Algebraic values of meromorphic functions on Riemann surfaces
type: Item
instance of: scholarly article
title: Algebraic values of meromorphic functions on Riemann surfaces
zbMATH Open document ID: 0624.10027
DOI: 10.1016/0022-314X(87)90028-X
author: Isao Wakabayashi
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Consider a Riemann surface R obtained from a compact surface by deleting a finite number of points, \(q_ 1,...,q_ m\). The main result of the paper under review asserts that if \(f_ 1\) and \(f_ 2\) are meromorphic functions on R, algebraically independent over the rationals, and if \(f_ 1\) (resp. \(f_ 2)\) is of order at most \(\rho_ k\) (resp. \(\lambda \rho_ k)\) at \(q_ k\), then \(card(S)\leq (1+\lambda)(\rho_ 1+...\rho_ m)d,\) where S is the set of points where \(f_ 1\) and \(f_ 2\) are holomorphic and have Taylor coefficients satisfying \(\delta^{k+1}(d^ kf_ i/dz^ k)\in I_ K,\delta\) being an integer depending on the point, and K being an algebraic number field of degree d over the rationals, with \(I_ K\) the algebraic integers of K. (For precise technical conditions and definition of the order, see the paper.) The result extends the Schneider-Lang theorem on algebraic values of meromorphic functions in the complex plane. A key ingredient in the proof is a generalization of Jensen's formula.
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 4012729
zbMATH Keywords: Schneider-Lang theorem
zbMATH Keywords: algebraic values of meromorphic functions
MaRDI profile type: Publication
cites work: Q5593095
cites work: Q4079677
cites work: Q3931656
cites work: Q5559456
cites work: Nombres transcendants
cites work: Q4141911
full work available at URL: 0022-314x(87)90028-x
OpenAlex ID: W2002662821
description: scientific article; zbMATH DE number 4012729