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p-adic confluence of q-difference equations

Part of: Differential and difference algebra Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  01 July 2008

Andrea Pulita
Andrea Pulita*
Affiliation:
SFB701, Spectral Structures and Topological Methods in Mathematics, Fakultät für Mathematik, Universität Bielefeld, PO Box 100 131, D-33501 Bielefeld, Germany (email: pulita@math.uni-bielefeld.de)
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Abstract

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We develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a 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 develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.

Keywords

p-adic q-difference equationsp-adic differential equationsconfluencedeformationunipotentp-adic local monodromy theorem

MSC classification

Secondary: 12H25: $p$-adic differential equations 12H05: Differential algebra 12H10: Difference algebra 12H20: Abstract differential equations 12H99: None of the above, but in this section 11S15: Ramification and extension theory 11S20: Galois theory
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 4 , July 2008 , pp. 867 - 919
Copyright
Copyright © Foundation Compositio Mathematica 2008