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Article Dans Une Revue Compositio Mathematica Année : 2008

p-adic confluence of q-difference equations

Résumé

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 $\xi$, and the stalk at a root of unity is a mixed object, formed by a differential equation and an action of $\sigma_\xi$.

Dates et versions

hal-00803815 , version 1 (22-03-2013)

Identifiants

Citer

Andrea Pulita. p-adic confluence of q-difference equations. Compositio Mathematica, 2008, Vol.144, p.867-919. ⟨10.1112/S0010437X07003454⟩. ⟨hal-00803815⟩
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