On an ultrahyperelliptic surface with Picard constant three
type: Item
instance of: scholarly article
title: On an ultrahyperelliptic surface with Picard constant three
zbMATH Open document ID: 0826.30026
author: Kiyoshi Niino
published in: Kodai Mathematical Journal
publication date: 1995-11-27T00:00:00Z
review text: Let \(S\) be an open Riemann surface, \(M(S)\) the family of nonconstant meromorphic functions and \(P(f)\) the number of values which are not taken by \(f \in M(S)\). The Picard constant \(P(S)\) of \(S\) is defined by \(P(S) = \{P(f) : f \in M (S)\}\). Let \(R\) be the ultrahyperelliptic surface defined by \(y^2 = G(z)\), where \(G(z) = 1 - 2 \beta_1 e^{H(z)} - 2 \beta_2 e^{L(z)} + \beta^2_1 e^{2H (z)} - 2 \beta_1 \beta_2 e^{H (z) + L(z)} + \beta^2_2 e^{2L(z)}\) with two nonconstant entire functions \(H\) and \(L\) \((H(0) = L(0) = 0)\) and two non- zero constants \(\beta_1\) and \(\beta_2\). Then we know that \(3 \leq P (R) < 4\) and \(P(R) = 4\) in the following four cases: (i) \(H = L\); (ii) \(H = 2L\), \(16 \beta_1 = \beta^2_2\); (iii) \(2H = L\), \(\beta^2_1 = 16 \beta_2\); (iv) \(H = - L\), \(16 \beta_1 \beta_2 = 1\). Is \(P(R) = 3\) right with the above four exceptional cases? \textit{M. Ozawa} [Kōdai Math. Sem. Rep. 19, 245-256 (1967; Zbl 0164.088)] proved that this is valid when \(H\) and \(L\) are polynomials. The author proves that this statement is true when \(L = \lambda H + K\), where \(\lambda\) is a rational number and \(K\) is an entire function satisfying \(m(r,e^K) = o(m(r,e^H))\), \(r \to \infty\), outside a set of \(r\) of finite measure.
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30F99
Mathematics Subject Classification ID: 14J99
zbMATH DE Number: 558890
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039852
OpenAlex ID: W1994264758
cites work: On the existence of analytic mappings between two ultrahyperelliptic surfaces
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cites work: On complex analytic mappings
cites work: On ultrahyperelliptic surfaces
cites work: On an ultrahyperelliptic surface whose Picard's constant is three
DOI: 10.2996/KMJ/1138039852
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description: scientific article; zbMATH DE number 558890