On p-adic meromorphic functions. (English) Zbl 0544.30039

Let D denote the open unit disc in \({\mathbb{C}}_ p ({\mathbb{C}}_ p\) denotes the completion of the algebraic closure of \({\mathbb{Q}}_ p)\). For a meromorphic function \(\phi\) on D and \(a\in {\mathbb{C}}_ p\) a certain real function \(T(\phi,a,t)\) of \(t>0\) is introduced which measures the set of zero’s of \(\phi\)-a and a growth of the function \(\phi\). This function \(T(\phi,a,t)\) is an analogue of a function in Nevanlinna theory. It is shown that, under the condition that \(T(\phi,a,t)\) is unbounded, the function \(\phi\) is determined by the divisors \(\phi^{-1}(a_ 1)\), \(\phi^{-1}(a_ 2)\), \(\phi^{-1}(a_ 3)\) where \(a_ 1,a_ 2,a_ 3\) are three different points of \({\mathbb{C}}_ p\). We note that \(T(\phi,a,t)\) has no clear meaning in the p-adic case and that the theorem above can easily be proved under the weaker assumption that \(\phi\) is not the quotient of two bounded holomorphic functions on D.
Reviewer: M.van der Put


30G06 Non-Archimedean function theory
12J10 Valued fields
12J15 Ordered fields
Full Text: DOI


[1] Y. Amice, Interpolation \(p\)-adique et transformation de Mellin-Mazur selon Hà Huy Khóai, Group d’étude d’analyse ultramétrique , 1978/1979, Jan. 1979.
[2] Y. Amice, Les nombres \(p\)-adiques , Presses Universitaires de France, Paris, 1975. · Zbl 0313.12104
[3] Ha Zuĭ Hoaĭ, \(p\)-adic interpolation , Mat. Zametki 26 (1979), no. 1, 101-112, 158. · Zbl 0421.12020
[4] Hà Huy Khoái, \(p\)-adic interpolation and the Mellin-Mazur transform , Acta Math. Vietnam. 5 (1980), no. 1, 77-99 (1981). · Zbl 0476.12016
[5] M. Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet , Inst. Hautes Études Sci. Publ. Math. (1962), no. 14, 47-75. · Zbl 0119.03701
[6] Ju. I. Manin, \(p\)-adic automorphic functions , Current problems in mathematics, Vol. 3 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1974, 5-92, 259. (loose errata). · Zbl 0375.14007
[7] B. Mazur, \(p\)-adic meromorphic continuation of Gauss sums , · Zbl 1105.11008
[8] R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes , Paris, 1929. · JFM 55.0773.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.