Abstract
Beginning with the work of Osgood [65], it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) has many similarities with Roth’s theorem on diophantine approximation.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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
Ahlfors, L.V.: The theory of meromorphic curves. Acta Soc. Sci. Fennicae. Nova Ser. A. 3(4), 1–31 (1941)
Artin, E: Algebraic numbers and algebraic functions. Gordon and Breach Science, New York (1967)
Ashline, G.L.: The defect relation of meromorphic maps on parabolic manifolds. Mem. Amer. Math. Soc. 139(665), 78 (1999). ISSN 0065-9266
Belyĭ, G.V.: Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk SSSR Ser. Mat. 43(2), 267–276, 479 (1979). ISSN 0373-2436
Bloch, A.: Sur les systèmes de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension. J. de Math. 5, 19–66 (1926)
Bombieri, E.: Roth’s theorem and the abc-conjecture. Preprint ETH Zürich (1994)
Borel, E.: Sur les zéros des fonctions entières. Acta Math. 20, 357–396 (1897)
Brownawell, W.D., Masser, D.W.: Vanishing sums in function fields. Math. Proc. Cambridge Philos. Soc. 100(3), 427–434 (1986).
Campana, F.: Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble) 54(3), 499–630 (2004). ISSN 0373-0956
Carlson, J., Griffiths, P.: A defect relation for equidimensional holomorphic mappings between algebraic varieties. Ann. Math. 95(2), 557–584 (1972) ISSN 0003-486X
Cartan, H.: Sur les zéros des combinaisons linéaires de p fonctions holomorphes données. Mathematica (Cluj) 7, 5–29 (1933)
Cherry, W., Ye, Z.: Nevanlinna’s theory of value distribution. Springer Monographs in Mathematics. Springer, Berlin (2001). ISBN 3-540-66416-5
Corvaja, P., Zannier, U.: A subspace theorem approach to integral points on curves. C. R. Math. Acad. Sci. Paris 334(4), 267–271 (2002). ISSN 1631-073X
Corvaja, P., Zannier, U.: On integral points on surfaces. Ann. Math. 160(2), 705–726 (2004). ISSN 0003-486X
Deligne, P.: Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, vol. 163. Springer, Berlin (1970)
Dufresnoy, J.: Théorie nouvelle des familles complexes normales. Applications à l’étude des fonctions algébroïdes. Ann. Sci. École Norm. Sup. 61(3), 1–44 (1944) ISSN 0012-9593
Eisenbud, D.: Commutative algebra (with a view toward algebraic geometry), vol. 150 In: Graduate texts in mathematics. Springer, New York (1995). ISBN 0-387-94268-8; 0-387-94269-6
Elkies, N.D.: ABC implies Mordell. Int. Math. Res. Notices, 1991(7), 99–109 (1991). ISSN 1073-7928
Evertse, J.-H.: On sums of S-units and linear recurrences. Compositio Math. 53(2), 225–244 (1984). ISSN 0010-437X
Evertse, J.-H., Ferretti, R.G.: A generalization of the subspace theorem with polynomials of higher degree. In: Diophantine approximation. Dev. Math., vol.16, pp. 175–198. Springer, NewYork (2008). ArXiv:math.NT/0408381
Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983). ISSN 0020-9910
Faltings, G.: Finiteness theorems for abelian varieties over number fields. In: Arithmetic geometry (Storrs, Conn., 1984), pp. 9–27. Springer, New York (1986). Translated from the German original [Invent.Math.73(3), 349–366 (1983); Invent.Math.75(2), 381 (1984); MR 85g:11026ab] by Edward Shipz
Faltings, G.: Diophantine approximation on abelian varieties. Ann. Math. (2) 133(3), 549–576 (1991). ISSN 0003-486X
Faltings, G.: The general case of S. Lang’s conjecture. In: Barsotti Symposium in algebraic geometry (Abano Terme, 1991), Perspect. Math., vol. 15, pp. 175–182. Academic, CA (1994)
Ferretti, R.G.: Mumford’s degree of contact and Diophantine approximations. Compositio Math. 121(3), 247–262 (2000). ISSN 0010-437X
Fujimoto, H.: Extensions of the big Picard’s theorem. Tôhoku Math. J. 24(2), 415–422 (1972). ISSN 0040-8735
