Generalizations of Siegel’s and Picard’s theorems

Abstract

We prove new theorems that are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ to affine curves. These include results on integral points over varying number fields of bounded degree and results on Kobayashi hyperbolicity. We give a number of new conjectures describing, from our point of view, how we expect Siegel’s and Picard’s theorems to optimally generalize to higher dimensions.

  • [Co] Go to document P. Corvaja and U. Zannier, "A subspace theorem approach to integral points on curves," C. R. Math. Acad. Sci. Paris, vol. 334, iss. 4, pp. 267-271, 2002.
    @article{Co, MRKEY = {1891001},
      AUTHOR = {Corvaja, Pietro and Zannier, Umberto},
      TITLE = {A subspace theorem approach to integral points on curves},
      JOURNAL = {C. R. Math. Acad. Sci. Paris},
      FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      VOLUME = {334},
      YEAR = {2002},
      NUMBER = {4},
      PAGES = {267--271},
      ISSN = {1631-073X},
      MRCLASS = {11G30},
      MRNUMBER = {2003d:11087},
      DOI = {10.1016/S1631-073X(02)02240-9},
      ZBLNUMBER = {1012.11051},
      MRREVIEWER = {Damien Roy},
      }
  • [Co5] Go to document P. Corvaja and U. Zannier, "On the number of integral points on algebraic curves," J. Reine Angew. Math., vol. 565, pp. 27-42, 2003.
    @article{Co5, MRKEY = {2024644},
      AUTHOR = {Corvaja, Pietro and Zannier, Umberto},
      TITLE = {On the number of integral points on algebraic curves},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {565},
      YEAR = {2003},
      PAGES = {27--42},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11G30 (11J25 14G25)},
      MRNUMBER = {2004j:11059},
      DOI = {10.1515/crll.2003.103},
      MRREVIEWER = {Clemens Fuchs},
      ZBLNUMBER = {02021326},
      }
  • [Co3] Go to document P. Corvaja and U. Zannier, "On a general Thue’s equation," Amer. J. Math., vol. 126, iss. 5, pp. 1033-1055, 2004.
    @article{Co3, MRKEY = {2089081},
      AUTHOR = {Corvaja, Pietro and Zannier, Umberto},
      TITLE = {On a general {T}hue's equation},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {126},
      YEAR = {2004},
      NUMBER = {5},
      PAGES = {1033--1055},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {11D59 (11G30 11J25)},
      MRNUMBER = {2005g:11049},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v126/126.5corvaja.pdf},
      ZBLNUMBER = {1125.11022},
      MRREVIEWER = {Clemens Fuchs},
      }
  • [Co2] Go to document P. Corvaja and U. Zannier, "On integral points on surfaces," Ann. of Math., vol. 160, iss. 2, pp. 705-726, 2004.
    @article{Co2, MRKEY = {2123936},
      AUTHOR = {Corvaja, Pietro and Zannier, Umberto},
      TITLE = {On integral points on surfaces},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {160},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {705--726},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G35 (11G30 14G25)},
      MRNUMBER = {2005m:11118},
      URL = {http://projecteuclid.org/getRecord?id=euclid.annm/1111770732},
      MRREVIEWER = {Damien Roy},
      ZBLNUMBER = {02157788},
      DOI = {10.4007/annals.2004.160.705},
      }
  • [Co6] P. Corvaja and U. Zannier, "On the integral points on certain surfaces," Int. Math. Res. Not., p. I, 2006.
    @article{Co6, MRKEY = {2219222},
      AUTHOR = {Corvaja, Pietro and Zannier, Umberto},
      TITLE = {On the integral points on certain surfaces},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2006},
      PAGES = {Art. ID 98623, 20},
      ISSN = {1073-7928},
      MRCLASS = {14G05 (11G35 14J20)},
      MRNUMBER = {2006m:14030},
      ZBLNUMBER = {1131.14023},
      MRREVIEWER = {Timothy D. Browning},
      }
  • [Fa] G. Faltings, "A new application of Diophantine approximations," in A Panorama of Number Theory or the View From Baker’s Garden, Wüstholz, G., Ed., Cambridge: Cambridge Univ. Press, 2002, pp. 231-246.
