l-adic Turrittin

Question

What would be an l-adic analogue of the Turrittin-Levelt decomposition theorem?

Turrittin-Levelt is a structure theorem of meromorphic connections on a complex curve in the formal neighbourhood of a singularity.

Answer 1

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!)

Levelt-Turrittin 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 1-dimensional 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 1-dimensional characters of inertia with tamely ramified characters.

However, this is false -- any such thing is a sum of 1-dimensional characters when restricted to the wild inertia subgroup. To construct something that doesn't have this property, take a finite totally ramified non-abelian 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 non-abelian 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.