Meromorphic Functions over Non-Archimedean Fields

Nevanlinna theory (or value distribution theory) in complex analysis is so beautiful that one would naturally be interested in determining how such a theory would look in the non­ Archimedean analysis and Diophantine approximations. There are two "main theorems" and defect relations that occupy a central place in N evanlinna theory. They generate a lot of applications in studying uniqueness of meromorphic functions, global solutions of differential equations, dynamics, and so on. In this book, we will introduce non-Archimedean analogues of Nevanlinna theory and its applications. In value distribution theory, the main problem is that given a holomorphic curve f : C -+ M into a projective variety M of dimension n and a family 01 of hypersurfaces on M, under a proper condition of non-degeneracy on f, find the defect relation. If 01 n is a family of hyperplanes on M = r in general position and if the smallest dimension of linear subspaces containing the image f(C) is k, Cartan conjectured that the bound of defect relation is 2n - k + 1. Generally, if 01 is a family of admissible or normal crossings hypersurfaces, there are respectively Shiffman's conjecture and Griffiths-Lang's conjecture. Here we list the process of this problem: A. Complex analysis: (i) Constant targets: R. Nevanlinna[98] for n = k = 1; H. Cartan [20] for n = k > 1; E. I. Nochka [99], [100],[101] for n > k ~ 1; Shiffman's conjecture partially solved by Hu-Yang [71J; Griffiths-Lang's conjecture (open).

Table of Contents:
1 Basic facts in p-adic analysis.- 1.1 p-adic numbers.- 1.2 Field extensions.- 1.3 Maximum term of power series.- 1.4 Weierstrass preparation theorem.- 1.5 Newton polygons.- 1.6 Non-Archimedean meromorphic functions.- 2 Nevanlinna theory.- 2.1 Characteristic functions.- 2.2 Growth estimates of meromorphic functions.- 2.3 Two main theorems.- 2.4 Notes on the second main theorem.- 2.5 ‘abc’ conjecture over function fields.- 2.6 Waring’s problem over function fields.- 2.7 Exponent of convergence of zeros.- 2.8 Value distribution of differential polynomials.- 3 Uniqueness of meromorphic functions.- 3.1 Adams-Straus’ uniqueness theorems.- 3.2 Multiple values of meromorphic functions.- 3.3 Uniqueness polynomials of meromorphic functions.- 3.4 Unique range sets of meromorphic functions.- 3.5 The Frank-Reinders’ technique.- 3.6 Some urscm for M(?) and A(?).- 3.7 Some ursim for meromorphic functions.- 3.8 Unique range sets for multiple values.- 4 Differential equations.- 4.1 Malmquist-type theorems.- 4.2 Generalized Malmquist-type theorems.- 4.3 Further results on Malmquist-type theorems.- 4.4 Admissible solutions of some differential equations.- 4.5 Differential equations of constant coefficients.- 5 Dynamics.- 5.1 Attractors and repellers.- 5.2 Riemann-Hurwitz relation.- 5.3 Fixed points of entire functions.- 5.4 Normal families.- 5.5 Montel’s theorems.- 5.6 Fatou-Julia theory.- 5.7 Properties of the Julia set.- 5.8 Iteration of z ? zd.- 5.9 Iteration of z ? z2 + c.- 6 Holomorphic curves.- 6.1 Multilinear algebra.- 6.2 The first main theorem of holomorphic curves.- 6.3 The second main theorem of holomorphic curves.- 6.4 Nochka weight.- 6.5 Degenerate holomorphic curves.- 6.6 Uniqueness of holomorphic curves.- 6.7 Second main theorem for hypersurfaces.- 6.8Holomorphic curves into projective varieties.- 7 Diophantine approximations.- 7.1 Schmidt’s subspace theorems.- 7.2 Vojta’s conjecture.- 7.3 General subspace theorems.- 7.4 Ru-Vojta’s subspace theorem for moving targets.- 7.5 Subspace theorem for degenerate mappings.- A The Cartan conjecture for moving targets.- A.1 Non-degenerate holomorphic curves.- A.2 The Steinmetz lemma.- A.3 A defect relation for moving targets.- A.4 The Ru-Stoll techniques.- A.5 Growth of the Steinmetz-Stoll mappings.- A.6 Moving targets in subgeneral position.- A.7 Moving targets in general position.- Symbols.