Nevanlinna Theory in Characteristic P and Applications

Abstract

Let K be a complete ultrametric algebraically closed field of characteristic p. We show that Nevanlinna’s main Theorem holds, with however some corrections. Then, many results obtained in characteristic zero have generalization. When p ≠ 0, we have to make new proofs in most of the cases. Many algebraic curves admit no parametrizations by meromorphic functions in K, or by unbounded meromorphic functions inside a disk, like in zero characteristic, provided we assume one of the function to have a non zero derivative. More generally, certain functional equations have no solution. In zero characteristic, results previously obtained are somewhat generalised, and then are extended to any characteristic. In functional equations f ^m + g ⁿ = 1, conclusions also are similar to those obtained in zero characteristic, provided we replace m, n by $\tilde m = m|m{|_p}$ , ñ = n|n| _p . We consider the Yoshida Equation in charactersitic p ≥ 0 and characterise all solutions when it has constant coefficients: this generalizes previous results in characteristic zero but with a more general form involving polynomials with a zero derivative. Proofs are given in a preprint where applications to the abc-problem are also obtained.