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.
-
[AbloFoka]
M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Second ed., Cambridge Univ. Press, Cambridge, 2003.@BOOK{AbloFoka,
author = {Ablowitz, Mark J. and Fokas, Athanassios S.},
title = {Complex Variables: Introduction and Applications},
series = {Cambridge Texts Appl. Math.},
edition = {Second},
publisher = {Cambridge Univ. Press, Cambridge},
year = {2003},
pages = {xii+647},
isbn = {0-521-53429-1},
mrclass = {30-01},
mrnumber = {1989049},
doi = {10.1017/CBO9780511791246},
url = {https://doi.org/10.1017/CBO9780511791246},
zblnumber = {1088.30001},
} -
[Aslanyan] V. Aslanyan, Ax-Schanuel type theorems and geometry of strongly minimal sets in differentially closed fields, 2016.
@MISC{Aslanyan,
author = {Aslanyan, V.},
title = {Ax-{S}chanuel type theorems and geometry of strongly minimal sets in differentially closed fields},
year = {2016},
arxiv = {1606.01778},
zblnumber = {},
} -
[BayTra] P. Bayer and A. Travesa, "Uniformization of triangle modular curves," Publ. Mat., iss. Proceedings of the Primeras Jornadas de Teor\’ıa de Números, pp. 43-106, 2007.
@ARTICLE{BayTra,
author = {Bayer, P. and Travesa, A.},
title = {Uniformization of triangle modular curves},
journal = {Publ. Mat.},
fjournal = {Publicacions Matemàtiques},
year = {2007},
number = {Proceedings of the Primeras Jornadas de Teor\'ıa de N\'{u}meros},
pages = {43--106},
issn = {0214-1493},
mrclass = {11F03 (11F06 11F30 20H10)},
mrnumber = {2499687},
mrreviewer = {Jannis A. Antoniadis},
doi = {10.5565/PUBLMAT_PJTN05_03},
url = {https://doi.org/10.5565/PUBLMAT_PJTN05_03},
zblnumber = {1185.11028},
} -
[BayTra1] P. Bayer and A. Travesa, "Uniformizing functions for certain Shimura curves, in the case $D=6$," Acta Arith., vol. 126, iss. 4, pp. 315-339, 2007.
@ARTICLE{BayTra1,
author = {Bayer, P. and Travesa, A.},
title = {Uniformizing functions for certain {S}himura curves, in the case {$D=6$}},
journal = {Acta Arith.},
fjournal = {Acta Arithmetica},
volume = {126},
year = {2007},
number = {4},
pages = {315--339},
issn = {0065-1036},
mrclass = {11G18 (14G35 33E30)},
mrnumber = {2289964},
mrreviewer = {Montserrat Alsina},
doi = {10.4064/aa126-4-3},
url = {https://doi.org/10.4064/aa126-4-3},
zblnumber = {1158.11031},
} -
[Galfin] G. Binyamini, "Bezout-type theorems for differential fields," Compos. Math., vol. 153, iss. 4, pp. 867-888, 2017.
@ARTICLE{Galfin,
author = {Binyamini, Gal},
title = {Bezout-type theorems for differential fields},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {153},
year = {2017},
number = {4},
pages = {867--888},
issn = {0010-437X},
mrclass = {12H05 (03C60 14G05 14H52)},
mrnumber = {3705244},
mrreviewer = {David A. Pierce},
doi = {10.1112/S0010437X17007035},
url = {https://doi.org/10.1112/S0010437X17007035},
zblnumber = {1376.12006},
} -
[bombpila] E. Bombieri and J. Pila, "The number of integral points on arcs and ovals," Duke Math. J., vol. 59, iss. 2, pp. 337-357, 1989.
@ARTICLE{bombpila,
author = {Bombieri, E. and Pila, J.},
title = {The number of integral points on arcs and ovals},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {59},
year = {1989},
number = {2},
pages = {337--357},
issn = {0012-7094},
mrclass = {11P21 (11D99)},
mrnumber = {1016893},
mrreviewer = {Ulrich Rausch},
doi = {10.1215/S0012-7094-89-05915-2},
url = {https://doi.org/10.1215/S0012-7094-89-05915-2},
zblnumber = {0718.11048},
} -
[Buium] A. Buium, "Geometry of differential polynomial functions. III. Moduli spaces," Amer. J. Math., vol. 117, iss. 1, pp. 1-73, 1995.
@ARTICLE{Buium,
author = {Buium, Alexandru},
title = {Geometry of differential polynomial functions. {III}. {M}oduli spaces},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {117},
year = {1995},
number = {1},
pages = {1--73},
issn = {0002-9327},
mrclass = {14K10 (11G10 12H05)},
mrnumber = {1314457},
mrreviewer = {Daniel Bertrand},
doi = {10.2307/2375035},
url = {https://doi.org/10.2307/2375035},
zblnumber = {0829.14020},
} -
[Bui] A. Buium, Differential Algebraic Groups of Finite Dimension, Springer-Verlag, Berlin, 1992, vol. 1506.
@BOOK{Bui,
author = {Buium, Alexandru},
title = {Differential Algebraic Groups of Finite Dimension},
series = {Lecture Notes in Math.},
volume = {1506},
publisher = {Springer-Verlag, Berlin},
year = {1992},
pages = {xvi+145},
isbn = {3-540-55181-6},
mrclass = {12H05 (14L17 14L40)},
mrnumber = {1176753},
mrreviewer = {G. Eisenreich},
doi = {10.1007/BFb0087235},
url = {https://doi.org/10.1007/BFb0087235},
zblnumber = {0756.14028},
} -
[CresHaj] T. Crespo and Z. Hajto, Algebraic Groups and Differential Galois Theory, Amer. Math. Soc., Providence, RI, 2011, vol. 122.
@BOOK{CresHaj,
author = {Crespo, Teresa and Hajto, Zbigniew},
title = {Algebraic Groups and Differential {G}alois Theory},
series = {Grad. Stud. Math.},
volume = {122},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2011},
pages = {xiv+225},
isbn = {978-0-8218-5318-4},
mrclass = {12H05 (20G15 34M15 34M50 68W30)},
mrnumber = {2778109},
mrreviewer = {David Bl\'{a}zquez-Sanz},
doi = {10.1090/gsm/122},
url = {https://doi.org/10.1090/gsm/122},
zblnumber = {1215.12001},
} -
[apAS] C. Daw and J. Ren, "Applications of the hyperbolic Ax-Schanuel conjecture," Compos. Math., vol. 154, iss. 9, pp. 1843-1888, 2018.
