Milnor $K$-group attached to a torus and 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.