arXiv subjects

Matti Würthen

Publications and source records attributed to Matti Würthen.

4 recordsLinked to original sources

arXiv2023

Log prismatic Dieudonné theory for log $p$-divisible groups over $\mathcal{O}_{K}$

Let $\mathcal{O}_{K}$ be a complete discrete valuation ring of mixed characteristic with perfect residue field, endowed with its canonical log-structure. We prove that log $p$-divisible groups over $\mathcal{O}_{K}$ correspond to Dieudonné crystals on the absolute log-prismatic site of $\mathcal{O}_{K}$ endowed with the Kummer log-flat topology. The proof uses log-descent to reduce the problem to the classical prismatic correspondence, recently established by Anschütz-Le Bras.

math.NT↗

arXiv2023

Divided prismatic Frobenius crystals of small height and the category $\mathcal{M}\mathcal{F}$

Let $\mathcal{X}$ be a smooth $p$-adic formal scheme over a mixed characteristic complete discrete valuation ring $\mathcal{O}_{K}$ with perfect residue field. We introduce a general category $\mathcal{M}\mathcal{F}_{[0, p-2]}^{tor-free}(\mathcal{X})$ of $p$-torsion free crystalline coefficient objects and show that this category is equivalent to the category of completed prismatic Frobenius crystals of height $p-2$, recently introduced by Du-Liu-Moon-Shimizu. In particular this shows that the category $\mathcal{M}\mathcal{F}^{tor-free}_{[0, p-2]}(\mathcal{X})$ is equivalent to the category of crystalline $\mathbb{Z}_p$-local systems on $\mathcal{X}$ with Hodge-Tate weights in $\{0,\ldots , p-2\}$, which generalizes the crystalline part of a theorem of Breuil-Liu to higher dimensions.

math.AG↗

arXiv2023

Log p-divisible groups associated to log 1-motives

We first provide a detailed proof of Kato's classification theorem of log $p$-divisible groups over a noetherian henselian local ring. Exploring Kato's idea further, we then define the notion of a standard extension of a classical finite étale group scheme (resp. classical étale $p$-divisible group) by a classical finite flat group scheme (resp. classical $p$-divisible group) in the category of finite Kummer flat group log schemes (resp. log $p$-divisible groups), with respect to a given chart on the base. These results are then used to prove that log $p$-divisible groups are formally log smooth. We then study the finite Kummer flat group log schemes $T_n(\mathbf{M}):=H^{-1}(\mathbf{M}\otimes_{\mathbb{Z}}^L\mathbb{Z}/n\mathbb{Z})$ (resp. the log $p$-divisible group $\mathbf{M}[p^{\infty}]$) of a log 1-motive $\mathbf{M}$ over an fs log scheme and show that they are étale locally standard extensions. Lastly, we give a proof of the Serre-Tate theorem for log abelian varieties with constant degeneration.

math.AG↗

arXiv2020

Vector bundles with numerically flat reduction on rigid analytic varieties and $p$-adic local systems

We show how to functorially attach continuous $p$-adic representations of the profinite fundamental group to vector bundles with numerically flat reduction on a proper rigid analytic variety over $\mathbb{C}_p$. This generalizes results by Deninger and Werner for vector bundles on smooth algebraic varieties. Our approach uses fundamental results on the pro-étale site of a rigid analytic variety introduced by Scholze. This enables us to get rid of the smoothness condition and to work in the analytic category. Moreover, under some mild conditions, the functor we construct gives a full embedding of the category of vector bundles with numerically flat reduction into the category of continuous $\mathbb{C}_p$-representations. This provides new insights into the $p$-adic Simpson correspondence in the case of a vanishing Higgs-field.

math.AG↗