Meromorphic Functions over Non-Archimedean Fields [Paperback]

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).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 Warings problem over function fields.- 2.7 Exponent of convergence of zerlsÅ