arXiv math/06127292006-12math.NT

\bs{p}-Adic Confluence of $\bs{q}$-Difference Equations

Andrea Pulita

备注：: 43 pages
DOI：: 10.1112/S0010437X07003454

英文摘要：

We develop the theory of $p$-adic confluence of $q$-difference equations. The main result is the surprising fact that, in the $p$-adic framework, a function is solution of a differential equation if and only if it is solution of a $q$-difference equation. This fact implies an equivalence, called ``Confluence'', between the category of differential equations and those of $q$-difference equations. We obtain this result by introducing a category of ``sheaves'' on the disk $\mathrm{D}^-(1,1)$, whose stalk at 1 is a differential equation, the stalk at $q$ is a $q$-difference equation if $q$ is not a root of unity $ξ$, and the stalk at a root of unity is a mixed object, formed by a differential equation and an action of $σ_ξ$.

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