What would be an ladic analogue of the TurrittinLevelt decomposition theorem?
TurrittinLevelt is a structure theorem of meromorphic connections on a complex curve in the formal neighbourhood of a singularity.
What would be an ladic analogue of the TurrittinLevelt decomposition theorem? TurrittinLevelt is a structure theorem of meromorphic connections on a complex curve in the formal neighbourhood of a singularity. 


Here's a naive formulation of an analogue, which is false. (This fits very well the conditional phrasing from your question, since it would be an analogue if it were true!) LeveltTurrittin says that after a finite extension of the disc, every de Rham local system is a direct sum of factors which are of the form a 1dimensional de Rham local system tensor something with tame ramification (i.e., regular). The naive $\ell$adic guess would be that an $\ell$adic representation of a local Galois group is, after restricting to a finite index subgroup, a sum of factors which are tensor products of 1dimensional characters of inertia with tamely ramified characters. However, this is false  any such thing is a sum of 1dimensional characters when restricted to the wild inertia subgroup. To construct something that doesn't have this property, take a finite totally ramified nonabelian extension of a your field which is wild (i.e., the Galois group is a $p$group) and an irreducible representation of the corresponding finite Galois group with nonabelian image. However, Google seems to think that there is a (true) analogue in the theory of $p$adic differential equations. I don't know that subject, so I'll leave any such description to an expert in that field. 