    @incollection{Fa, MRKEY = {1975455},
      AUTHOR = {Faltings, G.},
      EDITOR = {W{ü}stholz, Gisbert},
      TITLE = {A new application of {D}iophantine approximations},
      BOOKTITLE = {A Panorama of Number Theory or the View From {B}aker's Garden},
      PAGES = {231--246},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2002},
      MRCLASS = {11J68 (11G35 11J95 14G05)},
      MRNUMBER = {2004b:11100},
      ZBLNUMBER = {1080.14025},
      MRREVIEWER = {William A. Cherry},
      }
  • [Fa2] Go to document G. Faltings and G. Wüstholz, "Diophantine approximations on projective spaces," Invent. Math., vol. 116, iss. 1-3, pp. 109-138, 1994.
    @article{Fa2, MRKEY = {1253191},
      AUTHOR = {Faltings, Gerd and W{ü}stholz, Gisbert},
      TITLE = {Diophantine approximations on projective spaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {116},
      YEAR = {1994},
      NUMBER = {1-3},
      PAGES = {109--138},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11J68 (11G35 14K15)},
      MRNUMBER = {95g:11068},
      DOI = {10.1007/BF01231559},
      ZBLNUMBER = {0805.14011},
      MRREVIEWER = {J.-H. Evertse},
      }
  • [Fu2] Go to document H. Fujimoto, "Extensions of the big Picard’s theorem," Tôhoku Math. J., vol. 24, pp. 415-422, 1972.
    @article{Fu2, MRKEY = {0333253},
      AUTHOR = {Fujimoto, Hirotaka},
      TITLE = {Extensions of the big {P}icard's theorem},
      JOURNAL = {Tôhoku Math. J.},
      FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
      VOLUME = {24},
      YEAR = {1972},
      PAGES = {415--422},
      ISSN = {0040-8735},
      MRCLASS = {32H25},
      MRNUMBER = {48 \#11578},
      DOI = {10.2748/tmj/1178241480},
      ZBLNUMBER = {0244.32011},
      MRREVIEWER = {Shoshichi Kobayashi},
      }
  • [Ful] W. Fulton, Intersection Theory, Second ed., New York: Springer-Verlag, 1998.
    @book{Ful, MRKEY = {1644323},
      AUTHOR = {Fulton, William},
      TITLE = {Intersection Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      NUMBER = {2},
      EDITION = {{S}econd},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1998},
      PAGES = {xiv+470},
      ISBN = {3-540-62046-X; 0-387-98549-2},
      MRCLASS = {14C17 (14-02)},
      MRNUMBER = {99d:14003},
      ZBLNUMBER = {0885.14002},
      }
  • [Gr] Go to document M. L. Green, "Holomorphic maps into complex projective space omitting hyperplanes," Trans. Amer. Math. Soc., vol. 169, pp. 89-103, 1972.
    @article{Gr, MRKEY = {0308433},
      AUTHOR = {Green, Mark L.},
      TITLE = {Holomorphic maps into complex projective space omitting hyperplanes},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {169},
      YEAR = {1972},
      PAGES = {89--103},
      ISSN = {0002-9947},
      MRCLASS = {32A30 (32H25)},
      MRNUMBER = {46 \#7547},
      DOI = {10.2307/1996232},
      ZBLNUMBER = {0256.32015},
      MRREVIEWER = {R. O. Kujala},
      }
  • [Gr2] Go to document M. L. Green, "The hyperbolicity of the complement of $2n+1$ hyperplanes in general position in $P_{n}$ and related results," Proc. Amer. Math. Soc., vol. 66, iss. 1, pp. 109-113, 1977.
    @article{Gr2, MRKEY = {0457790},
      AUTHOR = {Green, Mark L.},
      TITLE = {The hyperbolicity of the complement of {$2n+1$} hyperplanes in general position in {$P\sb{n}$} and related results},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {66},
      YEAR = {1977},
      NUMBER = {1},
      PAGES = {109--113},
      ISSN = {0002-9939},
      MRCLASS = {32H20},
      MRNUMBER = {56 \#15994},
      DOI = {10.2307/2041540},
      ZBLNUMBER = {0366.32013},
      MRREVIEWER = {M. Herve},
      }
  • [Ha] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977.
