Compositio Mathematica

Research Article

Semistable reduction for overconvergent F-isocrystals, III: Local semistable reduction at monomial valuations

Kiran S. Kedlayaa1

a1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA (email:


We resolve the local semistable reduction problem for overconvergent F-isocrystals at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree zero). We first introduce a higher-dimensional analogue of the generic radius of convergence for a p-adic differential module, which obeys a convexity property. We then combine this convexity property with a form of the p-adic local monodromy theorem for so-called fake annuli.

(Received September 14 2007)

(Accepted June 09 2008)

2000 Mathematics Subject Classification

  • 14F30 (primary);
  • 14F40 (secondary)


  • rigid cohomology;
  • overconvergent isocrystals;
  • semistable reduction;
  • monomial valuations


The author was supported by NSF grant DMS-0400727, NSF CAREER grant DMS-0545904, and a Sloan Research Fellowship.