Skip to main content
SpringerLink
Log in

Nevanlinna Theory over Function Fields

  • Chapter
Nevanlinna Theory in Several Complex Variables and Diophantine Approximation

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 350))

  • 1966 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

Arakelov, S.Ju.

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

    MathSciNet  Google Scholar 

Bombieri, E.

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

    MathSciNet  MATH  Google Scholar 

Borel, E.

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

Chai, C.-L.

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

    Article  MATH  Google Scholar 

Coleman, R.F.

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

    MathSciNet  MATH  Google Scholar 

Corvaja, P. and Noguchi, J.

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

    Article  MathSciNet  MATH  Google Scholar 

Corvaja, P. and Zannier, U.

  1. On integral points on surfaces, Ann. Math. 160 (2004a), 705–726.

    Article  MathSciNet  MATH  Google Scholar 

Faltings, G.

  1. Arakelov’s theorem for abelian varieties, Invent. Math. 73 (1983a), 337–347.

    Article  MathSciNet  MATH  Google Scholar 

  2. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983b), 349–366.

    Article  MathSciNet  MATH  Google Scholar 

  3. Diophantine approximation on abelian varieties, Ann. Math. 133 (1991), 549–576.

    Article  MathSciNet  MATH  Google Scholar 

Granville, A. and Tucker, T.J.

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

    MathSciNet  MATH  Google Scholar 

Grauert, H.

  1. Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368.

    Article  MathSciNet  MATH  Google Scholar 

  2. Mordells Vermutung über rationale Punkte auf Algebraischen Kurven und Funktionenköper, Publ. Math. IHÉS 25 (1965), 131–149.

    Article  MathSciNet  Google Scholar 

Hindry, M. and Silverman, J.H.

  1. Diophantine Geometry: An Introduction, G.T.M. 201, Springer, Berlin, 2000.

    Book  MATH  Google Scholar 

Horst, C.

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

    Article  MathSciNet  MATH  Google Scholar 

Imayoshi, Y. and Shiga, H.

  1. 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. Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975), 153–166.

    MathSciNet  MATH  Google Scholar 

  2. Hyperbolic Complex Spaces, Grundl. Math. Wissen. 318, Springer, Berlin, 1998.

    Book  MATH  Google Scholar 

Kobayashi, S. and Ochiai, T.

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

    Article  MathSciNet  MATH  Google Scholar 

Lang, S.

  1. Integral points on curves, Publ. Math. IHÉS 6 (1960), 27–43.

    Article  Google Scholar 

  2. Higher dimensional Diophantine problems, Bull. Am. Math. Soc. 80 (1974), 779–787.

    Article  MATH  Google Scholar 

  3. Fundamentals of Diophantine Geometry, Springer, Berlin, 1983.

    Book  MATH  Google Scholar 

  4. Number Theory III, Encycl. Math. Sci. 60, Springer, Berlin, 1991.

    Book  MATH  Google Scholar 

Manin, Y.

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

    MathSciNet  MATH  Google Scholar 

Mason, R.C.

  1. Diophantine Equations over Function Fields, London Math. Soc. Lect. Notes 96, Cambridge University Press, Cambridge, 1984.

    Book  MATH  Google Scholar 

Miyano, T. and Noguchi, J.

  1. 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.

    Chapter  Google Scholar 

Mordell, L.J.

  1. 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. Curves and Their Jacobians, University of Michigan Press, Ann Arbor, 1975.

    MATH  Google Scholar 

Noguchi, J.

  1. A higher dimensional analogue of Mordell’s conjecture over function fields, Math. Ann. 258 (1981b), 207–212.

    Article  MathSciNet  MATH  Google Scholar 

  2. Hyperbolic fibre spaces and Mordell’s conjecture over function fields, Publ. Res. Inst. Math. Sci. 21 (1985a), 27–46.

    Article  MathSciNet  MATH  Google Scholar 

  3. Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces, Invent. Math. 93 (1988), 15–34.

