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Local solutions to positive characteristic non-Archimedean differential equations

Part of: Differential and difference algebra Field extensions Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  01 November 2007

João Pedro Pinto dos Santos
João Pedro Pinto dos Santos*
Affiliation:
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France (email: joao-pedro.dos-santos@univ-rennes1.fr)
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Abstract

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In the complex domain, one can integrate (solve) holomorphic ordinary differential equations (ODEs) near a non-singular point. We study the existence of solutions in the case of a positive characteristic base field k which is complete with respect to a non-Archimedean absolute value. ODEs are substituted by modules over a ring of analytic functions endowed with an action of all differential operators. The monodromy groups associated to the corresponding category are computed.

Keywords

differential operators in positive characteristicTannakian fundamental groupsrigid geometrydifferential Galois theory

MSC classification

Secondary: 12H05: Differential algebra 12F12: Inverse Galois theory 14G22: Rigid analytic geometry
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 6 , November 2007 , pp. 1465 - 1477
Copyright
Copyright © Foundation Compositio Mathematica 2007

References

This work is part of the author’s Cambridge PhD thesis supervised by N. I. Shepherd-Barron and financially supported by Trinity College.