Gasbarri, C.: The strong abc conjecture over function fields (after McQuillian and Yamanoi). Astérisque, (326): Exp. No. 989, viii, 219–256 (2010). Séminaire Bourbaki. Vol. 2007/2008 (2009)
Gillet, H., Soulé, C.: On the number of lattice points in convex symmetric bodies and their duals. Israel J. Math. 74(2–3), 347–357 (1991). ISSN 0021-2172
Goldberg, A.A., Ostrovskii, I.V.: Value distribution of meromorphic functions. In: Translations of mathematical monographs, vol. 236. American Mathematical Society, RI (2008). ISBN 978-0-8218-4265-2; Translated from the 1970 Russian original by Mikhail Ostrovskii, With an appendix by Alexandre Eremenko and James K. Langley
Green, M.: Holomorphic maps into complex projective space omitting hyperplanes. Trans. Amer. Math. Soc. 169, 89–103 (1972). ISSN 0002-9947
Green, M.: Some Picard theorems for holomorphic maps to algebraic varieties. Amer. J.Math. 97, 43–75 (1975). ISSN 0002-9327
Green, M., Griffiths, P.: Two applications of algebraic geometry to entire holomorphic mappings. In: The Chern symposium 1979 (Proc. Internat. Sympos., Berkeley, CA, 1979), pp.41–74. Springer, New York (1980)
Grothendieck, A. etal.: Revêtements étales et groupe fondamental. Springer, Berlin (1971). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud, Lecture Notes in Mathematics, vol. 224; arXiv:math.AG/0206203
Grothendieck, A., Dieudonné, J.-A.-E.: Éléments de géométrie algébrique. Publ. Math. IHES, 4, 8, 11, 17, 20, 24, 28, and 32, (1960–1967). ISSN 0073-8301.
Gunning, R.C.: Introduction to holomorphic functions of several variables. vol. II (Local theory). The Wadsworth and Brooks/Cole Mathematics Series. Wadsworth and Brooks/Cole Advanced Books and Software, CA (1990). ISBN 0-534-13309-6
Hartshorne, R.: Algebraic geometry. Springer, New York (1977). ISBN 0-387-90244-9; Graduate Texts in Mathematics, No. 52
Hayman, W.K.: Meromorphic functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964)
Hodge, W.V.D., Pedoe, D.: Methods of algebraic geometry, vol. II. Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties. Cambridge University Press, Cambridge (1952)
Iitaka, S.: Algebraic geometry, An introduction to birational geometry of algebraic varieties, Graduate texts in mathematics, vol.76. Springer, New York (1982). ISBN 0-387-90546-4; North-Holland Mathematical Library, 24
Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), pp. 191–224. Johns Hopkins University Press, MD (1989)
Kawamata, Y.: On Bloch’s conjecture. Invent. Math. 57(1), 97–100 (1980). ISSN 0020-9910
Koblitz, N.: p-adic numbers, p-adic analysis, and zeta-functions, Graduate texts in mathematics, vol.58, 2nd edn. Springer, New York (1984). ISBN 0-387-96017-1
Kunz, E.: Kähler differentials. Advanced Lectures in Mathematics. Friedr. Vieweg, Braunschweig (1986). ISBN 3-528-08973-3
Lang, S.: Integral points on curves. Inst. Hautes Études Sci. Publ. Math. 6, 27–43 (1960). ISSN 0073-8301
Lang, S.: Algebraic number theory. Addison-Wesley, MA (1970)
Lang, S.: Fundamentals of diophantine geometry. Springer, New York (1983). ISBN 0-387-90837-4
Lang, S.: Hyperbolic and diophantine analysis. Bull. Amer. Math. Soc. (N.S.) 14(2), 159–205 (1986). ISSN 0273-0979
Lang, S.: Number theory III: Diophantine geometry, Encyclopaedia of mathematical sciences, vol.60. Springer, Berlin (1991). ISBN 3-540-53004-5
Lang, S., Cherry, W.: Topics in Nevanlinna theory, Lecture Notes in mathematics, vol. 1433. Springer, Berlin (1990). ISBN 3-540-52785-0; With an appendix by Zhuan Ye
Levin, A.: The dimensions of integral points and holomorphic curves on the complements of hyperplanes. Acta Arith. 134(3), 259–270 (2008). ISSN 0065-1036
Levin, A.: Generalizations of Siegel’s and Picard’s theorems. Ann. Math. (2) 170(2),609–655 (2009). ISSN 0003-486X
