On the Hodge--Tate crystals over O_K

Abstract

We prove that a Hodge-Tate prismatic crystal on (O K ) ∆ is uniquely determined by a topologically "nilpotent" matrix. Using this matrix, we construct a Cp-representation of G K from a Hodge-Tate crystal in an explicit way. We then compute the cohomology of a Hodge-Tate crystal by using this matrix and obtain the cohomological dimension of a crystal.As an application, when p > 2, we show the crystalline Breuil-Kisin modules admit "nilpotent connections", which is a new result on (ϕ, τ )-modules.This "connection" is conjectured predicting Hodge-Tate weights of associated crystalline representations. We confirm the conjecture in the rank-one case, the Barsotti-Tate case and the extended Fontaine-Laffaille case. Moreover, we give another description of prismatic crystals, by which the crystalline Breuil-Kisin modules in the above three cases can induce some prismatic F -crystals.These are special cases of Bhatt-Scholze's equivalence in [BS21].

The prismatization of $p$-adic formal schemes

“…Higgs field which is nilpotent modulo p, with the twist {−1} indicating that Θ acts by −1 on Ω 1 X/W (k) {−1}. This description was recently also discovered in [19,20] under the name of "enhanced Higgs bundles" in the more general case where W (k)[1/p] is replaced by a possibly ramified discretely valued extension K/Q p with perfect residue field; we expect that this stronger statement can also be proven using the geometric approach used above by replacing G ♯ m with the group scheme G (which equals G ♯ a when K is ramified) arising from the calculation in Example 9.6, but we do not pursue this idea further here.…”

“…To prove this, we can assume without loss of generality that the set I is a singleton, in which case the desired result follows from Example A. 19.…”

P-adic Simpson correpondence via prismatic crystals

“…At the same time, for a small smooth formal scheme Spf(R) over A/I with (A, I) a transversal prism, in [Tian21], Tian established an equivalence between the category of Hodge-Tate crystals on the relative prismatic site (R/(A, I)) ∆ and the topologically quasi-nilpotent Higgs modules over R. Although their constructions only work for small affines, they still predict a relation between prismatic theory and p-adic non-abelian Hodge theory. For this sake, we hope our work (together with our previous paper [MW21b]) might be the valuable attempt in this direction. Some results in this paper are still restricted to the rational case.…”

“…For the other direction, we construct an inverse Simpson functor from the category of Higgs bundles on X with "arithmetic Sen operators" to the category of generalised representations on X proét by using the prismatic theory developed in [BS19], especially the category of Hodge-Tate crystals on (X) ∆ . The main ingredient is the local computation of absolute prismatic cohomology, which is a generalisation of our previous work in [MW21b].…”

De Rham prismatic crystals over $\mathcal{O}_K$

“…Towards this direction, Yu Min and Yupeng Wang study Hodge-Tate crystals on O K in [MW21]. Later Hui Gao relates rational Hodge-Tate crystals on O K to nearly Hodge-Tate representations in [Gao22].…”