Meromorphic Functions over Non-Archimedean Fields

1. Front Matter

   Pages i-viii

   PDF

2. Basic facts in p-adic analysis

   • Pei-Chu Hu, Chung-Chun Yang

   Pages 1-31

3. Nevanlinna theory

   • Pei-Chu Hu, Chung-Chun Yang

   Pages 33-75

4. Uniqueness of meromorphic functions

   • Pei-Chu Hu, Chung-Chun Yang

   Pages 77-113

5. Differential equations

   • Pei-Chu Hu, Chung-Chun Yang

   Pages 115-138

6. Dynamics

   • Pei-Chu Hu, Chung-Chun Yang

   Pages 139-175

7. Holomorphic curves

   • Pei-Chu Hu, Chung-Chun Yang

   Pages 177-223

8. Diophantine approximations

   • Pei-Chu Hu, Chung-Chun Yang

   Pages 225-241

9. Back Matter

   Pages 243-295

   PDF

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).

• Meromorphic function
• Nevanlinna theory
• differential equation
• distribution
• iteration

• Shandong University, Shandong, P.R. China

  Pei-Chu Hu

• University of Science and Technology, Clearwater Bay, Hong Kong, China

  Chung-Chun Yang

Policies and ethics