We prove in all generality that on a smooth complex quasi-projective variety $X$,   rigid connections  yield $F$-isocrystals on almost all good reductions $X_{\mathbb F_q}$ and that rigid local systems yield crystalline local systems on  $X_K$ for $K$ the field of fractions of the Witt vectors of a finite field $\mathbb F_q$, for almost all $X_{\mathbb F_q}$. This improves our earlier work where, if $X$ was not projective, we assumed a strong cohomological condition (which is fulfilled for Shimura varieties of real rank $\geq 2$), and we obtained only infinitely many $\mathbb F_q$ of growing characteristic. While the earlier proof was via characteristic $p$, the new one is purely $p$-adic and uses $p$-adic topology. 

We will explain the the definition of a perfectoid field and the tilting construction. We will then state the Fontaine-Wintenberger theorem and prove it in the algebraically closed case.