Levin, A., McKinnon, D., Winkelmann, J.: On the error terms and exceptional sets in conjectural second main theorems. Q. J. Math. 59(4), 487–498 (2008). ISSN 0033-5606, doi:10.1093/qmath/ham052, http://dx.doi.org/10.1093/qmath/ham052
Lu, S.S.-Y.: On meromorphic maps into varieties of log-general type. In: Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol.52, pp. 305–333. AMS, RI (1991)
Mahler, K.: On a theorem of Liouville in fields of positive characteristic. Canadian J. Math. 1, 397–400 (1949). ISSN 0008-414X
McQuillan, M.: Diophantine approximations and foliations. Inst. Hautes Études Sci. Publ. Math. 87, 121–174 (1998). ISSN 0073-8301
McQuillan, M.: Non-commutative Mori theory. IHES preprint IHES/M/00/15 (2000)
McQuillan, M.: Canonical models of foliations. Pure Appl. Math. Q. 4 (3, part 2), 877–1012 (2008). ISSN 1558-8599
McQuillan, M.: Old and new techniques in function field arithmetic. (2009, submitted)
Neukirch, J.: Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, vol. 322 [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1999). ISBN 3-540-65399-6; Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder
Nevanlinna, R.: Analytic functions. Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162. Springer, New York (1970)
Noguchi, J.: Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties. Nagoya Math. J. 83, 213–233 (1981) ISSN 0027-7630
Noguchi, J.: On Nevanlinna’s second main theorem. In: Geometric complex analysis (Hayama, 1995), pp. 489–503. World Scientific, NJ (1996)
Noguchi, J.: On holomorphic curves in semi-abelian varieties. Math. Z. 228(4), 713–721 (1998). ISSN 0025-5874
Osgood, C.F.: Effective bounds on the Diophantine approximation of algebraic functions over fields of arbitrary characteristic and applications to differential equations. Nederl. Akad. Wetensch. Proc. Ser. A 78 Indag. Math. 37, 105–119 (1975)
Osgood, C.F.: A number theoretic-differential equations approach to generalizing Nevanlinna theory. Indian J. Math. 23(1–3), 1–15 (1981). ISSN 0019-5324
Osgood, C.F.: Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better. J. Number Theor. 21(3), 347–389 (1985). ISSN 0022-314X
Roth, K.F.: Rational approximations to algebraic numbers. Mathematika 2,1–20; corrigendum, 168 (1955). ISSN 0025-5793
Ru, M.: On a general form of the second main theorem. Trans. Amer. Math. Soc. 349(12), 5093–5105 (1997). ISSN 0002-9947
Ru, M.: Nevanlinna theory and its relation to Diophantine approximation. World Scientific, NJ (2001). ISBN 981-02-4402-9
Ru, M.: Holomorphic curves into algebraic varieties. Ann. Math. (2) 169(1), 255–267 (2009). ISSN 0003-486X, doi:10.4007/annals.2009.169.255, http://dx.doi.org/10.4007/annals.2009.169.255
Schmidt, W.M.: Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467. Springer, Berlin (1991). ISBN 3-540-54058-X
Serre, J.-P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble 6, 1–42 (1955/1956). ISSN 0373-0956
Serre, J.-P.: Corps locaux. Hermann, Paris (1968). Deuxième édition, Publications de l’Université de Nancago, No. VIII
Serre, J.-P.: Lectures on the Mordell-Weil theorem. Aspects of Mathematics, E15. Friedr. Vieweg, Braunschweig (1989). ISBN 3-528-08968-7; Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt
Shabat, B.V.: Distribution of values of holomorphic mappings, Translations of mathematical monographs, vol.61. American Mathematical Society, RI (1985). ISBN 0-8218-4514-4; Translated from the Russian by J. R. King, Translation edited by Lev J. Leifman
Shiffman, B.: Introduction to the Carlson-Griffiths equidistribution theory. In: Value distribution theory (Joensuu, 1981), Lecture Notes in Math., vol. 981, pp. 44–89. Springer, Berlin (1983)
Silverman, J.H.: The theory of height functions. In: Arithmetic geometry (Storrs, Conn., 1984), pp. 151–166. Springer, New York (1986)
Siu, Y.-T.: Hyperbolicity problems in function theory. In: Five decades as a mathematician and educator, pp. 409–513. World Scientific, NJ (1995)
