## Fonctions et éléments algébriques. (Algebraic functions and elements).(French)Zbl 0591.12018

The aim of this paper is to study congruence properties of Taylor coefficients of algebraic functions. Let $${\mathbb{C}}_ p$$ be the completion of some algebraic closure of $${\mathbb{Q}}_ p$$. Let E be the completion of $${\mathbb{C}}_ p(x)$$ for the Gauss norm. An analytic function in the unit disk of $${\mathbb{C}}_ p$$ will be called algebraic function if it is algebraic over the field E (then it is bounded), and it will be called algebraic element if it is a uniform limit (on the unit disk) of algebraic functions.
The first step is to find a ”fine” primitive element for each finite extension of the field E. Then we can prove finiteness properties for the ring generated by the Taylor coefficients of an algebraic function (or element). Afterwards, although $${\mathbb{C}}_ p$$ is a non discrete valuation field, we can use the discrete valuation technique.
We define a Frobenius operator $$\phi$$ over the ring $${\mathcal B}$$ of bounded analytic functions in the unit disk of $${\mathbb{C}}_ p$$ ($$\phi$$ (f) looks like $$f(x^ p))$$. For a function f of $${\mathcal B}$$, we consider the condition (C): the vector space generated over E by the $$\phi^ n(f)$$ is of finite dimension. It is shown that the algebraic functions fulfill (C) (the vector space being E(f)) and that any function satisfying (C) is an algebraic element. Translated for a linear differential equation with coefficients in E, this result gives a deep link between the existence of a strong Frobenius structure (something like to be an F-crystal) and the algebraicity of the solutions.
Finally, it is proved that a function of $${\mathcal B}$$ is an algebraic element if and only if, for every n, the sequence of its Taylor coefficients can be obtained modulo $$p^ n$$ by a p-automata.
Many examples of algebraic elements are given (exponential, hypergeometric, Bessel functions, diagonals of rational fractions of several variables,...), showing that this notion appears in several branches of mathematics: algebraic geometry, combinatorics,...

### MSC:

 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 12H25 $$p$$-adic differential equations 68Q45 Formal languages and automata 34G10 Linear differential equations in abstract spaces 14F30 $$p$$-adic cohomology, crystalline cohomology
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