## 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

### MSC:

 30G06 Non-Archimedean function theory 12J10 Valued fields 12J15 Ordered fields

### Keywords:

 [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