Milnor $K$-group attached to a torus and Birch-Tate conjecture

TL;DR

This paper formulates and proves a conjecture linking the order of a Milnor K-group associated with a torus to the value of an Artin L-function at s=-1, generalizing the Birch-Tate conjecture.

Contribution

It introduces a new conjecture relating Milnor K-groups of tori to Artin L-functions and proves it under certain assumptions, extending classical results.

Findings

Conjecture relating Milnor K-groups and Artin L-functions formulated.

Proof of the conjecture provided under specific assumptions.

Special case reduces to the classical Birch-Tate conjecture.

Abstract

We formulate (and prove under a certain assumption) a conjecture relating the order of Somekawa's Milnor $K$-group attached to a torus $T$ and the value of the Artin $L$-function attached to the cocharacter group of $T$ (regarded as an Artin representation) at $s=-1$. The case $T=\G_m$ reduces to the classical Birch-Tate conjecture.