For an unramified extension $F/\mathbb{Q}_p$ with perfect residue field, by works of Fontaine, Colmez, Wach and Berger it is well-known that the category of Wach modules over a certain integral period ring $\mathbf{A}_F^+$ is equivalent to the category of lattices inside crystalline representations of $G_F$, i.e. the absolute Galois group of $F$. Moreover, by recent work of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_F$ are equivalent to the category of prismatic $F$-crystals over $O_F$, i.e. the ring of integers of $F$. The goal of this talk is to present a direct construction of the categorical equivalence between Wach modules over $\mathbf{A}_F^+$ and prismatic $F$-crystals over $O_F$. If time permits, we will also mention generalization of our construction to the relative case as well as relations between Wach modules and $q$-connections.

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