A p-adic local monodromy theorem

We produce a canonical filtration for locally free sheaves on an open p-adic annulus equipped with a Frobenius structure. Using this filtration, we deduce a conjecture of Crew on p-adic differential equations, analogous to Grothendieck's local monodromy theorem (also a consequence of results of Andr and of Mebkhout). Namely, given a finite locally free sheaf on an open p-adic annulus with a connection and a compatible Frobenius structure, the module admits a basis over a finite cover of the annulus on which the connection acts via a nilpotent matrix. 4. The special Newton polygon 4.1. Properties of eigenvectors 4.2. Existence of eigenvectors 4.3. Raising the Newton polygon 4.4. Construction of the special Newton polygon 5. The generic Newton polygon 5.1. Properties of eigenvectors 5.2. The Dieudonn-Manin classification 5.3. Slope filtrations 5.4. Comparison of the Newton polygons