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

TL;DR

This paper develops Frobenius and monodromy operators on the de Rham cohomology of rigid spaces with semistable reduction, introduces log rigid cohomology, and applies these to Drinfel'd's symmetric space, impacting the monodromy-weight conjecture proof.

Contribution

It generalizes the Hyodo-Kato isomorphism to non-proper, non-perfect residue field cases and applies it to Drinfel'd's symmetric space, advancing understanding of p-adic cohomology.

Findings

Defined Frobenius and monodromy operators on de Rham cohomology of rigid spaces.

Generalized Hyodo-Kato isomorphism to broader settings.

Applied results to prove cases of the monodromy-weight conjecture.

Abstract

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