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 $\mathrm{\#}{\mathbf{K}}_2({\mathbf{Z}}_K)$ to ${\zeta }_K(-1)$; we compare, for quadratic and cyclic cubic fields with $p\in \{2,3\}$, $\mathrm{\#}{\mathbf{K}}_2({\mathbf{Z}}_K)[p^{\mathrm{\infty }}]$ 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 ${\mathbf{K}}_2({\mathbf{Z}}_K)[2^{\mathrm{\infty }}]$ for an interesting family of real quadratic fields (Theorem B). Then, for $p\ge 5$, we give a lower bound of ${\mathrm{r}\mathrm{k}}_p({\mathbf{K}}_2({\mathbf{Z}}_K))$ 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

Mots-clés : 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