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The second main theorem for small functions and related problems. (English) Zbl 1203.30035
In this outstanding paper the author proves a Nevanlinna truncated second main theorem for \(q\) small functions and the height inequality for the case of curves over function fields, which is a generalization of a conjecture posed by P. Vojta. A fundamental role is played by Ahlfors’ theory of covering surfaces and the moduli space of \(q\)-pointed stable curves of genus zero as a target space.
From the introduction: One of the most interesting results in value distribution theory is the defect relation obtained by R. Nevanlinna: If \(f\) is a non-constant meromorphic function on the complex plane \(\mathbb C\), then for an arbitrary collection of distinct \(a_1,\dots, a_q\in\mathbb P^1\), the following defect relation holds: \[ \sum_{i=1}^q(\delta(a_i, f)+\theta(a_i, f)) \leq 2. \tag{1.1} \]
Here, as usual in Nevanlinna theory, the terms \((\delta(a_i, f)\) and \(\theta(a_i, f)\) are defined by \[ \delta(a_i, f)=\liminf_{r\to\infty}\left(1-\frac{N(r,a_i,f)}{T(r,f)}\right), \] \[ \theta(a_i, f) = \liminf_{r\to\infty}\frac{N(r, a_i, f) - \overline N(r, a_i, f)}{T(r,f)}, \] and hence satisfy \(0\leq \delta(a_i, f)\leq 1\) and \(0\leq \theta(a_i,f)\leq 1\). For the definitions of the terms \(T(r, f)\), \(N(r, a_i, f)\) and \(\overline N(r, a_i, f)\) are as usual.
A problem, suggested by Nevanlinna, is whether the defect relation is still true when we replace the constants \(a_i\) by an arbitrary collection of distinct small functions \(a_i\) with respect to \(f\). Here we say that a meromorphic function \(a\) on \(\mathbb C\) is a small function with respect to \(f\) if a satisfies the condition \(T(r, a)=o(T(r, f))\) as \(r\to\infty\). Nevanlinna pointed out that the case \(q=3\) for this question is valid, because we may reduce the problem to the case that \(a_1, a_2\) and \(a_3\) are all constants by using a Möbius transform. But for the case \(q>3\), this method does not work.
Later, N. Steinmetz [J. Reine Angew. Math. 368, 134–141 (1986; Zbl 0598.30045)] and C. F. Osgood [J. Number Theory 21, 347–389 (1985; Zbl 0575.10032)] proved that \(\sum_{i=1}^q(\delta(a_i, f)\leq 2\) for distinct small functions \(a_i\). Their methods, which may be regarded as generalizations of Nevanlinna’s original proof of (1.1), are based on the consideration of differential polynomials in \(f\) and \(a_i\), \(1 \leq i\leq q\). Though Nevanlinna used only the first-order derivative of \(f\), Steinmetz and Osgood used higher-order derivatives of \(f\). Hence the truncation level of the counting function is greater than one in general.
However, it is hoped that the generalization of (1.1) for small functions is true with the form including the term \(\theta(a_i, f)\). In this paper, we give a solution for this problem by the following theorem.
Theorem 1. Let \(Y\) and \(B\) be Riemann surfaces with proper, surjective holomorphic maps \(\pi_Y: Y\to\mathbb C\) and \(\pi_B: B\to\mathbb C\). Assume that \(\pi_Y\) factors through \(\pi_B\), i.e., there exists a proper, surjective holomorphic map \(\pi: Y\to B\) such that \(\pi_Y= \pi_B\circ\pi\). Let \(f\) be a nonconstant meromorphic function on \(Y\). Let \(a_1, \dots,a_q\) be distinct meromorphic functions on \(B\). Assume that \(f\neq a_i\circ\pi\) for \(i = 1,\dots, q\). Then for all \(c>0\), there exists a positive constant \(C(\varepsilon)>0\) such that the following inequality holds:
\[ (q-2-\varepsilon)T(r, f) \leq \sum_{i=1}^q \overline N(r, a_i\circ\pi, f)+N_{\text{ram}\,\pi_Y} (r)+C(\varepsilon) \left(\sum_{i=1}^q (T(r,a_i)+N_{\text{ram}\,\pi_B} (r)\right)\;\|.\tag{1.2} \]
Here the symbol \(\|\) means that the stated estimate holds when \(r\notin E\) for some exceptional set \(E\subset \mathbb R_{>0}\) with \(\int_E d \log \log r < \infty\). The term \(N_{\text{ram}\,\pi_Y} (r)\) counts the ramification points of \(\pi_Y\). In the case \(Y=\mathbb C\) and \(\pi_Y=\text{id}_{\mathbb C}\), we have \(N_{\text{ram}\,\pi_Y} (r)=0\). Similarly for \(N_{\text{ram}\,\pi_B}(r)\).
Theorem 2 gives an algebraic version, in which the domains of the functions considered are compact Riemann surfaces.
A generalization to the results of this paper are given in Int. J. Math. 17, No. 4, 417-440 (2006; Zbl 1101.30031).

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
14G05 Rational points
14G25 Global ground fields in algebraic geometry
11J97 Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.)
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