    Article  MathSciNet  MATH  Google Scholar 

  4. Meromorphic mappings into compact hyperbolic complex spaces and geometric Diophantine problems, Int. J. Math. 3 (1992), 277–289.

    Article  MathSciNet  MATH  Google Scholar 

  5. 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.

    MathSciNet  MATH  Google Scholar 

  6. An arithmetic property of Shirosaki’s hyperbolic projective hypersurface, Forum Math. 15 (2003a), 935–941.

    MathSciNet  MATH  Google Scholar 

  7. Nevanlinna Theory in Several Variables and Diophantine Approximation (in Japanese), Kyoritsu, Tokyo, 2003b.

    Google Scholar 

  8. 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.

    Chapter  Google Scholar 

Noguchi, J. and Winkelmann, J.

  1. Holomorphic curves and integral points off divisors, Math. Z. 239 (2002), 593–610.

    Article  MathSciNet  MATH  Google Scholar 

Oesterlé, J.

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

Sarnak, P. and Wang, L.

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

    MathSciNet  MATH  Google Scholar 

Schmidt, W.M.

  1. Diophantine Approximation, Lect. Notes Math. 785, Springer, Berlin, 1980.

    MATH  Google Scholar 

  2. Diophantine Approximations and Diophantine Equations, Lect. Notes Math. 1467, Springer, Berlin, 1991.

    MATH  Google Scholar 

Shafarevich, I.

  1. Algebraic numbers, Proc. Int. Congr. Math. 1962, pp. 163–176, Inst. Mittag-Leffler, 1963.

    Google Scholar 

Siegel, C.L.

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

    MATH  Google Scholar 

Stothers, W.W.

  1. Polynomial identities and Hauptmoduln, Quart. J. Math. Oxford (2) 32 (1981), 349–370.

    Article  MathSciNet  MATH  Google Scholar 

Suzuki, Makoto

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

    Article  MATH  Google Scholar 

Vojta, P.

  1. Diophantine Approximations and Value Distribution Theory, Lect. Notes Math. 1239, Springer, Berlin, 1987.

    MATH  Google Scholar 

  2. Siegel’s theorem in the compact case, Ann. Math. (2) 133 no. 3 (1991), 509–548.

    Article  MathSciNet  MATH  Google Scholar 

  3. Integral points on subvarieties of semiabelian varieties, I, Invent. Math. 126 (1996), 133–181.

    Article  MathSciNet  MATH  Google Scholar 

  4. Integral points on subvarieties of semiabelian varieties, II, Am. J. Math. 121 (1999), 283–313.

    Article  MathSciNet  MATH  Google Scholar 

Voloch, J.F.

  1. Diagonal equations over function fields, Bol. Soc. Bras. Mat. 16 (1985), 29–39.

    Article  MathSciNet  MATH  Google Scholar 

Waldschmidt, M.

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

    Book  MATH  Google Scholar 

Wang, J.T.-Y.

  1. The truncated second main theorem of function fields, J. Number Theory 58 (1996a), 139–157.

    Article  MathSciNet  MATH  Google Scholar 

  2. Effective Roth theorem of function fields, Rocky Mt. J. Math. 26 (1996b), 1225–1234.

    Article  MATH  Google Scholar 

Zaidenberg, M.G.

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Emeritus Junjiro Noguchi

  2. Tokyo Institute of Technology, Tokyo, Japan

    Emeritus Junjiro Noguchi

  3. Ruhr-University Bochum, Bochum, Germany

    Jörg Winkelmann

Authors
  1. Emeritus Junjiro Noguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jörg Winkelmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Japan

About this chapter

Cite this chapter

Noguchi, J., Winkelmann, J. (2014). Nevanlinna Theory over Function Fields. In: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation. Grundlehren der mathematischen Wissenschaften, vol 350. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54571-2_8

Download citation

Publish with us

Policies and ethics

Search

Navigation