**FRANCESCO BALDASSARRI** and **MAURIZIO CAILOTTO**

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

**Pages:** 33-65

**Received:** 7 September 2004

**Mathematics Subject Classification (2000):** 11T23 - 11S31 - 12H25 - 14F30

**Abstract**:
In this paper, we define the notion of *it Dwork family of logarithmic
F-crystals*, a typical example of which is the family of Gauss
hypergeometric differential
systems, viewed as parametrized by their exponents of algebraic monodromy.
The p-adic analytic dependence of the Frobenius operation upon those
exponents, is Dwork's "Boyarsky Principle". We discuss, in favorable
cases, the p-adic analytic continuation of the unit root F-subcrystal
in the open tube of a singularity, uniformly w.r.t. the exponents.Using
these ideas, we obtain a conceptual proof of the Koblitz-Diamond formula
p-adically analog to Gauss' evaluation of F(a,b,c;1).
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