Asymptotic cohomology of circular units

TL;DR

This paper computes the Tate cohomology groups of circular units in the cyclotomic Z_p-extension of abelian number fields, providing explicit asymptotic results without additional assumptions.

Contribution

It offers the first explicit computation of Tate cohomology groups of circular units in this setting, independent of extra conditions on the field or prime.

Findings

Computed Tate cohomology groups for circular units in cyclotomic extensions

Provided asymptotic formulas for Galois actions on circular units

Achieved results without restrictions on the number field or prime

Abstract

Let $F$ be a number field, abelian over the rational field, and fix a odd prime number $p$. Consider the cyclotomic $Z_p$-extension $F_\infty/F$ and denote $F_n$ the ${n}^{\rm th}$ finite subfield and $C_n$ its group of circular units. Then the Galois groups $G_{m,n}=\Gal(F_m/F_n)$ act naturally on the $C_m$'s (for any $m\geq n>> 0$). We compute the Tate cohomology groups $\Hha^i(G_{m,n}, C_m)$ for $i=-1,0$ without assuming anything else neither on $F$ nor on $p$.