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Non-Archimedean Nevanlinna theory
in several variables and
the non-Archimedean Nevanlinna inverse problem


Authors: William Cherry and Zhuan Ye
Journal: Trans. Amer. Math. Soc. 349 (1997), 5043-5071
MSC (1991): Primary 11J99, 11S80, 30D35, 32H30, 32P05
DOI: https://doi.org/10.1090/S0002-9947-97-01874-6
MathSciNet review: 1407485
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Abstract | References | Similar Articles | Additional Information

Abstract: Cartan's method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space, there exists a non-Archimedean analytic function with the given defects at the specified hyperplanes, and with no other defects.


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  • [A-S] W. W. Adams and E. G. Straus, Non-Archimedean Analytic Functions Taking the Same Values at the Same Points, Illinois J. Math. 15 (1971), 418-424. MR 43:3504
  • [Ah] L. Ahlfors, The theory of meromorphic curves, Acta Soc. Sci. Finn., N.S., A, III (1941), 1-31. MR 2:357b
  • [Am] Y. Amice, Les nombres p-adiques, Presses Universitaires de France, 1975. MR 56:5510
  • [BGR] S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean Analysis, Springer-Verlag, 1984. MR 86b:32031
  • [Bo 1] A. Boutabaa, Sur la théorie de Nevanlinna $p$-adique, Thése de Doctorat, Université Paris 7, 1991.
  • [Bo 2] A. Boutabaa, Theorie de Nevanlinna p-adique, Manuscripta Math. 67 (1990), 251-269. MR 91m:30039
  • [Bo 3] A. Boutabaa, Sur les courbes holomorphes $p$-adiques, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), 29-52. CMP 97:02
  • [C-G] J. Carlson and P. Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. Math. 95 (1972), 557-584. MR 47:497
  • [Ca] H. Cartan, Sur les zéros des combinaisons linéaires de $p$ fonctions holomorphes données, Mathematica 7 (1933), 5-31.
  • [Ch 1] W. Cherry, Hyperbolic $p$-Adic Analytic Spaces, Ph.D. Thesis, Yale University, 1993.
  • [Ch 2] W. Cherry, Non-Archimedean analytic curves in Abelian varieties, Math. Ann. 300 (1994), 393-404. MR 96i:14021
  • [Co 1] C. Corrales-Rodrigáñez, Nevanlinna Theory in the $p$-Adic Plane, Ph.D. Thesis, University of Michigan, 1986.
  • [Co 2] C. Corrales-Rodrigáñez, Nevanlinna Theory on the $p$-Adic Plane, Annales Polonici Mathematici LVII (1992), 135-147. MR 93h:30067
  • [Dr] D. Drasin, The inverse problem of the Nevanlinna theory, Acta Math. 138 (1976), 83-151. MR 58:28502
  • [Fu] H. Fujimoto, Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into $P^{N_1}(\mathbf {C})\times \cdots\times P^{N_k}(\mathbf {C}),$ Japan. J. Math. 11 (1985), 233-264. MR 88m:32049
  • [G-K] P. Griffiths and J. King, Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145-220. MR 55:721
  • [Kh 1] Hà Huy Khoái, On $p$-Adic Meromorphic Functions, Duke Math. J. 50 (1983), 695-711. MR 85d:11092
  • [Kh 2] Hà Huy Khoái, Heights for $p$-Adic Meromorphic Functions and Value Distribution Theory, Max-Planck-Institut Für Mathematik. 89-67 (1989). Cf. MR 96j:32044
  • [Kh 3] Hà Huy Khoái, Heights for $p$-Adic Holomorphic Functions of Several Variables, Max-Planck-Institut Für Mathematik. 89-83 (1989). Cf. MR 92f:32059
  • [Kn] H. Kneser, Zur Theorie der gebrochenen Funktionen mehrerer Veränderlicher, Jber. Deutsch. Math. Verein. 48 (1938/39), 1-28.
  • [K-Q] Hà Huy Khoái and My Vinh Quang, On $p$-adic Nevanlinna Theory, in Lecture Notes in Mathematics 1351, Springer-Verlag 1988, pp. 146-158. MR 90e:11153
  • [K-T] Hà Huy Khoái and Mai Van Tu, $p$-Adic Nevanlinna-Cartan Theorem, Internat. J. Math. 6 (1995), 719-731. MR 96k:32053
  • [La] S. Lang, Introduction to Complex Hyperbolic Spaces, Springer-Verlag, 1987. MR 88f:32065
  • [L-C] S. Lang and W. Cherry, Topics in Nevanlinna Theory, Springer Lecture Notes in Mathematics 1433, Springer-Verlag, 1990. MR 91k:32025
  • [Lü] W. Lütkebohmert, Letter to R. Remmert, 1995.
  • [Ne 1] R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Paris, 1929.
  • [Ne 2] R. Nevanlinna, Analytic Functions, Springer-Verlag, 1970. MR 43:5003
  • [Os 1] C.F. Osgood, A number theoretic-differential equations approach to generalizing Nevanlinna theory, Indian J. of Math. 23 (1981), 1-15. MR 85b:30043
  • [Os 2] C.F. Osgood, Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better, J. Number Theory 21 (1985), 347-389. MR 87f:11046
  • [St] W. Stoll, The Ahlfors-Weyl theory of meromorphic maps on parabolic manifolds, Lecture Notes in Mathematics 981, Springer-Verlag, 1983. MR 85c:32045
  • [Vi] A. Vitter, The lemma of the logarithmic derivative in several complex variables, Duke Math. J. 44 (1977), 89-104. MR 55:5903
  • [Vo] P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Mathematics 1239, Springer-Verlag, 1987. MR 91k:11049
  • [Wo] P.M. Wong, On the second main theorem of Nevanlinna theory, Amer. J. Math. 111 (1989), 549-583. MR 91b:32030
  • [W-S] P.M. Wong and W. Stoll, Second main theorem of Nevanlinna theory for non-equidimensional meromorphic maps, Amer. J. Math. 116 (1994), 1031-1071. MR 95g:32042
  • [Ye 1] Z. Ye, On Nevanlinna's Error Terms, Duke Math. J. 64 (1991), 243-260. MR 93a:30039
  • [Ye 2] Z. Ye, On Nevanlinna's second main theorem in projective space, Invent. Math. 122 (1995), 475-507. MR 96j:32030

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Additional Information

William Cherry
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: wcherry@math.lsa.umich.edu

Zhuan Ye
Affiliation: Department of Mathematics, Northern Illinois University, DeKalb, Illinois 60115
Email: ye@math.niu.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01874-6
Keywords: Non-Archimedean, Nevanlinna theory, $p$-adic, inverse problem, defect relations, projective space
Received by editor(s): October 14, 1995
Received by editor(s) in revised form: June 17, 1996
Additional Notes: Financial support for the first author was provided in part by NSF grant # DMS-9505041
Article copyright: © Copyright 1997 American Mathematical Society

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