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Logarithmic growth and Frobenius filtrations for solutions of p-adic differential equations

Part of: Differential and difference algebra

Published online by Cambridge University Press:  18 February 2009

Bruno Chiarellotto  and
Nobuo Tsuzuki
Bruno Chiarellotto
Affiliation:
Dipartimento Matematica Pura e Applicata, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy, ([email protected]).
Nobuo Tsuzuki
Affiliation:
Mathematical Institute, Tohoku University, 6-3 Aza-Aoba, Aramaki, Aoba-ku, Sendai, 980-8578, Japan, ([email protected]).
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Abstract

For a ∇-module M over the ring K[[x]]0 of bounded functions over a p-adic local field K we define the notion of special and generic log-growth filtrations on the base of the power series development of the solutions and horizontal sections. Moreover, if M also admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case of M a ϕ–∇-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, Mv, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork.

Keywords

p-adic differential equationslogarithmic growthNewton polygons

MSC classification

Secondary: 12H25: $p$-adic differential equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 3 , July 2009 , pp. 465 - 505
Copyright
Copyright © Cambridge University Press 2009

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