    @book{Ha, MRKEY = {0463157},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Algebraic Geometry},
      SERIES = {Grad. Texts Math.},
      FSERIES = {Graduate Texts in Mathematics},
      NUMBER = {52},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {xvi+496},
      ISBN = {0-387-90244-9},
      MRCLASS = {14-01},
      MRNUMBER = {57 \#3116},
      ZBLNUMBER = {0367.14001},
      MRREVIEWER = {Robert Speiser},
      }
  • [Ii] S. Iitaka, Algebraic Geometry: An Introduction to Birational Geometry of Algebraic Varieties, New York: Springer-Verlag, 1982.
    @book{Ii, MRKEY = {637060},
      AUTHOR = {Iitaka, Shigeru},
      TITLE = {Algebraic Geometry: {A}n Introduction to Birational Geometry of Algebraic Varieties},
      SERIES = {Graduate Texts in Math.},
      FSERIES = {Graduate Texts in Mathematics},
      NUMBER = {76},
      INVNOTE = {North-Holland Mathematical Library, 24},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {x+357},
      ISBN = {0-387-90546-4},
      MRCLASS = {14-01 (14-02 14E05)},
      MRNUMBER = {84j:14001},
      ZBLNUMBER = {0491.14006},
      MRREVIEWER = {Werner Kleinert},
      }
  • [Ka] Go to document Y. Kawamata, "A generalization of Kodaira-Ramanujam’s vanishing theorem," Math. Ann., vol. 261, iss. 1, pp. 43-46, 1982.
    @article{Ka, MRKEY = {675204},
      AUTHOR = {Kawamata, Yujiro},
      TITLE = {A generalization of {K}odaira-{R}amanujam's vanishing theorem},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {261},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {43--46},
      ISSN = {0025-5831},
      CODEN = {MAANA3},
      MRCLASS = {14J10 (14C20 14C40)},
      MRNUMBER = {84i:14022},
      DOI = {10.1007/BF01456407},
      ZBLNUMBER = {0476.14007},
      MRREVIEWER = {T. Matsusaka},
      }
  • [Kl] Go to document S. L. Kleiman, "Toward a numerical theory of ampleness," Ann. of Math., vol. 84, pp. 293-344, 1966.
    @article{Kl, MRKEY = {0206009},
      AUTHOR = {Kleiman, Steven L.},
      TITLE = {Toward a numerical theory of ampleness},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {84},
      YEAR = {1966},
      PAGES = {293--344},
      ISSN = {0003-486X},
      MRCLASS = {14.55 (14.10)},
      MRNUMBER = {34 \#5834},
      DOI = {10.2307/1970447},
      ZBLNUMBER = {0146.17001},
      MRREVIEWER = {J. W. Gray},
      }
  • [La] S. Lang, Fundamentals of Diophantine Geometry, New York: Springer-Verlag, 1983.
    @book{La, MRKEY = {715605},
      AUTHOR = {Lang, Serge},
      TITLE = {Fundamentals of {D}iophantine Geometry},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1983},
      PAGES = {xviii+370},
      ISBN = {0-387-90837-4},
      MRCLASS = {11-02 (11Dxx 11Gxx 14G25)},
      MRNUMBER = {85j:11005},
      ZBLNUMBER = {0528.14013},
      MRREVIEWER = {Gerd Faltings},
      }
  • [La2] S. Lang, Introduction to Complex Hyperbolic Spaces, New York: Springer-Verlag, 1987.
    @book{La2, MRKEY = {886677},
      AUTHOR = {Lang, Serge},
      TITLE = {Introduction to Complex Hyperbolic Spaces},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      PAGES = {viii+271},
      ISBN = {0-387-96447-9},
      MRCLASS = {32H15 (11J99 30D35 32-02 32H20 32H25)},
      MRNUMBER = {88f:32065},
      ZBLNUMBER = {0628.32001},
      MRREVIEWER = {Hirotaka Fujimoto},
      }
  • [La3] S. Lang, Number Theory, III: Diophantine Geometry, New York: Springer-Verlag, 1991.
