The second main theorem for small functions and related problems

Abstract

We shall establish the following three results in more general forms. (1) The second main theorem for small functions. Let f be a meromorphic function on the complex plane C. Let a1, . . . , aq be distinct meromorphic functions on C. Assume that ai are small with respect to f ; i.e., T (r, a) 0 (Corollary 1). Here as usual in Nevanlinna theory, the terms T (r, f) and N(r, ai, f) denote for the characteristic function and the truncated counting function, respectively. (2) Application to functional equations. Let KC be the field of meromorphic functions on C. For a function ψ : R>0 → R, put KψC = {a ∈ KC; T (r, a) ≤ O(ψ(r)) ||}, which is a subfield of KC. Then the following holds: Let F (x, y) ∈ KψC [x, y] be a polynomial in two variables over K ψ C . Assume that the curve F (x, y) = 0 over KψC has genus greater than one. If ζ1, ζ2 ∈ KC satisfy the functional equation F (ζ1, ζ2) = 0, then both ζ1 and ζ2 are contained in KψC (Corollary 2). (3) Height inequality for curves over function fields. Let k be a function field of one variable over C. Let X be a smooth projective curve over k, let D ⊂ X be a reduced divisor, let L be an ample line bundle on X and let e > 0. Then we have hk,KX (D)(P ) ≤ N (1) k,S(D,P ) + dk(P ) + ehk,L(P ) +Oe(1) for all P ∈ X(k)\D (Theorem 5). Here the notations are introduced in [V1], [V3] (see also section 9). Our proof uses Ahlfors’ theory of covering surfaces and the geometry of the moduli space of q-pointed stable curves of genus 0.