@ARTICLE{apAS,
author = {Daw, Christopher and Ren, Jinbo},
title = {Applications of the hyperbolic {A}x-{S}chanuel conjecture},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {154},
year = {2018},
number = {9},
pages = {1843--1888},
issn = {0010-437X},
mrclass = {11G18 (14G35)},
mrnumber = {3867286},
mrreviewer = {Martin Orr},
doi = {10.1112/s0010437x1800725x},
url = {https://doi.org/10.1112/s0010437x1800725x},
zblnumber = {1427.11057},
} -
[FreSan] J. Freitag and O. León Sánchez, "Effective uniform bounding in partial differential fields," Adv. Math., vol. 288, pp. 308-336, 2016.
@ARTICLE{FreSan,
author = {Freitag, James and León S\'{a}nchez, Omar},
title = {Effective uniform bounding in partial differential fields},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {288},
year = {2016},
pages = {308--336},
issn = {0001-8708},
mrclass = {03C60 (12H05 12H20)},
mrnumber = {3436387},
mrreviewer = {David A. Pierce},
doi = {10.1016/j.aim.2015.10.013},
url = {https://doi.org/10.1016/j.aim.2015.10.013},
zblnumber = {1401.12002},
} -
[FreSca] J. Freitag and T. Scanlon, "Strong minimality and the $j$-function," J. Eur. Math. Soc. (JEMS), vol. 20, iss. 1, pp. 119-136, 2018.
@ARTICLE{FreSca,
author = {Freitag, James and Scanlon, Thomas},
title = {Strong minimality and the {$j$}-function},
journal = {J. Eur. Math. Soc. (JEMS)},
fjournal = {Journal of the European Mathematical Society (JEMS)},
volume = {20},
year = {2018},
number = {1},
pages = {119--136},
issn = {1435-9855},
mrclass = {03C60 (03C45 12H05 33E30)},
mrnumber = {3743238},
mrreviewer = {G. Cherlin},
doi = {10.4171/JEMS/761},
url = {https://doi.org/10.4171/JEMS/761},
zblnumber = {06827896},
} -
[Gao] Z. Gao, "A special point problem of André-Pink-Zannier in the universal family of Abelian varieties," Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), vol. 17, iss. 1, pp. 231-266, 2017.
@ARTICLE{Gao,
author = {Gao, Ziyang},
title = {A special point problem of {A}ndré-{P}ink-{Z}annier in the universal family of {A}belian varieties},
journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)},
fjournal = {Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V},
volume = {17},
year = {2017},
number = {1},
pages = {231--266},
issn = {0391-173X},
mrclass = {11G18 (03C64 14G35)},
mrnumber = {3676047},
mrreviewer = {Min Ho Lee},
doi = {10.2422/2036-2145.201510_013},
url = {https://doi.org/10.2422/2036-2145.201510_013},
zblnumber = {1410.11067},
} -
[mixedALW] Z. Gao, "Towards the André-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points," J. Reine Angew. Math., vol. 732, pp. 85-146, 2017.
@ARTICLE{mixedALW,
author = {Gao, Ziyang},
title = {Towards the {A}ndré-{O}ort conjecture for mixed {S}himura varieties: the {A}x-{L}indemann theorem and lower bounds for {G}alois orbits of special points},
journal = {J. Reine Angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
volume = {732},
year = {2017},
pages = {85--146},
issn = {0075-4102},
mrclass = {14G35 (11G18 14C30)},
mrnumber = {3717089},
mrreviewer = {Rolf Berndt},
doi = {10.1515/crelle-2014-0127},
url = {https://doi.org/10.1515/crelle-2014-0127},
zblnumber = {1422.11140},
} -
[newML] D. Ghioca, F. Hu, T. Scanlon, and U. Zannier, "A variant of the Mordell-Lang conjecture," Math. Res. Lett., vol. 26, iss. 5, pp. 1383-1392, 2019.
@ARTICLE{newML,
author = {Ghioca, Dragos and Hu, Fei and Scanlon, Thomas and Zannier, Umberto},
title = {A variant of the {M}ordell-{L}ang conjecture},
journal = {Math. Res. Lett.},
fjournal = {Mathematical Research Letters},
volume = {26},
year = {2019},
number = {5},
pages = {1383--1392},
issn = {1073-2780},
mrclass = {11G10 (14K12)},
mrnumber = {4049814},
mrreviewer = {Daniel Loughran},
doi = {10.4310/MRL.2019.v26.n5.a7},
url = {https://doi.org/10.4310/MRL.2019.v26.n5.a7},
zblnumber = {1440.11099},
} -
[Gray] J. J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, second ed., Birkhäuser Boston, Inc., Boston, MA, 2008.
@BOOK{Gray,
author = {Gray, Jeremy J.},
title = {Linear Differential Equations and Group Theory from {R}iemann to {P}oincaré},
series = {Mod. Birkhäuser Class.},
edition = {second},
publisher = {Birkhäuser Boston, Inc., Boston, MA},
year = {2008},
pages = {xx+338},
isbn = {978-0-8176-4772-8},
mrclass = {34-03 (14-03 20-03 30-03 30E25 33-03 34Mxx)},
mrnumber = {2414794},
doi = {10.1007/978-0-8176-4773-5},
url = {https://doi.org/10.1007/978-0-8176-4773-5},
zblnumber = {1141.01010},
} -
[ML1] E. Hrushovski, "The Mordell-Lang conjecture for function fields," J. Amer. Math. Soc., vol. 9, iss. 3, pp. 667-690, 1996.
@ARTICLE{ML1,
author = {Hrushovski, Ehud},
title = {The {M}ordell-{L}ang conjecture for function fields},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {9},
year = {1996},
number = {3},
pages = {667--690},
issn = {0894-0347},
mrclass = {11U09 (03C60 11G10 14G99)},
mrnumber = {1333294},
mrreviewer = {Anand Pillay},
doi = {10.1090/S0894-0347-96-00202-0},
url = {https://doi.org/10.1090/S0894-0347-96-00202-0},
zblnumber = {0864.03026},
} -
[HruICM] E. Hrushovski, "Geometric model theory," in Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 281-302.