    @book{La3, MRKEY = {1112552},
      AUTHOR = {Lang, Serge},
      TITLE = {Number Theory, {\rm III}: {D}iophantine Geometry},
      SERIES = {Encycl. Math. Sci.},
      NUMBER = {60},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1991},
      PAGES = {xiv+296},
      ISBN = {3-540-53004-5},
      MRCLASS = {11Gxx (14Gxx)},
      MRNUMBER = {93a:11048},
      ZBLNUMBER = {0744.14012},
      MRREVIEWER = {Joseph H. Silverman},
      }
  • [Ru4] Go to document Y. Liu and M. Ru, "Degeneracy of holomorphic curves in surfaces," Sci. China Ser. A, vol. 48, iss. suppl., pp. 156-167, 2005.
    @article{Ru4, MRKEY = {2156497},
      AUTHOR = {Liu, Yuancheng and Ru, Min},
      TITLE = {Degeneracy of holomorphic curves in surfaces},
      JOURNAL = {Sci. China Ser. {\rm A}},
      FJOURNAL = {Science in China. Series A. Mathematics},
      VOLUME = {48},
      YEAR = {2005},
      NUMBER = {suppl.},
      PAGES = {156--167},
      ISSN = {1006-9283},
      MRCLASS = {32H25 (32H30)},
      MRNUMBER = {2006d:32019},
      DOI = {10.1007/BF02884702},
      MRREVIEWER = {Yoshihiro Aihara},
      ZBLNUMBER = {05210965},
      }
  • [Mo] Go to document S. Mori, "Threefolds whose canonical bundles are not numerically effective," Ann. of Math., vol. 116, iss. 1, pp. 133-176, 1982.
    @article{Mo, MRKEY = {662120},
      AUTHOR = {Mori, Shigefumi},
      TITLE = {Threefolds whose canonical bundles are not numerically effective},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {116},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {133--176},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {14J30 (14E35)},
      MRNUMBER = {84e:14032},
      DOI = {10.2307/2007050},
      ZBLNUMBER = {0557.14021},
      MRREVIEWER = {Ulf Persson},
      }
  • [No2] Go to document J. Noguchi, "Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties," Nagoya Math. J., vol. 83, pp. 213-233, 1981.
    @article{No2, MRKEY = {632655},
      AUTHOR = {Noguchi, Junjiro},
      TITLE = {Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {83},
      YEAR = {1981},
      PAGES = {213--233},
      ISSN = {0027-7630},
      CODEN = {NGMJA2},
      MRCLASS = {32H25 (32D15)},
      MRNUMBER = {84a:32040},
      URL = {http://projecteuclid.org/getRecord?id=euclid.nmj/1118786486},
      ZBLNUMBER = {0429.32003},
      MRREVIEWER = {Pierre Lelong},
      }
  • [No] Go to document J. Noguchi and J. Winkelmann, "Holomorphic curves and integral points off divisors," Math. Z., vol. 239, iss. 3, pp. 593-610, 2002.
    @article{No, MRKEY = {1893854},
      AUTHOR = {Noguchi, Junjiro and Winkelmann, J{ö}rg},
      TITLE = {Holomorphic curves and integral points off divisors},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {239},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {593--610},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {32H30 (11J97 32H25)},
      MRNUMBER = {2003b:32019},
      DOI = {10.1007/s002090100327},
      ZBLNUMBER = {1011.32012},
      MRREVIEWER = {Min Ru},
      }
  • [Ru] M. Ru, Nevanlinna Theory and its Relation to Diophantine Approximation, River Edge, NJ: World Sci. Publ. Co., 2001.
    @book{Ru, MRKEY = {1850002},
      AUTHOR = {Ru, Min},
      TITLE = {Nevanlinna Theory and its Relation to {D}iophantine Approximation},
      PUBLISHER = {World Sci. Publ. Co.},
      FPUBLISHER = {World Scientific Publishing Co.},
      ADDRESS = {River Edge, NJ},
      YEAR = {2001},
      PAGES = {xiv+323},
      ISBN = {981-02-4402-9},
      MRCLASS = {11J97 (11J68 32H30)},
      MRNUMBER = {2002g:11106},
      ZBLNUMBER = {0998.30030},
      MRREVIEWER = {Julie Tzu-Yueh Wang},
      }
  • [Ru3] Go to document M. Ru, "A defect relation for holomorphic curves intersecting hypersurfaces," Amer. J. Math., vol. 126, iss. 1, pp. 215-226, 2004.
