P-Adic Nevanlinna Theory and Functional Analysis

Abstract

Let K be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the p-adic Nevanlinna theory to functional equations of the form $g = R \circ f$, where R ∈ K(x), f, g are meromorphic functions in K, or in an “open disk”, g satisfying conditions on the order of its zeros and poles. In various cases we show that f and g must be constant when they are meromorphic in all K, or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular, we have an easy way to obtain again Picard-Berkovich’s theorem for curves of genus 1 and 2. These results apply to equations f ^m + g ⁿ = 1, when f, g are meromorphic functions, or entire functions in K or analytic functions in an “open disk”. We finally apply the method to Yoshida’s equation ${{y'}^{m}} = F(y),$, when F ∈ K(X), and we describe the only case where solutions exist: F must be a polynomial of the form A(y—a) ^d where m — d divides m, and then the solutions are the functions of the form $f\left( x \right) = \alpha + \lambda {\left( {x - \alpha } \right)^{\frac{m}{{m - d}}}}$ with ${\lambda ^{m - d}}{\left( {\frac{m}{{m - d}}} \right)^m} = A$.