Advertisement

Nevanlinna Theory over Function Fields

  • Junjiro Noguchi
  • Jörg Winkelmann
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 350)

Abstract

We develop here Nevanlinna theory described in Sects.  4.1 and  4.2 for holomorphic curves over algebraic function fields. This is understood as an approximation theory of algebraic functions by algebraic functions. Vojta (Diophantine Approximations and Value Distribution Theory, 1987) formulated Diophantine approximation theory from the viewpoint of Nevanlinna theory, noticing their analogies. From that viewpoint Nevanlinna theory is an approximation theory of complex numbers by transcendental meromorphic functions. This brought a new viewpoint to the both theories and has activated their research. The theory over algebraic function fields is considered to be situated in the middle of them.

Keywords

Line Bundle Diophantine Equation Counting Function Diophantine Approximation Nevanlinna Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

Arakelov, S.Ju.

  1. [71]
    Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR, Ser. Mat. 35 (1971), 1277–1302. MathSciNetGoogle Scholar

Bombieri, E.

  1. [90]
    The Mordell conjecture revisited, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 17 (1990), 615–640. MathSciNetMATHGoogle Scholar

Borel, E.

  1. [1897]
    Sur les zéros des fonctions entières, Acta Math. 20 (1897), 357–396. MathSciNetCrossRefMATHGoogle Scholar

Brownawell, W.D. and Masser, D.M.

  1. [86]
    Vanishing sums in function fields, Math. Proc. Camb. Philos. Soc. 100 (1986), 427–434. MathSciNetCrossRefMATHGoogle Scholar

Chai, C.-L.

  1. [91]
    A note on Manin’s theorem of the kernel, Am. J. Math. 113 no. 3 (1991), 387–389. CrossRefMATHGoogle Scholar

Coleman, R.F.

  1. [90]
    Manin’s proof of the Mordell conjecture over function fields, Enseign. Math. 36 (1990) 393–427. MathSciNetMATHGoogle Scholar

Corvaja, P. and Noguchi, J.

  1. [12]
    A new unicity theorem and Erdös’ problem for polarized semi-abelian varieties, Math. Ann. 353 (2012), 439–464. MathSciNetCrossRefMATHGoogle Scholar

Corvaja, P. and Zannier, U.

  1. [04a]
    On integral points on surfaces, Ann. Math. 160 (2004a), 705–726. MathSciNetCrossRefMATHGoogle Scholar

Faltings, G.

  1. [83a]
    Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983a), 337–347. MathSciNetCrossRefMATHGoogle Scholar
  2. [83b]
    Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983b), 349–366. MathSciNetCrossRefMATHGoogle Scholar
  3. [91]
    Diophantine approximation on abelian varieties, Ann. Math. 133 (1991), 549–576. MathSciNetCrossRefMATHGoogle Scholar

Granville, A. and Tucker, T.J.

  1. [02]
    It’s as easy as abc, Not. Am. Math. Soc. 49 no. 10 (2002), 1224–1231. MathSciNetMATHGoogle Scholar

Grauert, H.

  1. [62]
    Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368. MathSciNetCrossRefMATHGoogle Scholar
  2. [65]
    Mordells Vermutung über rationale Punkte auf Algebraischen Kurven und Funktionenköper, Publ. Math. IHÉS 25 (1965), 131–149. MathSciNetCrossRefGoogle Scholar

Hindry, M. and Silverman, J.H.

  1. [00]
    Diophantine Geometry: An Introduction, G.T.M. 201, Springer, Berlin, 2000. CrossRefMATHGoogle Scholar

Horst, C.

  1. [90]
    A finiteness criterion for compact varieties of surjective holomorphic mappings, Kodai Math. J. 13 (1990), 373–376. MathSciNetCrossRefMATHGoogle Scholar

Imayoshi, Y. and Shiga, H.

  1. [88]
    A finiteness theorem for holomorphic families of Riemann surfaces, Holomorphic Functions and Moduli Vol. II, D. Drasin (Ed.), Springer, Berlin, 1988. Google Scholar

Kobayashi, S.