    @article{Ru3, MRKEY = {2033568},
      AUTHOR = {Ru, Min},
      TITLE = {A defect relation for holomorphic curves intersecting hypersurfaces},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {126},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {215--226},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {32H25 (32H30)},
      MRNUMBER = {2004k:32026},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v126/126.1ru.pdf},
      ZBLNUMBER = {1044.32009},
      MRREVIEWER = {Yoshihiro Aihara},
      }
  • [Si2] Go to document J. H. Silverman, "Lower bounds for height functions," Duke Math. J., vol. 51, iss. 2, pp. 395-403, 1984.
    @article{Si2, MRKEY = {747871},
      AUTHOR = {Silverman, Joseph H.},
      TITLE = {Lower bounds for height functions},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {51},
      YEAR = {1984},
      NUMBER = {2},
      PAGES = {395--403},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11G10 (11G05 14G25 14K15)},
      MRNUMBER = {87d:11039},
      DOI = {10.1215/S0012-7094-84-05118-4},
      ZBLNUMBER = {0579.14035},
      MRREVIEWER = {J. Top},
      }
  • [Tu] Go to document X. Song and T. J. Tucker, "Dirichlet’s theorem, Vojta’s inequality, and Vojta’s conjecture," Compositio Math., vol. 116, iss. 2, pp. 219-238, 1999.
    @article{Tu, MRKEY = {1686848},
      AUTHOR = {Song, Xiangjun and Tucker, Thomas J.},
      TITLE = {Dirichlet's theorem, {V}ojta's inequality, and {V}ojta's conjecture},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {116},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {219--238},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {11G30 (11J68 11J97 14G40)},
      MRNUMBER = {2000d:11085},
      DOI = {10.1023/A:1000948001301},
      MRREVIEWER = {Yuri Bilu},
      ZBLNUMBER = {0933.11033},
      }
  • [Vo2] P. Vojta, Diophantine Approximations and Value Distribution Theory, New York: Springer-Verlag, 1987.
    @book{Vo2, MRKEY = {883451},
      AUTHOR = {Vojta, Paul},
      TITLE = {Diophantine Approximations and Value Distribution Theory},
      SERIES = {Lecture Notes in Math.},
      FSERIES = {Lecture Notes in Mathematics},
      NUMBER = {1239},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      PAGES = {x+132},
      ISBN = {3-540-17551-2},
      MRCLASS = {11G35 (11J68 14G25 30D35 32H30)},
      MRNUMBER = {91k:11049},
      ZBLNUMBER = {0609.14011},
      MRREVIEWER = {J. Silverman},
      }
  • [Vo6] Go to document P. Vojta, "A refinement of Schmidt’s subspace theorem," Amer. J. Math., vol. 111, iss. 3, pp. 489-518, 1989.
    @article{Vo6, MRKEY = {1002010},
      AUTHOR = {Vojta, Paul},
      TITLE = {A refinement of {S}chmidt's subspace theorem},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {111},
      YEAR = {1989},
      NUMBER = {3},
      PAGES = {489--518},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {11J13 (11G35 14G25)},
      MRNUMBER = {90f:11054},
      DOI = {10.2307/2374670},
      ZBLNUMBER = {0662.14002},
      MRREVIEWER = {Daniel Bertrand},
      }
  • [Vo8] P. Vojta, "Arithmetic discriminants and quadratic points on curves," in Arithmetic Algebraic Geometry, van der Geer, G., Oort, F., and Steenbrink, J., Eds., Boston: Birkhäuser, 1991, pp. 359-376.
    @incollection{Vo8, MRKEY = {1085268},
      AUTHOR = {Vojta, Paul},
      EDITOR = {van der Geer, G. and Oort, F. and Steenbrink, J.},
      TITLE = {Arithmetic discriminants and quadratic points on curves},
      BOOKTITLE = {Arithmetic Algebraic Geometry},
      SERIES = {Progr. Math.},
      NUMBER = {89},
      PAGES = {359--376},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston},
      YEAR = {1991},
      MRCLASS = {11G30 (11G35 14G40)},
      MRNUMBER = {92j:11059},
      ZBLNUMBER = {0749.14018},
      }
  • [Vo7] Go to document P. Vojta, "A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing," J. Amer. Math. Soc., vol. 5, iss. 4, pp. 763-804, 1992.
