Rigid analytic reconstruction of Hyodo--Kato theory
Veronika Ertl, Kazuki Yamada
TL;DR
This work develops a rigid-analytic reconstruction of Hyodo--Kato theory by formulating rigid Hyodo--Kato cohomology $R\Gamma_{\mathrm{HK}}^{\mathrm{rig}}(\mathcal{X})$ via a Kim--Hain complex on dagger spaces over the open unit disk and introducing a rigid Hyodo--Kato map $\Psi_{\pi,q}$ to de Rham cohomology. It proves independence from the uniformiser and clarifies the dependence on branches of the $p$-adic logarithm, establishing compatibility with the classical crystalline Hyodo--Kato theory. The framework relies on weak formal schemes and log structures to enable explicit, computable descriptions (including Čech cocycle presentations) and provides concrete examples, such as Tate curves, to illustrate the theory. The results connect rigid-analytic constructions with the semistable conjecture’s arithmetic objectives and indicate potential extensions to log overconvergent $F$-isocrystals, making Hyodo--Kato theory more amenable to computation and generalisation. The paper thus offers a flexible, explicit, and uniformisable approach to Hyodo--Kato theory in the rigid setting, with broad implications for $p$-adic Hodge theory and arithmetic geometry.
Abstract
We give a new and very intuitive construction of Hyodo--Kato cohomology and the Hyodo--Kato map, based on logarithmic rigid cohomology. We show that it is independent of the choice of a uniformiser and study its dependence on the choice of a branch of the $p$-adic logarithm. Moreover, we show the compatibility with the classical construction of Hyodo--Kato cohomology and the Hyodo--Kato map.)