Genus theory of $p$-adic pseudo-measures, Hilbert kernels and abelian $p$-ramification
Abstract
We consider, for real abelian fields $K$, the Birch–Tate formula linking $\# {\bf K}_2({\bf Z}_K)$ to $\zeta _K(-1)$; we compare, for quadratic and cyclic cubic fields with $p \in \{2, 3\}$, $\# {\bf K}_2({\bf Z}_K)[p^\infty ]$ with the order of the torsion group $\mathcal T_{K, p}$ of abelian $p$-ramification theory, given, for all $p$, by the residue of $\zeta _{K, p}(s)$ at $s=1$. This is done, when $p\,|\, [K : \mathbb Q]$, via the “genus theory” of $p$-adic pseudo-measures, inaugurated in the 1970/80’s (Theorem A). We apply this to prove a conjecture of Deng–Li giving the structure of ${\bf K}_2({\bf Z}_K)[2^\infty ]$ for an interesting family of real quadratic fields (Theorem B). Then, for $p \geq 5$, we give a lower bound of ${\rm rk}_p({\bf K}_2({\bf Z}_K))$ in cyclic $p$-extensions $K/\mathbb Q$ (Theorem C). Complements, numerical illustrations and PARI programs are given in the Appendices.
