Genus theory of $p$-adic pseudo-measures, Hilbert kernels and abelian $p$-ramification

Georges Gras¹

¹ 38520 Le Bourg d’Oisans, France <a href="https://orcid.org/0000-0002-1318-4414">ORCID: 0000-0002-1318-4414</a>

Acta Arithmetica, Tome 218 (2025) no. 1, pp. 25-63

Résumé

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. Published in Open Access (under CC-BY license).

Détail

DOI : 10.4064/aa240130-9-6

Keywords: consider real abelian fields birch tate formula linking zeta compare quadratic cyclic cubic fields infty order torsion group mathcal abelian p ramification theory given residue zeta done mathbb via genus theory p adic pseudo measures inaugurated theorem apply prove conjecture deng giving structure infty interesting family real quadratic fields theorem geq lower bound cyclic p extensions mathbb theorem complements numerical illustrations pari programs given appendices