    @article{Vo7, MRKEY = {1151542},
      AUTHOR = {Vojta, Paul},
      TITLE = {A generalization of theorems of {F}altings and {T}hue-{S}iegel-{R}oth-{W}irsing},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {5},
      YEAR = {1992},
      NUMBER = {4},
      PAGES = {763--804},
      ISSN = {0894-0347},
      MRCLASS = {11G30 (11J25 14G40 14H25)},
      MRNUMBER = {94a:11093},
      DOI = {10.2307/2152710},
      ZBLNUMBER = {0778.11037},
      MRREVIEWER = {Jeffrey D. Vaaler},
      }
  • [Vo1] Go to document P. Vojta, "Integral points on subvarieties of semiabelian varieties, I," Invent. Math., vol. 126, iss. 1, pp. 133-181, 1996.
    @article{Vo1, MRKEY = {1408559},
      AUTHOR = {Vojta, Paul},
      TITLE = {Integral points on subvarieties of semiabelian varieties, {I}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {126},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {133--181},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14G05 (11G10)},
      MRNUMBER = {98a:14034},
      DOI = {10.1007/s002220050092},
      ZBLNUMBER = {1011.11040},
      MRREVIEWER = {Alexandru Buium},
      }
  • [Vo3] Go to document P. Vojta, "On Cartan’s theorem and Cartan’s conjecture," Amer. J. Math., vol. 119, iss. 1, pp. 1-17, 1997.
    @article{Vo3, MRKEY = {1428056},
      AUTHOR = {Vojta, Paul},
      TITLE = {On {C}artan's theorem and {C}artan's conjecture},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {119},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {1--17},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {32H30 (11J13)},
      MRNUMBER = {97m:32041},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.1vojta.pdf},
      ZBLNUMBER = {0877.11040},
      MRREVIEWER = {William A. Cherry},
      }
  • [Vo5] P. Vojta, "Nevanlinna theory and Diophantine approximation," in Several Complex Variables, Schneider, M. and Siu, Y., Eds., Cambridge: Cambridge Univ. Press, 1999, pp. 535-564.
    @incollection{Vo5, MRKEY = {1748613},
      AUTHOR = {Vojta, Paul},
      EDITOR = {Schneider, Michael and Siu, Yum-Tong},
      TITLE = {Nevanlinna theory and {D}iophantine approximation},
      BOOKTITLE = {Several Complex Variables},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      NUMBER = {37},
      PAGES = {535--564},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1999},
      MRCLASS = {11J97 (30D35 32H30)},
      MRNUMBER = {2001j:11072},
      ZBLNUMBER = {0960.32013},
      MRREVIEWER = {William A. Cherry},
      }
  • [Co4] Go to document U. Zannier, "On the integral points on the complement of ramification-divisors," J. Inst. Math. Jussieu, vol. 4, iss. 2, pp. 317-330, 2005.
    @article{Co4, MRKEY = {2135140},
      AUTHOR = {Zannier, Umberto},
      TITLE = {On the integral points on the complement of ramification-divisors},
      JOURNAL = {J. Inst. Math. Jussieu},
      FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institute de Mathématiques de Jussieu},
      VOLUME = {4},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {317--330},
      ISSN = {1474-7480},
      MRCLASS = {11D72 (11G35 14G99)},
      MRNUMBER = {2005m:11055},
      DOI = {10.1017/S1474748005000071},
      ZBLNUMBER = {1089.11021},
      MRREVIEWER = {Clemens Fuchs},
      }
  • [Za] Go to document O. Zariski, "The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface," Ann. of Math., vol. 76, pp. 560-615, 1962.
    @article{Za, MRKEY = {0141668},
      AUTHOR = {Zariski, Oscar},
      TITLE = {The theorem of {R}iemann-{R}och for high multiples of an effective divisor on an algebraic surface},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {76},
      YEAR = {1962},
      PAGES = {560--615},
      ISSN = {0003-486X},
      MRCLASS = {14.20 (14.15)},
      MRNUMBER = {25 \#5065},
      DOI = {10.2307/1970376},
      ZBLNUMBER = {0124.37001},
      MRREVIEWER = {Y. Nakai},
      }

Authors

Aaron Levin

Centro di Ricerca Matematica Ennio De Giorgi
Scuola Normale Superiore
56100 Pisa
Italy