# Cup products in the etale cohomology of number fields

## TL;DR

This paper explores cup product pairings in étale cohomology of number fields, linking Kim's invariants with McCallum and Sharifi's pairings through Ext groups, and provides formulas relating Kim's invariants to Artin maps.

## Contribution

It introduces a new pairing combining Kim's invariants with McCallum and Sharifi's, using Ext groups, and derives a formula for Kim's invariant in cyclic unramified Kummer extensions.

## Key findings

- A pairing that unifies Kim's and McCallum-Sharifi's invariants via Ext groups.
- A formula expressing Kim's invariant in terms of Artin maps.
- Existence of infinitely many number fields with trivial and non-trivial Kim invariants for cyclic groups.

## Abstract

This paper concerns cup product pairings in \'etale cohomology related to work of M. Kim and of W. McCallum and R. Sharifi. We will show that by considering Ext groups rather than cohomology groups, one arrives at a pairing which combines invariants defined by Kim with a pairing defined by McCallum and Sharifi. We also prove a formula for Kim's invariant in terms of Artin maps in the case of cyclic unramified Kummer extensions. One consequence is that for all $n > 1$, there are infinitely many number fields $F$ over which there are both trivial and non-trivial Kim invariants associated to cyclic groups of order $n$.