@INPROCEEDINGS{HruICM,
author = {Hrushovski, Ehud},
title = {Geometric model theory},
booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {I} ({B}erlin, 1998)},
journal = {Doc. Math.},
fjournal = {Documenta Mathematica},
year = {1998},
number = {Extra Vol. I},
pages = {281--302},
issn = {1431-0635},
mrclass = {03C45 (03C60 11U09 14G99)},
mrnumber = {1648035},
mrreviewer = {O. V. Belegradek},
url = {https://www.elibm.org/article/10011725},
zblnumber = {0906.03035},
} -
[HruSok] E. Hrushovski and Z. Sokolović, Strongly minimal sets in differentially closed fields, 1994.
@MISC{HruSok,
author = {Hrushovski, Ehud and Sokolović, Z.},
title = {Strongly minimal sets in differentially closed fields},
note = {unpublished manuscript},
year = {1994},
zblnumber = {},
} -
[Hump] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York-Berlin, 1978, vol. 9.
@BOOK{Hump,
author = {Humphreys, James E.},
title = {Introduction to {L}ie Algebras and Representation Theory},
series = {Grad. Texts in Math.},
volume = {9},
note = {Second printing, revised},
publisher = {Springer-Verlag, New York-Berlin},
year = {1978},
pages = {xii+171},
isbn = {0-387-90053-5},
mrclass = {17Bxx},
mrnumber = {0499562},
mrreviewer = {I. P. Shestakov},
zblnumber = {0447.17001},
} -
[Katok] S. Katok, Fuchsian Groups, Univ. of Chicago Press, Chicago, IL, 1992.
@BOOK{Katok,
author = {Katok, Svetlana},
title = {Fuchsian Groups},
series = {Chicago Lectures Math.},
publisher = {Univ. of Chicago Press, Chicago, IL},
year = {1992},
pages = {x+175},
isbn = {0-226-42582-7; 0-226-42583-5},
mrclass = {20H10 (30F35)},
mrnumber = {1177168},
mrreviewer = {I. Kra},
zblnumber = {0753.30001},
} -
[HyperALW] B. Klingler, E. Ullmo, and A. Yafaev, "The hyperbolic Ax-Lindemann-Weierstrass conjecture," Publ. Math. Inst. Hautes Études Sci., vol. 123, pp. 333-360, 2016.
@ARTICLE{HyperALW,
author = {Klingler, B. and Ullmo, E. and Yafaev, A.},
title = {The hyperbolic {A}x-{L}indemann-{W}eierstrass conjecture},
journal = {Publ. Math. Inst. Hautes \'{E}tudes Sci.},
fjournal = {Publications Mathématiques. Institut de Hautes \'{E}tudes Scientifiques},
volume = {123},
year = {2016},
pages = {333--360},
issn = {0073-8301},
mrclass = {14G35 (03C64 11J91 14P10 32H02 32M15)},
mrnumber = {3502100},
mrreviewer = {G. K. Sankaran},
doi = {10.1007/s10240-015-0078-9},
url = {https://doi.org/10.1007/s10240-015-0078-9},
zblnumber = {1372.14016},
} -
[Kovacic] J. J. Kovacic, "An algorithm for solving second order linear homogeneous differential equations," J. Symbolic Comput., vol. 2, iss. 1, pp. 3-43, 1986.
@ARTICLE{Kovacic,
author = {Kovacic, Jerald J.},
title = {An algorithm for solving second order linear homogeneous differential equations},
journal = {J. Symbolic Comput.},
fjournal = {Journal of Symbolic Computation},
volume = {2},
year = {1986},
number = {1},
pages = {3--43},
issn = {0747-7171},
mrclass = {12H05 (34A30)},
mrnumber = {0839134},
doi = {10.1016/S0747-7171(86)80010-4},
url = {https://doi.org/10.1016/S0747-7171(86)80010-4},
zblnumber = {0603.68035},
} -
[Lehner] J. Lehner, Discontinuous groups and automorphic functions, Amer. Math. Soc., Providence, R.I., 1964, vol. VIII.
@BOOK{Lehner,
author = {Lehner, Joseph},
title = {Discontinuous groups and automorphic functions},
series = {Math. Surveys},
volume = {VIII},
publisher = {Amer. Math. Soc., Providence, R.I.},
year = {1964},
pages = {xi+425},
mrclass = {30.49 (10.22)},
mrnumber = {0164033},
mrreviewer = {R. A. Rankin},
zblnumber = {0178.42902},
} -
[Marg] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin, 1991, vol. 17.
@BOOK{Marg,
author = {Margulis, G. A.},
title = {Discrete Subgroups of Semisimple {L}ie Groups},
series = {Ergeb. Math. Grenzgeb.},
volume = {17},
publisher = {Springer-Verlag, Berlin},
year = {1991},
pages = {x+388},
isbn = {3-540-12179-X},
mrclass = {22E40 (20Hxx 22-02 22D40)},
mrnumber = {1090825},
mrreviewer = {Gopal Prasad},
doi = {10.1007/978-3-642-51445-6},
url = {https://doi.org/10.1007/978-3-642-51445-6},
zblnumber = {0732.22008},
} -
[Mark] D. Marker, "Manin kernels," in Connections Between Model Theory and Algebraic and Analytic Geometry, Dept. Math., Seconda Univ. Napoli, Caserta, 2000, vol. 6, pp. 1-21.
@INCOLLECTION{Mark,
author = {Marker, David},
title = {Manin kernels},
booktitle = {Connections {B}etween {M}odel {T}heory and {A}lgebraic and {A}nalytic {G}eometry},
series = {Quad. Mat.},
volume = {6},
pages = {1--21},
publisher = {Dept. Math., Seconda Univ. Napoli, Caserta},
year = {2000},
mrclass = {12H05 (03C60)},
mrnumber = {1930680},
mrreviewer = {Grégory Berhuy},
zblnumber = {1100.14522},
} -
[MarkI] D. Marker, M. Messmer, and A. Pillay, Model Theory of Fields, Second ed., Association for Symbolic Logic, La Jolla, CA; A K Peters, Ltd., Wellesley, MA, 2006, vol. 5.
@BOOK{MarkI,
author = {Marker, David and Messmer, Margit and Pillay, Anand},
title = {Model Theory of Fields},
series = {Lecture Notes in Logic},
volume = {5},
edition = {Second},
publisher = {Association for Symbolic Logic, La Jolla, CA; A K Peters, Ltd., Wellesley, MA},
year = {2006},
pages = {xii+155},
isbn = {978-1-56881-282-3; 1-56881-282-5},
mrclass = {03C60 (03C45 12H05 14K99)},
mrnumber = {2215060},
zblnumber = {1104.12006},
doi = {10.1017/9781316716991},
url = {https://doi.org/10.1017/9781316716991},
} -
[masser2010torsion] D. Masser and U. Zannier, "Torsion anomalous points and families of elliptic curves," Amer. J. Math., vol. 132, iss. 6, pp. 1677-1691, 2010.
