic theory of $q$-difference equations: the $q$-analogue of Grothendieck-Katzâs conjecture on $p$-curvatures, Invent. Math. 150 (2002), no. 3, 517â578. MR 1946552, DOI 10.1007/s00222-002-0241-z
- Lucia Di Vizio and Charlotte Hardouin, Courbures, groupes de Galois gĂ©nĂ©riques et $D$-groupoĂŻde de Galois dâun systĂšme aux $q$-diffĂ©rences, C. R. Math. Acad. Sci. Paris 348 (2010), no. 17-18, 951â954 (French, with English and French summaries). MR 2721777, DOI 10.1016/j.crma.2010.08.001
- Lucia Di Vizio and Charlotte Hardouin, Descent for differential Galois theory of difference equations: confluence and $q$-dependence, Pacific J. Math. 256 (2012), no. 1, 79â104. MR 2928542, DOI 10.2140/pjm.2012.256.79
- L. Di Vizio, J.-P. Ramis, J. Sauloy, and C. Zhang, Ăquations aux $q$-diffĂ©rences, Gaz. Math. 96 (2003), 20â49 (French). MR 1988639
- Charles Favre and Juan Rivera-Letelier, Ăquidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann. 335 (2006), no. 2, 311â361 (French, with English and French summaries). MR 2221116, DOI 10.1007/s00208-006-0751-x
- Henri Gillet, Sergey Gorchinskiy, and Alexey Ovchinnikov, Parameterized Picard-Vessiot extensions and Atiyah extensions, Adv. Math. 238 (2013), 322â411. MR 3033637, DOI 10.1016/j.aim.2013.02.006
- Henri Gillet, Differential algebraâa scheme theory approach, Differential algebra and related topics (Newark, NJ, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 95â123. MR 1921696, DOI 10.1142/9789812778437_{0}003
- A. Granier, Un groupoïde de galois pour les équations aux $q$-différences, Ph.D. thesis, Université Toulouse III Paul Sabatier, 2009.
- Anne Granier, Un $D$-groupoĂŻde de Galois local pour les systĂšmes aux $q$-diffĂ©rences fuchsiens, C. R. Math. Acad. Sci. Paris 348 (2010), no. 5-6, 263â265 (French, with English and French summaries). MR 2600119, DOI 10.1016/j.crma.2010.01.030
- Anne Granier, A Galois $D$-groupoid for $q$-difference equations, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1493â1516 (2012) (English, with English and French summaries). MR 2951501, DOI 10.5802/aif.2648
- Charlotte Hardouin, Iterative $q$-difference Galois theory, J. Reine Angew. Math. 644 (2010), 101â144. MR 2671776, DOI 10.1515/CRELLE.2010.053
- P. Hendriks, Algebraic Aspects of Linear Differential and Difference Equations, Ph.D. thesis, University of Groningen., 1996.
- Charlotte Hardouin and Michael F. Singer, Differential Galois theory of linear difference equations, Math. Ann. 342 (2008), no. 2, 333â377. MR 2425146, DOI 10.1007/s00208-008-0238-z
- Charlotte Hardouin, Jacques Sauloy, and Michael F. Singer, Galois theories of linear difference equations: an introduction, Mathematical Surveys and Monographs, vol. 211, American Mathematical Society, Providence, RI, 2016. Papers from the courses held at the CIMPA Research School in Santa Marta, July 23âAugust 1, 2012. MR 3410204, DOI 10.1090/surv/211
- Moshe Kamensky, Model theory and the Tannakian formalism, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1095â1120. MR 3280038, DOI 10.1090/S0002-9947-2014-06062-5
- Irving Kaplansky, An introduction to differential algebra, Publications de lâInstitut MathĂ©matique de lâUniversitĂ© de Nancago [Publications of the Mathematical Institute of the University of Nancago], No. 5, Hermann, Paris, 1957. ActualitĂ©s Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1251. MR 93654
- Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Ătudes Sci. Publ. Math. 39 (1970), 175â232. MR 291177
- Nicholas M. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), no. 2, 203â239 (English, with French summary). MR 667751
- E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 568864
- Jerald J. Kovacic, Differential schemes, Differential algebra and related topics (Newark, NJ, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 71â94. MR 1921695, DOI 10.1142/9789812778437_{0}002
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
- Alexander Levin, Difference algebra, Algebra and Applications, vol. 8, Springer, New York, 2008. MR 2428839, DOI 10.1007/978-1-4020-6947-5
- Bernard Malgrange, Le groupoĂŻde de Galois dâun feuilletage, Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva, 2001, pp. 465â501 (French). MR 1929336
- B. Malgrange, Pseudogroupes de lie et théorie de galois différentielle, Preprint, 2009.
