On the Hodge--Tate crystals over O_K
Abstract
We prove that a Hodge--Tate prismatic crystal on (O_K)_{\Prism} is uniquely determined by a topologically "nilpotent" operator. Using this operator, we construct a C_p-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 operator and obtain the cohomological dimension of a crystal. In particular, we conjecture that this operator is essentially the classical Sen operator. As applications, under some mild assumption, we show the crystalline Breuil--Kisin modules admit "nilpotent connections" and give an explicit description of prismatic crystals. This "connection" is conjectured predicting the Hodge--Tate weights of associated crystalline representations.
Keywords
Cite
@article{arxiv.2112.10140,
title = {On the Hodge--Tate crystals over O_K},
author = {Yu Min and Yupeng Wang},
journal= {arXiv preprint arXiv:2112.10140},
year = {2023}
}
Comments
All results in this preprint are now contained (as a subset) in the preprint 2311.07024, except Section 5, which will be discussed elsewhere. Please read that preprint instead. We apologise for confusions