logo AFST

Some functional transcendence results around the Schwarzian differential equation.
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 5, pp. 1265-1300.

Dans cet article, nous démontrons des variantes du théorème d’Ax–Lindemann–Weierstrass (ALW) pour des fonctions analytiques satisfaisant des équations différentielles de type « Schwarzienne ». Dans des travaux antérieurs, nous avons prouvé le théorème ALW pour les uniformisantes de groupes fuchsiens de genre zéro. Dans ce travail, nous généralisons ce résultat de plusieurs manières en utilisant des techniques variées provenant de la théorie des modèles, de la théorie de Galois différentielle et de la géométrie complexe.

This paper centers around proving variants of the Ax–Lindemann–Weierstrass (ALW) theorem for analytic functions which satisfy Schwarzian differential equations. In previous work, the authors proved the ALW theorem for the uniformizers of genus zero Fuchsian groups, and in this work, we generalize that result in several ways using a variety of techniques from model theory, differential Galois theory and complex geometry.

Publié le :
DOI : https://doi.org/10.5802/afst.1661
Classification : 11F03,  12H05,  03C60
@article{AFST_2020_6_29_5_1265_0,
     author = {David Bl\'azquez-Sanz and Guy Casale and James Freitag and Joel Nagloo},
     title = {Some functional transcendence results around the Schwarzian differential equation.},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1265--1300},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 29},
     number = {5},
     year = {2020},
     doi = {10.5802/afst.1661},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1661/}
}
David Blázquez-Sanz; Guy Casale; James Freitag; Joel Nagloo. Some functional transcendence results around the Schwarzian differential equation.. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 5, pp. 1265-1300. doi : 10.5802/afst.1661. https://afst.centre-mersenne.org/articles/10.5802/afst.1661/

[1] Mark J. Ablowitz; Athanassios S. Fokas Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2003 | Zbl 1088.30001

[2] John T. Baldwin; Alistair H. Lachlan On strongly minimal sets, J. Symb. Log., Volume 36 (1971), pp. 79-96 | Article | MR 286642 | Zbl 0217.30402

[3] Gennadii V Belyĭ On Galois extensions of a maximal cyclotomic field, Math. USSR, Izv., Volume 14 (1980) no. 2, pp. 247-256 | Article | MR 534593 | Zbl 0429.12004

[4] L. Blum Generalized algebraic theories: a model theorectic approach (1968) (Ph. D. Thesis)

[5] Guy Casale; James Freitag; Joel Nagloo Ax–Lindemann–Weierstrass with derivatives and the genus 0 Fuchsian groups, Ann. Math., Volume 192 (2020) no. 3, pp. 721-765 | Article | MR 4172620 | Zbl 07285353

[6] James Freitag; Thomas Scanlon Strong minimality and the j-function, J. Eur. Math. Soc., Volume 20 (2018) no. 1, pp. 119-136 | Article | MR 3743238 | Zbl 06827896

[7] Leon Greenberg Maximal Fuchsian groups, Bull. Am. Math. Soc., Volume 69 (1963), pp. 569-573 | Article | MR 148620 | Zbl 0115.06701

[8] Bradd Hart; Matthew Valeriote Lectures on algebraic model theory, Fields Institute Monographs, 15, American Mathematical Society, 2002 | MR 1873592 | Zbl 0980.00023

[9] Moshe Kamensky; Anand Pillay Interpretations and Differential Galois Extensions, Int. Math. Res. Not., Volume 2016 (2016) no. 24, pp. 7390-7413 | Article | MR 3632087 | Zbl 1404.12006

[10] Tosihusa Kimura On Riemann’s equations which are solvable by quadratures, Funkc. Ekvacioj, Volume 12 (1969), pp. 269-281 | MR 277789 | Zbl 0198.11601

[11] Bruno Klingler; Emmanuel Ullmo; Andrei Yafaev Bi-algebraic geometry and the André–Oort conjecture, Algebraic Geometry: Salt Lake City 2015, Part 2 (Utah, 2015) (Proceedings of Symposia in Pure Mathematics), Volume 97, American Mathematical Society; Clay Mathematics Institute, 2016, pp. 319-360 | Zbl 07272616

[12] Ellis R. Kolchin Galois theory of differential fields, Am. J. Math., Volume 75 (1953) no. 4, pp. 753-824 | Article | MR 58591 | Zbl 0052.27301

[13] Ellis R. Kolchin Differential algebra and algebraic groups, Pure and Applied Mathematics, 54, Academic Press Inc., 1973 | MR 568864 | Zbl 0264.12102

[14] Jerald J. Kovacic An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput., Volume 2 (1986), pp. 3-43 | Article | MR 839134 | Zbl 0603.68035

