Note

Syntomic Tate Twists over p-adic Local Fields and the Cyclotomic Torsion Groups

Author: Wei Yang · Added: June 29, 2026

Article date: June 2026.

Abstract

Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}_K$ . We compute the BMS syntomic complexes

\mathbb{Z}_p(i)(\mathcal{O}_K) = \mathrm{R}\Gamma_{\mathrm{syn}}(\mathrm{Spec}\,\mathcal{O}_K,\mathbb{Z}_p(i))

for all integer weights $i \in \mathbb{Z}$ . For $i < 0$ the complexes vanish. For $i \geq 0$ the calculation reduces, through the comparison of Bhatt—Mathew with Geisser—Sato—Schneider $p$ -adic etale Tate twists, to local Galois cohomology.

The answer is expressed in terms of the completed unit group, the rank $[K:\mathbb{Q}_p]$ , and the finite cyclotomic torsion groups

W_m(K)=H^0(K,\mathbb{Q}_p/\mathbb{Z}_p(m)).

The note also gives a complete calculation of $W_m(K)$ from the image of the $p$ -adic cyclotomic character.