Number Theory Seminar

Speaker: Abhinandan (Tokyo)

Title: Crystalline representations, Wach modules and prismatic F-crystals

Abstract: For the field of p-adic numbers Q_p and its absolute Galois group G_Qp, by the works of Fontaine, Colmez, Wach and Berger, it is well known that the category of lattices inside crystalline representations of G_Qp is equivalent to the category of Wach modules, i.e. certain (φ, Γ)-modules, over the integral period ring A_Qp⁺. Moreover, by the recent work of Bhatt and Scholze, it is also known that the category of lattices inside crystalline representations of G_Qp is equivalent to the category of prismatic F-crystals over the absolute prismatic site of the p-adic formal scheme Spf(Z_p). The goal of this talk is to present various functors relating these categories, in particular, a new and direct construction of the categorical equivalence between Wach modules over A_Qp⁺ and prismatic F-crystals over the absolute prismatic site of Spf(Z_p). If time permits, we will also mention the generalisation of our construction to the relative setting, as well as, relationships between relative Wach modules, q-connections and filtered (φ, ∂)-modules.

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