A general theory of André’s solution algebras

Nagy, Levente; Szamuely, Tamás

Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2103-2129.

Nous étendons la théorie des algèbres de solutions d’Yves André en théorie de Galois différentielle à un contexte général tannakien. Comme application nous obtenons des analogues de sa correspondance entre corps de solutions et sous-groupes observables du groupe de Galois différentiel pour les équations différentielles itérées en caractéristique positive et pour les équations aux différences. L’utilisation des algèbres de solutions dans le cadre de l’algèbre aux différences permet également un nouveau point de vue sur des résultats récents de Philippon et d’Adamczewski–Faverjon en théorie de la transcendance.

We extend Yves André’s theory of solution algebras in differential Galois theory to a general Tannakian context. As applications, we establish analogues of his correspondence between solution fields and observable subgroups of the Galois group for iterated differential equations in positive characteristic and for difference equations. The use of solution algebras in the difference algebraic context also allows a new approach to recent results of Philippon and Adamczewski–Faverjon in transcendence theory.

Classification : 18M25, 12H20, 12H10, 11J91
Keywords: Tannakian categories, iterative differential modules, difference modules, solution algebras, Mahler functions
Mot clés : catégories tannakiennes, modules différentiel itérés, modules aux differences, algèbres de solutions, fonctions de Mahler

[1] Adamczewski, Boris; Faverjon, Colin Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques, Proc. Lond. Math. Soc., Volume 115 (2017), pp. 55-90 | DOI | Zbl

[2] Alper, Jarod D.; Easton, Robert W. Recasting results in equivariant geometry: affine cosets, observable subgroups and existence of good quotients, Transform. Groups, Volume 17 (2012) no. 1, pp. 1-20 | DOI | MR | Zbl

[3] André, Yves Différentielles non commutatives et théorie de Galois différentielle ou aux différences, Ann. Sci. Éc. Norm. Supér., Volume 34 (2001) no. 5, pp. 685-739 | DOI | Numdam | MR | Zbl

[4] André, Yves Une introduction aux motifs. Motifs purs, motifs mixtes, périodes, Panoramas et Synthèses, 17, Société Mathématique de France, 2004 | Zbl

[5] André, Yves Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 2, pp. 449-467 | DOI | MR | Zbl

[6] Artin, Michael; Grothendieck, Alexander; Verdier, Jean-Louis Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, 269, Springer, 1972 Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat | MR | Zbl

[7] Beukers, Frits A refined version of the Siegel–Shidlovskii theorem, Ann. Math., Volume 163 (2006) no. 1, pp. 369-379 | DOI | MR | Zbl

[8] Brandenburg, Martin; Chirvasitu, Alexandru Tensor functors between categories of quasi-coherent sheaves, J. Algebra, Volume 399 (2014), pp. 675-692 | DOI | MR | Zbl

[9] Deligne, Pierre R. Le groupe fondamental de la droite projective moins trois points, Galois groups over $\mathbb{Q}$ (Berkeley, CA, 1987) (Mathematical Sciences Research Institute Publications), Volume 16, Springer, 1989, pp. 79-297 | DOI | MR | Zbl

[10] Deligne, Pierre R. Catégories tannakiennes, The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II (Progress in Mathematics), Volume 87, Birkhäuser, 1990, pp. 111-195 | MR | Zbl

[11] Deligne, Pierre R.; Milne, James S. Tannakian categories, Hodge cycles, motives, and Shimura varieties (Lecture Notes in Mathematics), Volume 900, Springer, 1982, pp. 101–-228 | DOI | MR | Zbl

[12] Demazure, Michel; Gabriel, Pierre Groupes algébriques. Tome I: Géométrie algébrique, généralités. groupes commutatifs, Masson & Cie; North-Holland, 1970 (“Avec un appendice Corps de classes local” par Michiel Hazewinkel.)) | MR | Zbl

[13] Fontaine, Jean-Marc Représentations $p$-adiques des corps locaux. I, The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II (Progress in Mathematics), Volume 87, Birkhäuser, 1990, pp. 249-309 | MR | Zbl

[14] Görtz, Ulrich; Wedhorn, Torsten Algebraic geometry I. Schemes. With examples and exercises, Advanced Lectures in Mathematics, Vieweg + Teubner Verlag, 2010 | DOI | MR | Zbl

[15] Grosshans, Frank D. Algebraic homogeneous spaces and invariant theory, Lecture Notes in Mathematics, 1673, Springer, 1997 | DOI | MR | Zbl

[16] Kashiwara, Masaki; Schapira, Pierre Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, 332, Springer, 2006 | MR | Zbl

[17] Matzat, Bernd Heinrich Differential Galois theory in positive characteristic (2001) (https://typo.iwr.uni-heidelberg.de/fileadmin/groups/arithgeo/templates/data/Matzat/difgal.html)

[18] Matzat, Bernd Heinrich; van der Put, Marius Iterative differential equations and the Abhyankar conjecture, J. Reine Angew. Math., Volume 557 (2003), pp. 1-52 | DOI | MR | Zbl

[19] Maurischat, Andreas Galois theory for iterative connections and nonreduced Galois groups, Trans. Am. Math. Soc., Volume 362 (2010) no. 10, pp. 5411-5453 | DOI | MR | Zbl

[20] Maurischat, Andreas A categorical approach to Picard-Vessiot theory, Theory Appl. Categ., Volume 32 (2017), 14, pp. 488-525 | MR | Zbl

[21] Nagy, Levente A general theory of solution algebras in differential and difference Galois theory, Ph. D. Thesis, Central European University, Hungary (2018) (http://goya.ceu.hu/record=b1248095)

[22] Nishioka, Kumiko New approach in Mahler’s method, J. Reine Angew. Math., Volume 407 (1990), pp. 202-219 | MR | Zbl

[23] Ovchinnikov, Alexey; Wibmer, Michael $\sigma$-Galois theory of linear difference equations, Int. Math. Res. Not., Volume 2015 (2015) no. 12, pp. 3962-4018 | MR | Zbl

[24] Papanikolas, Matthew A. Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math., Volume 171 (2008) no. 1, pp. 123-174 | DOI | MR | Zbl

[25] Philippon, Patrice Groupes de Galois et nombres automatiques, J. Lond. Math. Soc., Volume 92 (2015) no. 3, pp. 596-614 | DOI | MR | Zbl

[26] van der Put, Marius; Singer, Michael F. Galois theory of difference equations, Lecture Notes in Mathematics, 1666, Springer, 1997 | DOI | MR | Zbl

[27] van der Put, Marius; Singer, Michael F. Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, 2003 | DOI | MR | Zbl