Equations diff\'erentielles $p$-adiques et S\'eries Gevrey arithm\'etiques

Let $K$ be a number field. This paper is devoted to a $p$-adic study of the algebraic differential equation of $K[x,d\mathrm{/}dx]$ with minimal degree in $x$ annihilating a given $E$-function. Precisely, we prove a conjecture of yves Andr'e (conjecture 4.7 of math.NT/0003238)