Some results in view of Nevanlinna theory

Here we discuss and survey some results on rational points of algebraic varieties and Nevanlinna theory in relation to Lang’s conjectures and Vojta’s. We will also announce new results in the case of function fields and also the second main theorem for holomorphic curves into semiabelian varieties.

[1]  The second main theorem for holomorphic curves into semi-Abelian varieties , 2002 .

[2]  S. Lang,et al.  Integral points on curves , 1960 .

[3]  G. Faltings.  Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , 1983 .

[4]  M. Shirosaki.  Another proof of the defect relation for moving targets , 1991 .

[5]  Holomorphic curves and integral points off divisors , 1999, math/9902014.

[6]  M. Ru.  Integral points and the hyperbolicity of the complement of hypersurfaces. , 1993 .

[7]  M. Ru,et al.  Integral points of Pn−{2n+1 hyperplanes in general position} , 1991 .

[8]  J. Noguchi.  A higher dimensional analogue of Mordell's conjecture over function fields , 1981 .

[9]  S. Lang.  Number Theory III , 1991 .

[10]  M. Green.  Holomorphic maps into complex projective space omitting hyperplanes , 1972 .

[11]  Yu. I. Manin,et al.  Rational points on algebraic curves over function elds , 1996 .

[12]  Julietzu-Yueh Wang.  An Effective Roth's Theorem for Function Fields , 1996 .

[13]  J. Noguchi.  Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces , 1987 .

[14]  The value distribution of holomorphic curves into semi-Abelian varieties , 1999, math/9912086.

[15]  A. Bloch.  Sur les systèmes de fonctions uniformes satisfaisant à l'équation d'une variété algébrique dont l'irrégularité dépasse la dimension , 1926 .

[16]  Gerd Faltings,et al.  Diophantine approximation on abelian varieties , 1991 .

[17]  J. Noguchi.  Holomorphic curves in algebraic varieties , 1977 .

[18]  Paul Vojta.  Diophantine Approximations and Value Distribution Theory , 1987 .

[19]  J. Noguchi.  Hyperbolic Fibre Spaces and Mordell's Conjecture over Function Fields , 1985 .

[20]  Shôshichi Kobayashi.  Hyperbolic complex spaces , 1998 .

[21]  Michael McQuillan,et al.  A Toric Extension of Faltings' 'Diophantine Approximation on Abelian Varieties' , 2001 .

[22]  T. Ochiai.  On holomorphic curves in algebraic varieties with ample irregularity , 1977 .

[23]  Y. Siu,et al.  Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees , 1996, math/9610203.

[24]  S. Lang.  Introduction to Complex Hyperbolic Spaces , 1987 .

[25]  S. Lang,et al.  Introduction to Transcendental Numbers , 1967 .

[26]  Y. Siu,et al.  A generalized Bloch's theorem and the hyperbolicity of the complement of an ample divisor in an Abelian variety , 1996 .

[27]  J. Noguchi.  Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties , 1981, Nagoya Mathematical Journal.

[28]  Integral points on subvarieties of semiabelian varieties, I , 1996 .

[29]  P. Griffiths.  Holomorphic mappings: Survey of some results and discussion of open problems , 1972 .

[30]  Julietzu-Yueh Wang.  S-INTEGRAL POINTS OF PN - 2N + 1 HYPERPLANES IN GENERAL POSITION) OVER NUMBER FIELDS AND FUNCTION FIELDS , 1996 .

[31]  H. Fujimoto.  Extensions of the big Picard's theorem , 1972 .

[32]  C. B. Allendoerfer,et al.  Hyperbolic manifolds and holomorphic mappings , 1970 .

[33]  H. Grauert.  Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper , 1965 .

[34]  S. Lang.  Higher dimensional diophantine problems , 1974 .

[35]  J. Noguchi.  MEROMORPHIC MAPPINGS INTO COMPACT HYPERBOLIC COMPLEX SPACES AND GEOMETRIC DIOPHANTINE PROBLEMS , 1992 .