@ARTICLE{masser2010torsion,
author = {Masser, D. and Zannier, U.},
title = {Torsion anomalous points and families of elliptic curves},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {132},
year = {2010},
number = {6},
pages = {1677--1691},
issn = {0002-9327},
mrclass = {11G05},
mrnumber = {2766181},
mrreviewer = {Hizuru Yamagishi},
zblnumber = {1225.11078},
url = {https://muse.jhu.edu/article/404145/},
} -
[Matsuda] M. Matsuda, Lectures on Algebraic Solutions of Hypergeometric Differential Equations, Kinokuniya Company Ltd., Tokyo, 1985, vol. 15.
@BOOK{Matsuda,
author = {Matsuda, Michihiko},
title = {Lectures on Algebraic Solutions of Hypergeometric Differential Equations},
series = {Lect. Mat.},
volume = {15},
publisher = {Kinokuniya Company Ltd., Tokyo},
year = {1985},
pages = {xi+114},
mrclass = {14D05 (32G34 33C05)},
mrnumber = {1104881},
mrreviewer = {G. Cherlin},
zblnumber = {0606.12015},
url = {http://hdl.handle.net/2433/84920},
} -
[Mcshane] G. McShane, "Geodesic intersections and isoxial Fuchsian groups," Ann. Fac. Sci. Toulouse Math. (6), vol. 28, iss. 3, pp. 471-489, 2019.
@ARTICLE{Mcshane,
author = {McShane, Greg},
title = {Geodesic intersections and isoxial {F}uchsian groups},
journal = {Ann. Fac. Sci. Toulouse Math. (6)},
fjournal = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
volume = {28},
year = {2019},
number = {3},
pages = {471--489},
issn = {0240-2963},
mrclass = {57M50 (32G15 37F32 57K20)},
mrnumber = {4044432},
mrreviewer = {Subhojoy Gupta},
doi = {10.5802/afst.1606},
url = {https://doi.org/10.5802/afst.1606},
zblnumber = {1435.30127},
} -
[Mochi] S. Mochizuki, "Correspondences on hyperbolic curves," J. Pure Appl. Algebra, vol. 131, iss. 3, pp. 227-244, 1998.
@ARTICLE{Mochi,
author = {Mochizuki, Shinichi},
title = {Correspondences on hyperbolic curves},
journal = {J. Pure Appl. Algebra},
fjournal = {Journal of Pure and Applied Algebra},
volume = {131},
year = {1998},
number = {3},
pages = {227--244},
issn = {0022-4049},
mrclass = {14H35 (14E10 14H10)},
mrnumber = {1637015},
mrreviewer = {Atanas Iliev},
doi = {10.1016/S0022-4049(97)00078-9},
url = {https://doi.org/10.1016/S0022-4049(97)00078-9},
zblnumber = {0965.14013},
} -
[ASmpt] N. Mok, J. Pila, and J. Tsimerman, "Ax-Schanuel for Shimura varieties," Ann. of Math. (2), vol. 189, iss. 3, pp. 945-978, 2019.
@ARTICLE{ASmpt,
author = {Mok, Ngaiming and Pila, Jonathan and Tsimerman, Jacob},
title = {Ax-{S}chanuel for {S}himura varieties},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {189},
year = {2019},
number = {3},
pages = {945--978},
issn = {0003-486X},
mrclass = {14G35 (03C64)},
mrnumber = {3961087},
mrreviewer = {Sebastian Eterović},
doi = {10.4007/annals.2019.189.3.7},
url = {https://doi.org/10.4007/annals.2019.189.3.7},
zblnumber = {07107179},
} -
[Mum] D. Mumford, Geometric Invariant Theory, Springer-Verlag, New York, 1965, vol. 34.
@BOOK{Mum,
author = {Mumford, David},
title = {Geometric Invariant Theory},
series = {Ergeb. Math. Grenzgeb.},
volume = {34},
publisher = {Springer-Verlag, New York},
year = {1965},
pages = {vi+145},
mrclass = {14.55},
mrnumber = {0214602},
mrreviewer = {F. Oort},
zblnumber = {0147.39304},
} -
[NagPil] J. Nagloo and A. Pillay, "On algebraic relations between solutions of a generic Painlevé equation," J. Reine Angew. Math., vol. 726, pp. 1-27, 2017.
@ARTICLE{NagPil,
author = {Nagloo, Joel and Pillay, Anand},
title = {On algebraic relations between solutions of a generic {P}ainlevé equation},
journal = {J. Reine Angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
volume = {726},
year = {2017},
pages = {1--27},
issn = {0075-4102},
mrclass = {34M55 (12H05)},
mrnumber = {3641651},
mrreviewer = {Gleb A. Pogudin},
zblnumber = {1385.34068},
} -
[Nish] K. Nishioka, "A conjecture of Mahler on automorphic functions," Arch. Math. (Basel), vol. 53, iss. 1, pp. 46-51, 1989.
@ARTICLE{Nish,
author = {Nishioka, Keiji},
title = {A conjecture of {M}ahler on automorphic functions},
journal = {Arch. Math. (Basel)},
fjournal = {Archiv der Mathematik},
volume = {53},
year = {1989},
number = {1},
pages = {46--51},
issn = {0003-889X},
mrclass = {11F03 (11F27)},
mrnumber = {1005168},
mrreviewer = {Marvin I. Knopp},
doi = {10.1007/BF01194871},
url = {https://doi.org/10.1007/BF01194871},
zblnumber = {0684.10022},
} -
[NishiokaI] K. Nishioka, "Painlevé’s theorem on automorphic functions," Manuscripta Math., vol. 66, iss. 4, pp. 341-349, 1990.
@ARTICLE{NishiokaI,
author = {Nishioka, Keiji},
title = {Painlevé's theorem on automorphic functions},
journal = {Manuscripta Math.},
fjournal = {Manuscripta Mathematica},
volume = {66},
year = {1990},
number = {4},
pages = {341--349},
issn = {0025-2611},
mrclass = {12H05 (34C20)},
mrnumber = {1035631},
mrreviewer = {G. Eisenreich},
doi = {10.1007/BF02568502},
url = {https://doi.org/10.1007/BF02568502},
zblnumber = {0688.34004},
} -
[NishiokaII] K. Nishioka, "Painlevé’s theorem on automorphic functions. II," Funkcial. Ekvac., vol. 35, iss. 3, pp. 597-602, 1992.
