Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

Abstract

We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields.

Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem that have interesting consequences. In particular, we apply our results to give a complete proof of an assertion of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely, in the case of orbits of commensurators of Fuchsian groups.

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      author = {Matsuda, Michihiko},
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      volume = {15},
      publisher = {Kinokuniya Company Ltd., Tokyo},
      year = {1985},
      pages = {xi+114},
      mrclass = {14D05 (32G34 33C05)},
      mrnumber = {1104881},
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      author = {McShane, Greg},
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      journal = {Ann. Fac. Sci. Toulouse Math. (6)},
      fjournal = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
      volume = {28},
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      issn = {0240-2963},
      mrclass = {57M50 (32G15 37F32 57K20)},
      mrnumber = {4044432},
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      doi = {10.5802/afst.1606},
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      number = {3},
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      issn = {0003-486X},
      mrclass = {14G35 (03C64)},
      mrnumber = {3961087},
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      journal = {Arch. Math. (Basel)},
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      doi = {10.1007/BF01194871},
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      title = {Painlevé's theorem on automorphic functions. {II}},
      journal = {Funkcial. Ekvac.},
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      title = {The {H}amiltonians associated to the {P}ainlevé equations},
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      doi = {10.1007/978-1-4612-1532-5_13},
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      author = {Painlevé, P.},
      title = {Le\c cons sur la théorie analytique des équations différentielles professées à {S}tockholm (1895)},
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      mrclass = {03C64 (11F46 11G18 14G35 14K25)},
      mrnumber = {3039679},
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      doi = {10.1215/00127094-2080018},
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      year = {2013},
      number = {3-4},
      pages = {553--565},
      issn = {0029-4527},
      mrclass = {03C64 (11J91)},
      mrnumber = {3091671},
      doi = {10.1215/00294527-2143853},
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      number = {3},
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      mrclass = {11G18 (03C64 11U09 14G35)},
      mrnumber = {2800724},
      mrreviewer = {Alexandra Shlapentokh},
      doi = {10.4007/annals.2011.173.3.11},
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      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2009},
      number = {13},
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      issn = {1073-7928},
      mrclass = {14G05 (03C64 11G35 11J95)},
      mrnumber = {2520786},
      mrreviewer = {Hizuru Yamagishi},
      doi = {10.1093/imrn/rnp022},
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      fjournal = {Duke Mathematical Journal},
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      year = {2016},
      number = {13},
      pages = {2587--2605},
      issn = {0012-7094},
      mrclass = {11G18 (03C98 14F10)},
      mrnumber = {3546969},
      mrreviewer = {Alexandra Shlapentokh},
      doi = {10.1215/00127094-3620005},
      url = {https://doi.org/10.1215/00127094-3620005},
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      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
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      year = {2014},
      number = {2},
      pages = {659--681},
      issn = {0003-486X},
      mrclass = {14K10 (03C64 11G18)},
      mrnumber = {3152943},
      mrreviewer = {Jae-Hyun Yang},
      doi = {10.4007/annals.2014.179.2.5},
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      year = {2006},
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      pages = {591--616},
      issn = {0012-7094},
      mrclass = {03C64 (11G99 11U09)},
      mrnumber = {2228464},
      mrreviewer = {Alexandra Shlapentokh},
      doi = {10.1215/S0012-7094-06-13336-7},
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      volume = {32},
      note = {Oxford Science Publications},
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      issn = {0010-437X},
      mrclass = {14K15 (13N10)},
      mrnumber = {2004123},
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      doi = {10.1112/S0010437X03000186},
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      title = {Differential algebraic groups and the number of countable differentially closed fields},
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      note = {second edition, (Marker, Messmer, Pillay)},
      series = {Lecture Notes in Logic},
      volume = {5},
      publisher = {ASL-CUP},
      year = {2006},
      zblnumber = {},
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      doi = {10.1017/9781316716991.004},
      url = {https://doi.org/10.1017/9781316716991.004},
      }
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Authors

Guy Casale

Université Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France

James Freitag

University of Illinois Chicago, Department of Mathematics, Statistics and Computer Science, Chicago, IL, USA

Joel Nagloo

Department of Mathematics and Computer Science, Bronx Community College CUNY, Bronx, NY, USA