By Pei-Chu Hu,Chung-Chun Yang
Nevanlinna idea (or worth distribution idea) in advanced research is so appealing that one could evidently have an interest in choosing how one of these thought may glance within the non Archimedean research and Diophantine approximations. There are "main theorems" and illness family members that occupy a significant position in N evanlinna conception. They generate loads of purposes in learning forte of meromorphic features, international suggestions of differential equations, dynamics, and so forth. during this ebook, we'll introduce non-Archimedean analogues of Nevanlinna idea and its purposes. In price distribution thought, the most challenge is that given a holomorphic curve f : C -+ M right into a projective type M of size n and a family members 01 of hypersurfaces on M, lower than a formal of non-degeneracy on f, locate the illness relation. If 01 n is a family members of hyperplanes on M = r normally place and if the smallest size of linear subspaces containing the picture f(C) is okay, Cartan conjectured that the sure of illness relation is 2n - ok + 1. ordinarily, if 01 is a relatives of admissible or common crossings hypersurfaces, there are respectively Shiffman's conjecture and Griffiths-Lang's conjecture. the following we checklist the method of this challenge: A. advanced research: (i) consistent ambitions: R. Nevanlinna for n = ok = 1; H. Cartan  for n = okay > 1; E. I. Nochka , , for n > okay ~ 1; Shiffman's conjecture in part solved through Hu-Yang [71J; Griffiths-Lang's conjecture (open).