  Gevrey series are ubiquitous in analysis; any series satisfying some
(possibly non-linear) analytic differential equation is Gevrey of some rational
order. The present work stems from two observations: 1) the classical Gevrey
series, e.g. generalized hypergeometric series with rational parameters, enjoy
arithmetic counterparts of the Archimedean Gevrey condition; 2) the
differential operators which occur in classical treatises on special functions
have a rather simple structure: they are either Fuchsian, or have only two
singularities, 0 and infinity, one of them regular, the other irregular with a
single slope...
  The main idea of the paper is that the arithmetic property 1) accounts for
the global analytic property 2): the existence of an injective arithmetic
Gevrey solution at one point determines to a large extent the global behaviour
of a differential operator with polynomial coefficients.
  Proofs use both p-adic and complex analysis, and a detailed arithmetic study
of the Laplace transform.