@ARTICLE{NishiokaII,
author = {Nishioka, Keiji},
title = {Painlevé's theorem on automorphic functions. {II}},
journal = {Funkcial. Ekvac.},
fjournal = {Funkcialaj Ekvacioj. Serio Internacia},
volume = {35},
year = {1992},
number = {3},
pages = {597--602},
issn = {0532-8721},
mrclass = {12H05 (34A20 34A34)},
mrnumber = {1199477},
mrreviewer = {G. Eisenreich},
url = {http://fe.math.kobe-u.ac.jp/FE/Free/vol35/fe35-3-13.pdf},
zblnumber = {0781.34013},
} -
[Okamato] K. Okamoto, "The Hamiltonians associated to the Painlevé equations," in The Painlevé Property, Springer, New York, 1999, pp. 735-787.
@INCOLLECTION{Okamato,
author = {Okamoto, Kazuo},
title = {The {H}amiltonians associated to the {P}ainlevé equations},
booktitle = {The {P}ainlevé Property},
series = {CRM Ser. Math. Phys.},
pages = {735--787},
publisher = {Springer, New York},
year = {1999},
mrclass = {34M55 (33E17 37J05 37J35)},
mrnumber = {1714704},
doi = {10.1007/978-1-4612-1532-5_13},
url = {https://doi.org/10.1007/978-1-4612-1532-5_13},
zblnumber = {1017.37032},
} -
[Orr1] M. Orr, The André-Pink conjecture : Hecke orbits and weakly special subvarieties.
@misc{Orr1,
author = {Orr, Martin},
title = {The {A}ndré-{P}ink conjecture : {H}ecke orbits and weakly special subvarieties},
note={Ph. D thesis, Université Paris Sud - Paris XI, 2013, NNT : 2013PA112189. tel-00879010},
sortyear={2013},
url = {https://tel.archives-ouvertes.fr/tel-00879010/},
} -
[Orr2] M. Orr, "Families of abelian varieties with many isogenous fibres," J. Reine Angew. Math., vol. 705, pp. 211-231, 2015.
@ARTICLE{Orr2,
author = {Orr, Martin},
title = {Families of abelian varieties with many isogenous fibres},
journal = {J. Reine Angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
volume = {705},
year = {2015},
pages = {211--231},
issn = {0075-4102},
mrclass = {14K05 (14K10 14K22)},
mrnumber = {3377393},
mrreviewer = {H. Lange},
doi = {10.1515/crelle-2013-0058},
url = {https://doi.org/10.1515/crelle-2013-0058},
zblnumber = {1349.14143},
} -
[pain] P. Painlevé, "Le\c cons sur la théorie analytique des équations différentielles professées à Stockholm (1895)," in Oeuvres de Paul Painlevé. Tome I, Éditions du Centre National de la Recherche Scientifique, Paris, 1973, p. 825.
@INCOLLECTION{pain,
author = {Painlevé, P.},
title = {Le\c cons sur la théorie analytique des équations différentielles professées à {S}tockholm (1895)},
booktitle = {Oeuvres de {P}aul {P}ainlevé. {T}ome {I}},
publisher = {Éditions du Centre National de la Recherche Scientifique, Paris},
year = {1973},
pages = {825 pp.},
zblnumber = {},
} -
[PS1] Y. Peterzil and S. Starchenko, "Uniform definability of the Weierstrass $\wp$ functions and generalized tori of dimension one," Selecta Math. (N.S.), vol. 10, iss. 4, pp. 525-550, 2004.
@ARTICLE{PS1,
author = {Peterzil, Ya'acov and Starchenko, Sergei},
title = {Uniform definability of the {W}eierstrass {$\wp$} functions and generalized tori of dimension one},
journal = {Selecta Math. (N.S.)},
fjournal = {Selecta Mathematica. New Series},
volume = {10},
year = {2004},
number = {4},
pages = {525--550},
issn = {1022-1824},
mrclass = {03C64 (03C98 03H05 14H52 30G06 33E05)},
mrnumber = {2134454},
mrreviewer = {Ricardo Bianconi},
doi = {10.1007/s00029-005-0393-y},
url = {https://doi.org/10.1007/s00029-005-0393-y},
zblnumber = {1071.03022},
} -
[PS4] Y. Peterzil and S. Starchenko, "Complex analytic geometry and analytic-geometric categories," J. Reine Angew. Math., vol. 626, pp. 39-74, 2009.
@ARTICLE{PS4,
author = {Peterzil, Ya'acov and Starchenko, Sergei},
title = {Complex analytic geometry and analytic-geometric categories},
journal = {J. Reine Angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
volume = {626},
year = {2009},
pages = {39--74},
issn = {0075-4102},
mrclass = {32C15 (03C64 32B20)},
mrnumber = {2492989},
mrreviewer = {Fernando Sanz},
doi = {10.1515/CRELLE.2009.002},
url = {https://doi.org/10.1515/CRELLE.2009.002},
zblnumber = {1168.03029},
} -
[PS3] Y. Peterzil and S. Starchenko, "Tame complex analysis and o-minimality," in Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II-IV: Invited Lectures, , 2010.
@INCOLLECTION{PS3,
author = {Peterzil, Ya'acov and Starchenko, Sergei},
title = {Tame complex analysis and o-minimality},
booktitle = {Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II-IV: Invited Lectures},
year = {2010},
zblnumber = {1246.03061},
} -
[PS2] Y. Peterzil and S. Starchenko, "Definability of restricted theta functions and families of abelian varieties," Duke Math. J., vol. 162, iss. 4, pp. 731-765, 2013.
@ARTICLE{PS2,
author = {Peterzil, Ya'acov and Starchenko, Sergei},
title = {Definability of restricted theta functions and families of abelian varieties},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {162},
year = {2013},
number = {4},
pages = {731--765},
issn = {0012-7094},
mrclass = {03C64 (11F46 11G18 14G35 14K25)},
mrnumber = {3039679},
mrreviewer = {Jean-Philippe Rolin},
doi = {10.1215/00127094-2080018},
url = {https://doi.org/10.1215/00127094-2080018},
zblnumber = {1284.03215},
} -
[Pila] J. Pila, "Modular Ax-Lindemann-Weierstrass with derivatives," Notre Dame J. Form. Log., vol. 54, iss. 3-4, pp. 553-565, 2013.
