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

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 ψ-∇-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, M^v, 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.