Algebraic values of meromorphic functions on Riemann surfaces
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.
algebraic values of meromorphic functions
scientific article; zbMATH DE number 4012729