On the cohomology of negative Tate twists via cyclotomic descent

TL;DR

This paper demonstrates that the Galois cohomology of negative Tate twists can be understood through a universal cyclotomic complex, with explicit descriptions in the case of $Q_p/Z_p$.

Contribution

It introduces a cyclotomic descent approach to organize negative Tate twists via a universal complex, providing explicit cohomology descriptions.

Findings

Negative Tate twists are localized on specific cyclotomic branches.

Cohomology can be recovered by specializing the Iwasawa variable.

Explicit descriptions of $H^1$ and $H^2$ are obtained for $Q_p/Z_p$.

Abstract

We show that the Galois cohomology of negative Tate twists can be organized by a single universal cyclotomic complex over the cyclotomic tower of $\mathbb{Q}$. Using cyclotomic descent and Teichm\"uller branch decomposition, we prove that a negative twist contributes only on the corresponding branch and is recovered by specializing the Iwasawa variable at a single point; equivalently, it is computed as the fiber of $\gamma-u^{-m}$, or $T=u^{-m}-1$ in Iwasawa coordinates. In the case $\mathbb{Q}_p/\mathbb{Z}_p$, this gives explicit descriptions of $H^1$ and $H^2$ in terms of the quotient and torsion of the $S$-ramified Iwasawa module.