Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification

Abstract

We consider, for real abelian fields K, the Birch--Tate formula linking the tame kernel \#K\_2(Z\_K) to $\zeta$\_K(-1); we compare, for quadratic and cyclic cubic fields with p=2,3, \#K\_2(\BZ\_K)[p^$\infty$] to the order of the torsion group T\_{K, p} of abelian p-ramification theory given by the residue of $\zeta$\_{K, p}(s) at s=1. This is done via the ``genus theory'' of p-adic pseudo-measures, inaugurated in the 1970/80's and the fact that T\_{K, p} only depends on the p-class group and on the normalized p-adic regulator of K (Theorem A). We apply this to prove a conjecture of Deng--Li giving the structures of K\_2(Z\_K)[2^$\infty$] for an interesting family of real quadratic fields (Theorem B). Then, for p>3, we give a lower bound of the p-rank of K\_2(\BZ\_K) in cyclic p-extensions (Theorem C). Complements, PARI programs and tables are given in an Appendix.