- Andrey Minchenko and Alexey Ovchinnikov, Zariski closures of reductive linear differential algebraic groups, Adv. Math. 227 (2011), no. 3, 1195â1224. MR 2799605, DOI 10.1016/j.aim.2011.03.002
- Shuji Morikawa, On a general difference Galois theory. I, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 7, 2709â2732 (English, with English and French summaries). MR 2649331
- Shuji Morikawa and Hiroshi Umemura, On a general difference Galois theory. II, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 7, 2733â2771 (English, with English and French summaries). MR 2649332
- B. Heinrich Matzat and Marius van der Put, Iterative differential equations and the Abhyankar conjecture, J. Reine Angew. Math. 557 (2003), 1â52. MR 1978401, DOI 10.1515/crll.2003.032
- Alexey Ovchinnikov, Differential Tannakian categories, J. Algebra 321 (2009), no. 10, 3043â3062. MR 2512641, DOI 10.1016/j.jalgebra.2009.02.008
- Alexey Ovchinnikov, Tannakian categories, linear differential algebraic groups, and parametrized linear differential equations, Transform. Groups 14 (2009), no. 1, 195â223. MR 2480859, DOI 10.1007/s00031-008-9042-9
- Anand Pillay, Finite-dimensional differential algebraic groups and the Picard-Vessiot theory, Differential Galois theory (BÄdlewo, 2001) Banach Center Publ., vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, pp. 189â199. MR 1972454, DOI 10.4064/bc58-0-14
- Henri PoincarĂ©, Sur une classe nouvelle de transcendantes uniformes, Journal de MathĂ©matiques Pures et AppliquĂ©es 6 (1890), 313â365.
- C. Praagman, The formal classification of linear difference operators, Nederl. Akad. Wetensch. Indag. Math. 45 (1983), no. 2, 249â261. MR 705431
- Jacques Sauloy, SystĂšmes aux $q$-diffĂ©rences singuliers rĂ©guliers: classification, matrice de connexion et monodromie, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 4, 1021â1071 (French, with English and French summaries). MR 1799737
- J. Sauloy, Galois theory of Fuchsian $q$-difference equations, Annales Scientifiques de lâĂcole Normale SupĂ©rieure. QuatriĂšme SĂ©rie 36 (2004), no. 6, 925â968.
- Jacques Sauloy, La filtration canonique par les pentes dâun module aux $q$-diffĂ©rences et le graduĂ© associĂ©, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 1, 181â210 (French, with English and French summaries). MR 2069126
- Jean-Pierre Serre, GĂ©omĂ©trie algĂ©brique et gĂ©omĂ©trie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1â42 (French). MR 82175
- Hiroshi Umemura, Picard-Vessiot theory in general Galois theory, Algebraic methods in dynamical systems, Banach Center Publ., vol. 94, Polish Acad. Sci. Inst. Math., Warsaw, 2011, pp. 263â293. MR 2905461, DOI 10.4064/bc94-0-19
- Marius van der Put and Marc Reversat, Galois theory of $q$-difference equations, Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), no. 3, 665â718 (English, with English and French summaries). MR 2379057
- Marius van der Put and Michael F. Singer, Galois theory of difference equations, Lecture Notes in Mathematics, vol. 1666, Springer-Verlag, Berlin, 1997. MR 1480919, DOI 10.1007/BFb0096118
- William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117
- Anna Zdunik, Harmonic measure on the Julia set for polynomial-like maps, Invent. Math. 128 (1997), no. 2, 303â327. MR 1440307, DOI 10.1007/s002220050142