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).
