Logarithmic growth and Frobenius filtrations for solutions ofp-adic differential equations

Abstract

AbstractFor a ∇-moduleMover the ringK[[x]]0of bounded functions over ap-adic local fieldKwe 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, ifMalso admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case ofMa ϕ–∇-module of rank 2, the Frobenius polygon forMand 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.