  1. [75]
    Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975), 153–166. MathSciNetMATHGoogle Scholar
  2. [98]
    Hyperbolic Complex Spaces, Grundl. Math. Wissen. 318, Springer, Berlin, 1998. CrossRefMATHGoogle Scholar

Kobayashi, S. and Ochiai, T.

  1. [75]
    Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), 7–16. MathSciNetCrossRefMATHGoogle Scholar

Lang, S.

  1. [60]
    Integral points on curves, Publ. Math. IHÉS 6 (1960), 27–43. CrossRefGoogle Scholar
  2. [74]
    Higher dimensional Diophantine problems, Bull. Am. Math. Soc. 80 (1974), 779–787. CrossRefMATHGoogle Scholar
  3. [83]
    Fundamentals of Diophantine Geometry, Springer, Berlin, 1983. CrossRefMATHGoogle Scholar
  4. [91]
    Number Theory III, Encycl. Math. Sci. 60, Springer, Berlin, 1991. CrossRefMATHGoogle Scholar

Manin, Y.

  1. [63]
    Rational points of algebraic curves over function fields, Izv. Akad. Nauk SSSR, Ser. Mat. 27 (1963), 1395–1440. MathSciNetMATHGoogle Scholar

Mason, R.C.

  1. [84]
    Diophantine Equations over Function Fields, London Math. Soc. Lect. Notes 96, Cambridge University Press, Cambridge, 1984. CrossRefMATHGoogle Scholar

Miyano, T. and Noguchi, J.

  1. [91]
    Moduli spaces of harmonic and holomorphic mappings and Diophantine geometry, Prospects in Complex Geometry, Proc. 25th Taniguchi International Symposium, Katata/Kyoto, 1989, Lect. Notes Math. 1468, pp. 227–253, Springer, Berlin, 1991. CrossRefGoogle Scholar

Mordell, L.J.

  1. [22]
    On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc. Camb. Philos. Soc. 21 (1922), 179–192. Google Scholar

Mumford, D.

  1. [75]
    Curves and Their Jacobians, University of Michigan Press, Ann Arbor, 1975. MATHGoogle Scholar

Noguchi, J.

  1. [81b]
    A higher dimensional analogue of Mordell’s conjecture over function fields, Math. Ann. 258 (1981b), 207–212. MathSciNetCrossRefMATHGoogle Scholar
  2. [85a]
    Hyperbolic fibre spaces and Mordell’s conjecture over function fields, Publ. Res. Inst. Math. Sci. 21 (1985a), 27–46. MathSciNetCrossRefMATHGoogle Scholar
  3. [88]
    Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces, Invent. Math. 93 (1988), 15–34. MathSciNetCrossRefMATHGoogle Scholar
  4. [92]
    Meromorphic mappings into compact hyperbolic complex spaces and geometric Diophantine problems, Int. J. Math. 3 (1992), 277–289. MathSciNetCrossRefMATHGoogle Scholar
  5. [97]
    Nevanlinna–Cartan theory over function fields and a Diophantine equation, J. Reine Angew. Math. 487 (1997), 61–83; Correction to the paper: Nevanlinna–Cartan theory over function fields and a Diophantine equation, J. Reine Angew. Math. 497 (1998), 235. MathSciNetMATHGoogle Scholar
  6. [03a]
    An arithmetic property of Shirosaki’s hyperbolic projective hypersurface, Forum Math. 15 (2003a), 935–941. MathSciNetMATHGoogle Scholar
  7. [03b]
    Nevanlinna Theory in Several Variables and Diophantine Approximation (in Japanese), Kyoritsu, Tokyo, 2003b. Google Scholar
  8. [09]
    Value distribution and distribution of rational points, Spectral Analysis in Geometry and Number Theory, M. Kotani et al. (Eds.), Contemp. Math. 484, pp. 165–176, Am. Math. Soc., Providence, 2009. CrossRefGoogle Scholar

Noguchi, J. and Winkelmann, J.

  1. [02]
    Holomorphic curves and integral points off divisors, Math. Z. 239 (2002), 593–610. MathSciNetCrossRefMATHGoogle Scholar

Oesterlé, J.

  1. [88]
    Nouvelles approches du “théorème” de Fermat, Sem. Bourbaki 1987/88, Astérisque 161–162, Exp. No. 694, pp. 165–186, 1988. Google Scholar

Parshin, A.N.