@ARTICLE{Pila,
author = {Pila, Jonathan},
title = {Modular {A}x-{L}indemann-{W}eierstrass with derivatives},
journal = {Notre Dame J. Form. Log.},
fjournal = {Notre Dame Journal of Formal Logic},
volume = {54},
year = {2013},
number = {3-4},
pages = {553--565},
issn = {0029-4527},
mrclass = {03C64 (11J91)},
mrnumber = {3091671},
doi = {10.1215/00294527-2143853},
url = {https://doi.org/10.1215/00294527-2143853},
zblnumber = {1355.11078},
} -
[PilaAO] J. Pila, "O-minimality and the André-Oort conjecture for $\Bbb C^n$," Ann. of Math. (2), vol. 173, iss. 3, pp. 1779-1840, 2011.
@ARTICLE{PilaAO,
author = {Pila, Jonathan},
title = {O-minimality and the {A}ndré-{O}ort conjecture for {$\Bbb C^n$}},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {173},
year = {2011},
number = {3},
pages = {1779--1840},
issn = {0003-486X},
mrclass = {11G18 (03C64 11U09 14G35)},
mrnumber = {2800724},
mrreviewer = {Alexandra Shlapentokh},
doi = {10.4007/annals.2011.173.3.11},
url = {https://doi.org/10.4007/annals.2011.173.3.11},
zblnumber = {1243.14022},
} -
[PilaIMRN] J. Pila, "Rational points of definable sets and results of André-Oort-Manin-Mumford type," Int. Math. Res. Not. IMRN, iss. 13, pp. 2476-2507, 2009.
@ARTICLE{PilaIMRN,
author = {Pila, Jonathan},
title = {Rational points of definable sets and results of {A}ndré-{O}ort-{M}anin-{M}umford type},
journal = {Int. Math. Res. Not. IMRN},
fjournal = {International Mathematics Research Notices. IMRN},
year = {2009},
number = {13},
pages = {2476--2507},
issn = {1073-7928},
mrclass = {14G05 (03C64 11G35 11J95)},
mrnumber = {2520786},
mrreviewer = {Hizuru Yamagishi},
doi = {10.1093/imrn/rnp022},
url = {https://doi.org/10.1093/imrn/rnp022},
zblnumber = {1243.14021},
} -
[AXS] J. Pila and J. Tsimerman, "Ax-Schanuel for the $j$-function," Duke Math. J., vol. 165, iss. 13, pp. 2587-2605, 2016.
@ARTICLE{AXS,
author = {Pila, Jonathan and Tsimerman, Jacob},
title = {Ax-{S}chanuel for the {$j$}-function},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {165},
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},
zblnumber = {1419.11093},
} -
[AgALW] J. Pila and J. Tsimerman, "Ax-Lindemann for $\mathcal{A}_g$," Ann. of Math. (2), vol. 179, iss. 2, pp. 659-681, 2014.
@ARTICLE{AgALW,
author = {Pila, Jonathan and Tsimerman, Jacob},
title = {Ax-{L}indemann for {$\mathcal{A}_g$}},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {179},
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},
url = {https://doi.org/10.4007/annals.2014.179.2.5},
zblnumber = {1305.14020},
} -
[PW] J. Pila and A. J. Wilkie, "The rational points of a definable set," Duke Math. J., vol. 133, iss. 3, pp. 591-616, 2006.
@ARTICLE{PW,
author = {Pila, J. and Wilkie, A. J.},
title = {The rational points of a definable set},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {133},
year = {2006},
number = {3},
pages = {591--616},
issn = {0012-7094},
mrclass = {03C64 (11G99 11U09)},
mrnumber = {2228464},
mrreviewer = {Alexandra Shlapentokh},
doi = {10.1215/S0012-7094-06-13336-7},
url = {https://doi.org/10.1215/S0012-7094-06-13336-7},
zblnumber = {1217.11066},
} -
[PZ] J. Pila and U. Zannier, "Rational points in periodic analytic sets and the Manin-Mumford conjecture," Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., vol. 19, iss. 2, pp. 149-162, 2008.
@ARTICLE{PZ,
author = {Pila, Jonathan and Zannier, Umberto},
title = {Rational points in periodic analytic sets and the {M}anin-{M}umford conjecture},
journal = {Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.},
fjournal = {Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni},
volume = {19},
year = {2008},
number = {2},
pages = {149--162},
issn = {1120-6330},
mrclass = {11J95 (11G10 14G05)},
mrnumber = {2411018},
mrreviewer = {Pavlos Tzermias},
doi = {10.4171/RLM/514},
url = {https://doi.org/10.4171/RLM/514},
zblnumber = {1164.11029},
} -
[GST] A. Pillay, Geometric Stability Theory, The Clarendon Press, Oxford Univ. Press, New York, 1996, vol. 32.
@BOOK{GST,
author = {Pillay, Anand},
title = {Geometric Stability Theory},
series = {Oxford Logic Guides},
volume = {32},
note = {Oxford Science Publications},
publisher = {The Clarendon Press, Oxford Univ. Press, New York},
year = {1996},
pages = {x+361},
isbn = {0-19-853437-X},
mrclass = {03C45 (03-02 20A15)},
mrnumber = {1429864},
mrreviewer = {Alexandre Ivanov},
zblnumber = {0871.03023},
} -
[PilZie] A. Pillay and M. Ziegler, "Jet spaces of varieties over differential and difference fields," Selecta Math. (N.S.), vol. 9, iss. 4, pp. 579-599, 2003.
@ARTICLE{PilZie,
author = {Pillay, Anand and Ziegler, Martin},
title = {Jet spaces of varieties over differential and difference fields},
journal = {Selecta Math. (N.S.)},
fjournal = {Selecta Mathematica. New Series},
volume = {9},
year = {2003},
number = {4},
pages = {579--599},
issn = {1022-1824},
mrclass = {12H05},
mrnumber = {2031753},
mrreviewer = {Mykola Ya. Komarnitskiĭ},
doi = {10.1007/s00029-003-0339-1},
url = {https://doi.org/10.1007/s00029-003-0339-1},
zblnumber = {1060.12003},
} -
[ML2] A. Pillay, "Mordell-Lang conjecture for function fields in characteristic zero, revisited," Compos. Math., vol. 140, iss. 1, pp. 64-68, 2004.
