A $p$-adic local monodromy theorem

Abstract

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 André 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.