Log-crystalline representations and $(\varphi, Γ)$-modules

Abstract: Let $F$ be an absolutely unramified mixed characteristic local field with rings of integers $O_F$. For a small affine algebra $R$ over $O_F$, we consider the logarithmic étale fundamental group $G_R$ of its generic fibre equipped with a "horizontal" log-structure, in the sense of Fujiwara--Kato. We study $p$-adic representations of $G_R$ and obtain a classification of these representations in terms of étale $(\varphi, Γ_R)$-modules. Moreover, we define and study the notion of log-crystalline representations of $G_R$, a generalisation of crystalline representations from the non-logarithmic/smooth case. Furthermore, in the logarithmic setting, we show that log-crystalline representations of $G_R$ are equivalent to Wach modules for $R$, extending our previous results from the (non-logarithmic) crystalline case.