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

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

2004-09-13

arxiv.org/abs/math/0409207v1

Mathematics

Number Theory

20 pages, Plain TeX

Scientific paper

We define the notion of {\it Dwork family of logarithmic $F$-crystals}, a typical example of which is the family of Gauss hypergeometricdifferential 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. 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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