@ARTICLE{ML2,
author = {Pillay, Anand},
title = {Mordell-{L}ang conjecture for function fields in characteristic zero, revisited},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {140},
year = {2004},
number = {1},
pages = {64--68},
issn = {0010-437X},
mrclass = {14K15 (13N10)},
mrnumber = {2004123},
mrreviewer = {Dan Abramovich},
doi = {10.1112/S0010437X03000186},
url = {https://doi.org/10.1112/S0010437X03000186},
zblnumber = {1084.14043},
} -
[Pil] A. Pillay, "Differential algebraic groups and the number of countable differentially closed fields," in Model Theory of Fields, ASL-CUP, 2006, vol. 5, pp. 114-134.
@INCOLLECTION{Pil,
author = {Pillay, A.},
title = {Differential algebraic groups and the number of countable differentially closed fields},
booktitle = {Model Theory of Fields},
note = {second edition, (Marker, Messmer, Pillay)},
series = {Lecture Notes in Logic},
volume = {5},
publisher = {ASL-CUP},
year = {2006},
zblnumber = {},
pages = {114--134},
doi = {10.1017/9781316716991.004},
url = {https://doi.org/10.1017/9781316716991.004},
} -
[SagleWalde] A. A. Sagle and R. E. Walde, Introduction to Lie Groups and Lie Algebras, Academic Press, New York, 1973, vol. 51.
@BOOK{SagleWalde,
author = {Sagle, Arthur A. and Walde, Ralph E.},
title = {Introduction to {L}ie Groups and {L}ie {A}lgebras},
series = {Pure Appl. Math.},
volume = {51},
publisher = {Academic Press, New York},
year = {1973},
pages = {ix+361},
mrclass = {22EXX (17BXX)},
mrnumber = {0360927},
mrreviewer = {N. R. Wallach},
zblnumber = {0252.22001},
} -
[HPSG] H. P. de Saint-Gervais, Uniformisation Des Surfaces de Riemann, ENS Éditions, Lyon, 2010.
@BOOK{HPSG,
author = {de Saint-Gervais, Henri Paul},
title = {Uniformisation Des Surfaces de {R}iemann},
note = {Retour sur un théorème centenaire. [A look back at a 100-year-old theorem], The name of Henri Paul de Saint-Gervais covers a group composed of fifteen mathematicians : Aurélien Alvarez, Christophe Bavard, François Béguin, Nicolas Bergeron, Maxime Bourrigan, Bertrand Deroin, Sorin Dumitrescu, Charles Frances, \'{E}tienne Ghys, Antonin Guilloux, Frank Loray, Patrick Popescu-Pampu, Pierre Py, Bruno Sévennec, and Jean-Claude Sikorav},
publisher = {ENS \'{E}ditions, Lyon},
year = {2010},
pages = {544},
isbn = {978-2-84788-233-9},
mrclass = {30F10 (01A55 14H55 30-03 30Fxx 32Q30)},
mrnumber = {2768303},
mrreviewer = {Athanase Papadopoulos},
zblnumber = {1228.30001},
} -
[Sarnak] P. Sarnak, Torsion points on varieties and homology of abelian covers, 1988.
@MISC{Sarnak,
author = {Sarnak, P.},
title = {Torsion points on varieties and homology of abelian covers},
note = {unpublished manuscript},
year = {1988},
zblnumber = {},
} -
[Shim] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten, Publishers, Tokyo; Princeton Univ. Press, Princeton, N.J., 1971, vol. 11.
@book {Shim,
author = {Shimura, Goro},
title = {Introduction to the Arithmetic Theory of Automorphic Functions},
titlenote = {Kanô Memorial Lectures, No. 1},
series = {Publ. Math.l Soc. Japan},
volume={11},
publisher={Iwanami Shoten, Publishers, Tokyo; Princeton Univ. Press, Princeton, N.J.},
year = {1971},
pages = {xiv+267},
mrclass = {10D10 (12-02 14K22)},
mrnumber = {0314766},
mrreviewer = {A. N. Andrianov},
zblnumber = {0221.10029},
} -
[Take] K. Takeuchi, "Arithmetic triangle groups," J. Math. Soc. Japan, vol. 29, iss. 1, pp. 91-106, 1977.
@ARTICLE{Take,
author = {Takeuchi, Kisao},
title = {Arithmetic triangle groups},
journal = {J. Math. Soc. Japan},
fjournal = {Journal of the Mathematical Society of Japan},
volume = {29},
year = {1977},
number = {1},
pages = {91--106},
issn = {0025-5645},
mrclass = {10D05},
mrnumber = {0429744},
mrreviewer = {Roy Fuller},
doi = {10.2969/jmsj/02910091},
url = {https://doi.org/10.2969/jmsj/02910091},
zblnumber = {0344.20035},
} -
[take2] K. Takeuchi, "Commensurability classes of arithmetic triangle groups," J. Fac. Sci. Univ. Tokyo Sect. IA Math., vol. 24, iss. 1, pp. 201-212, 1977.
@ARTICLE{take2,
author = {Takeuchi, Kisao},
title = {Commensurability classes of arithmetic triangle groups},
journal = {J. Fac. Sci. Univ. Tokyo Sect. IA Math.},
fjournal = {Journal of the Faculty of Science. Univ. of Tokyo. Section IA. Mathematics},
volume = {24},
year = {1977},
number = {1},
pages = {201--212},
issn = {0040-8980},
mrclass = {10D05 (20H10)},
mrnumber = {0463116},
mrreviewer = {K.-B. Gundlach},
zblnumber = {0365.20055},
} -
[Tu] F. Tu, "Schwarzian differential equations associated to Shimura curves of genus zero," Pacific J. Math., vol. 269, iss. 2, pp. 453-489, 2014.
@ARTICLE{Tu,
author = {Tu, Fang-Ting},
title = {Schwarzian differential equations associated to {S}himura curves of genus zero},
journal = {Pacific J. Math.},
fjournal = {Pacific Journal of Mathematics},
volume = {269},
year = {2014},
number = {2},
pages = {453--489},
issn = {0030-8730},
mrclass = {11F12 (11G18)},
mrnumber = {3238486},
mrreviewer = {Marc Masdeu},
doi = {10.2140/pjm.2014.269.453},
url = {https://doi.org/10.2140/pjm.2014.269.453},
zblnumber = {1308.11047},
} -
[ALWcom] E. Ullmo and A. Yafaev, "Hyperbolic Ax-Lindemann theorem in the cocompact case," Duke Math. J., vol. 163, iss. 2, pp. 433-463, 2014.
