Frobenius and monodromy operators in rigid analysis, and Drinfel’d’s symmetric space

Abstract

We define Frobenius and monodromy operators on the de Rham cohomology of K K -dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction Y Y , over a complete discrete valuation ring K K of mixed characteristic. For this we introduce log rigid cohomology and generalize the so-called Hyodo-Kato isomorphism to versions for non-proper Y Y , for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of Y Y . We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel’d’s symmetric space X X and its quotients. Our results are used in a critical way in the recent proof of the monodromy-weight conjecture for quotients of X X given by de Shalit (2005).