Siu, Y.-T., Yeung, S.-K.: A generalized Bloch’s theorem and the hyperbolicity of the complement of an ample divisor in an abelian variety. Math. Ann. 306(4), 743–758 (1996). ISSN 0025-5831
Siu, Y.-T., Yeung, S.-K.: Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees. Amer. J. Math. 119(5), 1139–1172 (1997). ISSN 0002-9327
Stewart, C.L., Tijdeman, R.: On the Oesterlé-Masser conjecture. Monatsh. Math. 102(3), 251–257 (1986). ISSN 0026-9255
Stoll, W.: Value distribution of holomorphic maps into compact complex manifolds. Lecture Notes in Mathematics, vol. 135. Springer, Berlin (1970)
Szpiro, L., Ullmo, E., Zhang, S.: Équirépartition des petits points. Invent. Math. 127(2), 337–347 (1997). ISSN 0020-9910
vander Poorten, A.J., Schlickewei, H.P.: The growth conditions for recurrence sequences. Macquarie Math. Reports, 82-0041 (1982)
van Frankenhuijsen, M.: The ABC conjecture implies Vojta’s height inequality for curves. J.Number Theor. 95(2), 289–302 (2002). ISSN 0022-314X
van Frankenhuijsen, M.: ABC implies the radicalized Vojta height inequality for curves. J.Number Theor. 127(2), 292–300 (2007). ISSN 0022-314X
Vojta, P.: Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239. Springer, Berlin (1987). ISBN 3-540-17551-2
Vojta, P.: A refinement of Schmidt’s subspace theorem. Amer. J. Math. 111(3), 489–518 (1989). ISSN 0002-9327
Vojta, P.: Integral points on subvarieties of semiabelian varieties. I. Invent. Math. 126(1), 133–181 (1996). ISSN 0020-9910
Vojta, P.: On Cartan’s theorem and Cartan’s conjecture. Amer. J. Math. 119(1),1–17 (1997). ISSN 0002-9327
Vojta, P.: A more general abc conjecture. Int. Math. Res. Notices, 1998(21), 1103–1116 (1998). ISSN 1073-7928
Vojta, P.: Integral points on subvarieties of semiabelian varieties. II. Amer. J. Math. 121(2), 283–313 (1999). ISSN 0002-9327
Vojta, P.: On the ABC conjecture and Diophantine approximation by rational points. Amer. J.Math. 122(4), 843–872 (2000). ISSN 0002-9327
Vojta, P.: Correction to: On the ABC conjecture and Diophantine approximation by rational points [Amer. J. Math. 122(4), 843–872 (2000); MR1771576 (2001i:11094)]. Amer. J. Math. 123(2), 383–384 (2001). ISSN 0002-9327
Vojta, P.: Jets via Hasse-Schmidt derivations. In: Diophantine geometry, CRM Series, vol.4, pp. 335–361. Ed. Norm., Pisa (2007). arXiv:math.AG/0407113
Vojta, P.: Nagata’s embedding theorem. arXiv:0706.1907 (2007, to appear)
Vojta, P.: On McQuillan’s tautological inequality and the Weyl-Ahlfors theory of associated curves. arXiv:0706.3044 (2008, to appear)
Weil, A.: L’arithmétique sur les courbes algébriques. Acta Math. 52, 281–315 (1928)
Weil, A.: Arithmetic on algebraic varieties. Ann. Math. 53(2), 412–444 (1951). ISSN 0003-486X
Weyl, H., Weyl, J.: Meromorphic curves. Ann. Math. (2) 39(3), 516–538 (1938). ISSN 0003-486X
Wong, P.-M.: On the second main theorem of Nevanlinna theory. Amer. J. Math. 111(4), 549–583 (1989). ISSN 0002-9327
Yamanoi, K.: Algebro-geometric version of Nevanlinna’s lemma on logarithmic derivative and applications. Nagoya Math. J. 173, 23–63 (2004). ISSN 0027-7630
Yamanoi, K.: The second main theorem for small functions and related problems. Acta Math. 192(2), 225–294 (2004). ISSN 0001-5962
Acknowledgements
Partially supported by NSF grant DMS-0500512
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Vojta, P. (2011). Diophantine Approximation and Nevanlinna Theory. In: Corvaja, P., Gasbarri, C. (eds) Arithmetic Geometry. Lecture Notes in Mathematics(), vol 2009. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15945-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-15945-9_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15944-2
Online ISBN: 978-3-642-15945-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)