Bounds for computing the tame kernel

Abstract:

The tame kernel of the $K_2$ of a number field $F$ is the kernel of some explicit map $K_2F\to \bigoplus k_v^*$, where the product runs over all finite primes $v$ of $F$ and $k_v$ is the residue class field at $v$. When $S$ is a set of primes of $F$, containing the infinite ones, we can consider the $S$-unit group $U_S$ of $F$. Then $U_S\otimes U_S$ has a natural image in $K_2F$. The tame kernel is contained in this image if $S$ contains all finite primes of $F$ up to some bound. This is a theorem due to Bass and Tate. An explicit bound for imaginary quadratic fields was given by Browkin. In this article we give a bound, valid for any number field, that is smaller than Browkin’s bound in the imaginary quadratic case and has better asymptotics. A simplified version of this bound says that we only have to include in $S$ all primes with norm up to $4|\Delta |^{3/2}$, where $\Delta$ is the discriminant of $F$. Using this bound, one can find explicit generators for the tame kernel, and a “long enough” search would also yield all relations. Unfortunately, we have no explicit formula to describe what “long enough” means. However, using theorems from Keune, we can show that the tame kernel is computable.