  1. [68]
    Algebraic curves over function fields, I, Izv. Akad. Nauk SSSR, Ser. Mat. 32 (1968), 1145–1170. Google Scholar

Ru, M. and Wong, P.-M.

  1. [91]
    Integral points of P n−{2n+1 hyperplanes in general position}, Invent. Math. 106 (1991), 195–216. MathSciNetCrossRefMATHGoogle Scholar

Saito, M.-H. and Zucker, S.

  1. [91]
    Classification of nonrigid families of K3 surfaces and a finiteness theorem of Arakelov type, Math. Ann. 289 (1991), 1–31. MathSciNetCrossRefMATHGoogle Scholar

Sarnak, P. and Wang, L.

  1. [95]
    Some hypersurfaces in P 4 and the Hasse-principle, C.R. Math. Acad. Sci. Paris, Sér. I 321 (1995), 319–322. MathSciNetMATHGoogle Scholar

Schmidt, W.M.

  1. [80]
    Diophantine Approximation, Lect. Notes Math. 785, Springer, Berlin, 1980. MATHGoogle Scholar
  2. [91]
    Diophantine Approximations and Diophantine Equations, Lect. Notes Math. 1467, Springer, Berlin, 1991. MATHGoogle Scholar

Shafarevich, I.

  1. [63]
    Algebraic numbers, Proc. Int. Congr. Math. 1962, pp. 163–176, Inst. Mittag-Leffler, 1963. Google Scholar

Siegel, C.L.

  1. [26]
    The integer solutions of the equation y 2=ax n+bx n−1+⋯+k, J. Lond. Math. Soc. 1 (1926), 66–68. MATHGoogle Scholar

Stothers, W.W.

  1. [81]
    Polynomial identities and Hauptmoduln, Quart. J. Math. Oxford (2) 32 (1981), 349–370. MathSciNetCrossRefMATHGoogle Scholar

Suzuki, Makoto

  1. [94]
    Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and hyperbolic fibre spaces, J. Math. Soc. Jpn. 46 (1994), 681–698. CrossRefMATHGoogle Scholar

Vojta, P.

  1. [87]
    Diophantine Approximations and Value Distribution Theory, Lect. Notes Math. 1239, Springer, Berlin, 1987. MATHGoogle Scholar
  2. [91]
    Siegel’s theorem in the compact case, Ann. Math. (2) 133 no. 3 (1991), 509–548. MathSciNetCrossRefMATHGoogle Scholar
  3. [96]
    Integral points on subvarieties of semiabelian varieties, I, Invent. Math. 126 (1996), 133–181. MathSciNetCrossRefMATHGoogle Scholar
  4. [99]
    Integral points on subvarieties of semiabelian varieties, II, Am. J. Math. 121 (1999), 283–313. MathSciNetCrossRefMATHGoogle Scholar

Voloch, J.F.

  1. [85]
    Diagonal equations over function fields, Bol. Soc. Bras. Mat. 16 (1985), 29–39. MathSciNetCrossRefMATHGoogle Scholar

Waldschmidt, M.

  1. [00]
    Diophantine Approximation on linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Grundl. Math. Wiss. 326, Springer, Berlin, 2000. CrossRefMATHGoogle Scholar

Wang, J.T.-Y.

  1. [96a]
    The truncated second main theorem of function fields, J. Number Theory 58 (1996a), 139–157. MathSciNetCrossRefMATHGoogle Scholar
  2. [96b]
    Effective Roth theorem of function fields, Rocky Mt. J. Math. 26 (1996b), 1225–1234. CrossRefMATHGoogle Scholar

Zaidenberg, M.G.

  1. [90]
    A function-field analog of the Mordell conjecture: A noncompact version, Math. USSR, Izv. 35 (1990), 61–81. MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Junjiro Noguchi
    • 1
    • 2
  • Jörg Winkelmann
    • 3
  1. 1.The University of TokyoTokyoJapan
  2. 2.Tokyo Institute of TechnologyTokyoJapan
  3. 3.Ruhr-University BochumBochumGermany

Personalised recommendations