  We establish a generalization of the p-adic local monodromy theorem (of
Andre, Mebkhout, and the author) in which differential equations on rigid
analytic annuli are replaced by differential equations on so-called fake
annuli. The latter correspond loosely to completions of a Laurent polynomial
ring with respect to a monomial valuation. The result represents a step towards
a higher-dimensional version of the p-adic local monodromy theorem (the
"problem of semistable reduction"); it can also be viewed as a slightly novel
presentation of the original p-adic local monodromy theorem.
