p-adic formulas and unit root F-subcrystals of the hypergeometric system

This article is dedicated to the study of $p$-adic analytic continuation of the unit root $F$-subcrystal of a logarithmic $F$-crystal in the open tube of a singularity. We will show how the possibility of extending that unit root crystal as a {\it non-singular} crystal in more singular classes, originates non-trivial formulas of analytic continuation of classical functions. We improve on Katz' treatment at least in that we allow logarithmic singularities. A typical example of a formula of this type is the Koblitz-Diamond formula for the analytic continuation of a Dwork function related to the classical Gauss hypergeometric function.