[15] Joseph Lehner Discontinuous groups and automorphic functions, Mathematical Surveys, 8, American Mathematical Society, 1964 | MR 164939 | Zbl 0178.42902

[16] Andy R. Magid Lectures on Differential Galois Theory, University Lecture Series, 7, American Mathematical Society, 1994 | MR 1301076 | Zbl 0855.12001

[17] Grigoriĭ A. Margulis Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 17, Springer, 1990

[18] David Marker Strongly minimal sets and geometry, Colloquium ’95 (Haifa) (Lecture Notes in Logic), Volume 11, Springer, 1998, pp. 191-213 | Article | MR 1678361 | Zbl 0895.03013

[19] David Marker Introduction to model theory, Model theory, algebra, and geometry (Mathematical Sciences Research Institute Publications), Volume 39, Cambridge University Press, 2000, pp. 15-35 | MR 1773700 | Zbl 0961.03027

[20] Ben Moonen Linearity properties of Shimura varieties. I, J. Algebr. Geom., Volume 7 (1998) no. 3, pp. 539-567 | MR 1618140 | Zbl 0956.14016

[21] Joel Nagloo Model Theory, Algebra and Differential Equations (2014) (Ph. D. Thesis) | MR 3389520

[22] Joel Nagloo; Anand Pillay On the algebraic independence of generic Painlevé transcendents, Compos. Math., Volume 150 (2014) no. 4, pp. 668-678 | Article | Zbl 1309.14022

[23] Joel Nagloo; Anand Pillay On Algebraic relations between solutions of a generic Painlevé equation, J. Reine Angew. Math., Volume 726 (2017), pp. 1-27 | Article | Zbl 1385.34068

[24] Keiji Nishioka A conjecture of Mahler on automorphic functions, Arch. Math., Volume 53 (1989) no. 1, pp. 46-51 | Article | MR 1005168 | Zbl 0684.10022

[25] Paul Painlevé Leçons de Stokholm (1875), Oeuvres complètes Tome 1, Volume 1, éditions du CNRS, 1972

[26] Jonathan Pila o-minimality and the André–Oort conjecture for n , Ann. Math., Volume 173 (2011) no. 3, pp. 1779-1840 | Article | Zbl 1243.14022

[27] Anand Pillay Geometric stability theory, Oxford Logic Guides, 32, Oxford University Press, 1996 | MR 1429864 | Zbl 0871.03023

[28] Anand Pillay Stable embeddedness and NIP, J. Symb. Log., Volume 76 (2011) no. 2, pp. 665-672 | Article | MR 2830421 | Zbl 1220.03020

[29] Joseph Ritt Differential algebra, Colloquium Publications, 33, American Mathematical Society, 1950 | MR 35763 | Zbl 0037.18402

[30] A. Robinson On the concept of a differentially closed field, 1959 (Office of Scientific Research, US Air Force) | Zbl 0221.12054

[31] Henri P. de Saint-Gervais Uniformization of Riemann Surfaces. Revisiting a hundred-year-old-problem, Heritage of European Mathematics, European Mathematical Society, 2016 | Zbl 1332.30001

[32] Goro Shimura Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan. Kanô Memorial Lectures, 11, Princeton University Press, 1994 | MR 1291394 | Zbl 0872.11023

[33] David Singerman Finitely maximal Fuchsian groups, J. Lond. Math. Soc., Volume 6 (1972), pp. 29-38 | Article | MR 322165 | Zbl 0251.20052

[34] David Singerman Riemann surfaces, Belyi functions and hypermaps, Topics on Riemann surfaces and Fuchsian groups (London Mathematical Society Lecture Note Series), Volume 287, Cambridge University Press, 2001, pp. 43-68 | Article | MR 1842766 | Zbl 1015.14015

[35] Kisao Takeuchi Arithmetic triangle groups, J. Math. Soc. Japan, Volume 29 (1977), pp. 91-106 | MR 429744 | Zbl 0344.20035

[36] Hiroshi Umemura On the irreducibility of the first differential equation of Painlevé, Algebraic geometry and commutative algebra. Vol II, Konokuniya Company Ltd., 1988, pp. 771-789 | Article | Zbl 0704.12007

[37] Hiroshi Umemura Second proof of the irreducibility of the first differential equation of Painlevé, Nagoya Math. J. (1990) no. 117, pp. 125-171 | Article | Zbl 0688.34006

[38] Marie-France Vignéras Arithmétique des algébres de quaternions, Lecture Notes in Mathematics, 800, Springer, 1980 | Zbl 0422.12008

[39] Masaaki Yoshida Hypergeometric Functions, My Love, Aspects of Mathematics, E32, Springer, 1997 | Zbl 0889.33008

[40] Robert J. Zimmer Ergodic theory and semisimple groups, Monographs in Mathematics, 81, Birkhäuser, 1984 | MR 776417 | Zbl 0571.58015