$p$-typical curves on $p$-adic Tate twists and de Rham-Witt forms

TL;DR

The paper resolves Artin–Mazur's question by showing that animated de Rham–Witt forms $\mathbb{L}W\Omega^{n-1}_S$ are naturally isomorphic to $p$-typical curves on the Tate twists $\mathbf{Z}_p(n)[n]$ for any quasisyntomic $\mathbf{F}_p$-algebra $S$, via a motivic filtration framework on $\mathrm{TR}$ and $\mathrm{TC}$ inspired by Bhatt–Morrow–Scholze. It develops an animated Cartier–de Rham–Witt theory, establishes fpqc descent and pro-descent results, and defines the $p$-typical curves $\mathbb{D}(\mathbf{Z}_p(n)[n]_S)$ with Frobenius and Verschiebung, proving $\mathbb{L}W\Omega^{n-1}_S \simeq \mathbb{D}(\mathbf{Z}_p(n)[n]_S)$. The main technical advance is enhancing Hesselholt's isomorphism with a filtered, descent-friendly picture that identifies the graded pieces of $\mathrm{TR}$ with $\mathbb{L}W\Omega^n_S[n]$, and then passing to limits over truncated polynomials to obtain the global isomorphisms. This yields a uniform framework connecting crystalline slope decompositions to de Rham–Witt theory and provides a conceptual reconstruction of $\mathbb{L}W\Omega^*_S$ from $p$-adic Tate twists, with potential applications to $p$-adic Hodge theory and arithmetic geometry.

Abstract

We show that de Rham--Witt forms are naturally isomorphic to $p$-typical curves on $p$-adic Tate twists, which answers a question of Artin--Mazur from 1977 pursued in the earlier work of Bloch and Kato. We show this by more generally equipping a related result of Hesselholt on topological cyclic homology with the motivic filtrations introduced by Bhatt--Morrow--Scholze.

Theorems & Definitions (89)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Remark 1.4
• Proposition 1.5
• Theorem 1.6
• Proposition 1.7
• Proposition 1.8
• Proposition 2.2
• proof