@ARTICLE{ALWcom,
author = {Ullmo, Emmanuel and Yafaev, Andrei},
title = {Hyperbolic {A}x-{L}indemann theorem in the cocompact case},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {163},
year = {2014},
number = {2},
pages = {433--463},
issn = {0012-7094},
mrclass = {14G35 (03C64 11J91 14P10 32M15)},
mrnumber = {3161318},
mrreviewer = {Jae-Hyun Yang},
doi = {10.1215/00127094-2410546},
url = {https://doi.org/10.1215/00127094-2410546},
zblnumber = {1375.14096},
} -
[Umemura-1] H. Umemura, "Algebro-geometric problems arising from Painlevé’s works," in Algebraic and Topological Theories, Kinokuniya, Tokyo, 1986, pp. 467-495.
@INCOLLECTION{Umemura-1,
author = {Umemura, Hiroshi},
title = {Algebro-geometric problems arising from {P}ainlevé's works},
booktitle = {Algebraic and Topological Theories},
venue = {{K}inosaki, 1984},
pages = {467--495},
publisher = {Kinokuniya, Tokyo},
year = {1986},
mrclass = {14-02 (14-03)},
mrnumber = {1102272},
zblnumber = {0800.14001},
} -
[Umemura0] H. Umemura, "Birational automorphism groups and differential equations," Nagoya Math. J., vol. 119, pp. 1-80, 1990.
@ARTICLE{Umemura0,
author = {Umemura, Hiroshi},
title = {Birational automorphism groups and differential equations},
journal = {Nagoya Math. J.},
fjournal = {Nagoya Mathematical Journal},
volume = {119},
year = {1990},
pages = {1--80},
issn = {0027-7630},
mrclass = {12H05 (34A20)},
mrnumber = {1071899},
mrreviewer = {Keiji Nishioka},
doi = {10.1017/S0027763000003111},
url = {https://doi.org/10.1017/S0027763000003111},
zblnumber = {0714.12009},
} -
[Umemura1] H. Umemura, "On the irreducibility of the first differential equation of Painlevé," in Algebraic Geometry and Commutative Algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 771-789.
@INCOLLECTION{Umemura1,
author = {Umemura, Hiroshi},
title = {On the irreducibility of the first differential equation of {P}ainlevé},
booktitle = {Algebraic Geometry and Commutative Algebra, {V}ol. {II}},
pages = {771--789},
publisher = {Kinokuniya, Tokyo},
year = {1988},
mrclass = {12H05 (34A20 34C20)},
mrnumber = {0977782},
mrreviewer = {Keiji Nishioka},
zblnumber = {0704.12007},
} -
[umemura] H. Umemura, "Second proof of the irreducibility of the first differential equation of Painlevé," Nagoya Math. J., vol. 117, pp. 125-171, 1990.
@ARTICLE{umemura,
author = {Umemura, Hiroshi},
title = {Second proof of the irreducibility of the first differential equation of {P}ainlevé},
journal = {Nagoya Math. J.},
fjournal = {Nagoya Mathematical Journal},
volume = {117},
year = {1990},
pages = {125--171},
issn = {0027-7630},
mrclass = {12H05 (34A20 34C20)},
mrnumber = {1044939},
mrreviewer = {Keiji Nishioka},
doi = {10.1017/S0027763000001835},
url = {https://doi.org/10.1017/S0027763000001835},
zblnumber = {0688.34006},
} -
[tameT] L. van den Dries, Tame Topology and o-minimal Structures, Cambridge Univ. Press, Cambridge, 1998, vol. 248.
@BOOK{tameT,
author = {{van den Dries},
Lou},
title = {Tame Topology and o-minimal Structures},
series = {London Math. Soc. Lect. Note Ser.},
volume = {248},
publisher = {Cambridge Univ. Press, Cambridge},
year = {1998},
pages = {x+180},
isbn = {0-521-59838-9},
mrclass = {03-02 (03C50 03C60 14P10 52-02 54-02 55-02 57-02)},
mrnumber = {1633348},
mrreviewer = {O. V. Belegradek},
doi = {10.1017/CBO9780511525919},
url = {https://doi.org/10.1017/CBO9780511525919},
zblnumber = {0953.03045},
} -
[Venkov] A. B. Venkov, Spectral Theory of Automorphic Functions and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1990, vol. 51.
@BOOK{Venkov,
author = {Venkov, Alexei B.},
title = {Spectral Theory of Automorphic Functions and its Applications},
series = {Math. Appl. (Soviet Series)},
volume = {51},
note = {translated from the Russian by N. B. Lebedinskaya},
publisher = {Kluwer Academic Publishers Group, Dordrecht},
year = {1990},
pages = {xiv+176},
isbn = {0-7923-0487-X},
mrclass = {11F72 (11F12)},
mrnumber = {1135112},
mrreviewer = {Daniel Bump},
doi = {10.1007/978-94-009-1892-4},
url = {https://doi.org/10.1007/978-94-009-1892-4},
zblnumber = {0719.11030},
} -
[Vigneras] M. Vignéras, Arithmétique des Algèbres de Quaternions, Springer, Berlin, 1980, vol. 800.
@BOOK{Vigneras,
author = {Vignéras, Marie-France},
title = {Arithmétique des Algèbres de Quaternions},
series = {Lecture Notes in Math.},
volume = {800},
publisher = {Springer, Berlin},
year = {1980},
pages = {vii+169},
isbn = {3-540-09983-2},
mrclass = {12A80 (16A18)},
mrnumber = {0580949},
mrreviewer = {J. H. H. Chalk},
doi = {10.1007/BFb0091027},
url = {https://doi.org/10.1007/BFb0091027},
zblnumber = {0422.12008},
} -
[Wilkie] A. J. Wilkie, "Diophantine properties of sets definable in o-minimal structures," J. Symbolic Logic, vol. 69, iss. 3, pp. 851-861, 2004.
@ARTICLE{Wilkie,
author = {Wilkie, A. J.},
title = {Diophantine properties of sets definable in o-minimal structures},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {69},
year = {2004},
number = {3},
pages = {851--861},
issn = {0022-4812},
mrclass = {03C64 (03C50 14P10 32B20)},
mrnumber = {2078926},
mrreviewer = {Patrick Speissegger},
doi = {10.2178/jsl/1096901771},
url = {https://doi.org/10.2178/jsl/1096901771},
zblnumber = {1081.03038},
}
Full Article