Tate kernels, etale K-theory and the Gross kernel

Abstract

For an odd prime $p$ and a number field $F$ containing a $p$th root of unity, we study generalised Tate kernels, $D_F^{[i,n]}$, for $i\in \mathbb{Z}$ and $n\ge 1$, having the properties that if $i\ge 2$ and if either $p$ does not divide $i$ or ${\mu }_{p^n}\subset F$ then there are natural isomorphisms $D_F^{[i,n]}\cong K^{\mbox{\tiny ét}}_{2i-1}(O_F^S)/p^n$, and that they are periodic modulo a power of $p$ which depends on $F$ and $n$. Our main result is that if the Gross-Jaulent conjecture holds for $(F,p)$ then there is a natural isomorphism $D_F^{[i,n]}\cong {\mathcal{E}}_F/p^n$ where ${\mathcal{E}}_F$ is the Gross kernel. We apply this result to compute lower bounds for capitulation kernels in even étale $K$-theory.

26pp. Draft preprint from 2005. I don't intend to submit it to a journal. This preprint continues to be cited from time to time, however

Cited by in corpus (1)

• Galois Codescent For Motivic Tame Kernels