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Crystalline representations, Wach modules and Prismatic F-crystals

组织者

刁晗生 , 胡悦科 , 埃马纽埃尔·勒库图里耶 , 凯撒·鲁普

演讲者

Abhinandan

时间

2024年03月25日 10:00 至 11:00

地点

Online

线上

Zoom 271 534 5558 (YMSC)

摘要

For an absolutely unramified extension $K/Q_p$ with perfect residue field, by the works of Fontaine, Colmez, Wach, and Berger, it is well known that the category of Wach modules over a certain integral period ring is equivalent to the category of lattices inside crystalline representations of $G_K$ (the absolute Galois group of $K$). Moreover, by the recent works of Bhatt and Scholze, we also know that lattices inside crystalline representations of $G_K$ are equivalent to the category of prismatic $F$-crystals on the absolute prismatic site of the ring of integers of $K$. The goal of this talk is to present a generalisation of these results to a "small" relative base ring and discuss a direct construction of the categorical equivalence between relative Wach modules and prismatic $F$-crystals over the absolute prismatic site of the base ring. If time permits, we will also mention relationships between relative Wach modules, $q$-connections and filtered phi-modules with connections.