Genus theory of p-adic pseudo-measures -- Tame kernels and abelian p-ramification
Résumé
We consider the Birch--Tate formula, linking the order of the tame Hilbert kernel K_2Z_k to ζ(-1), for real abelian fields k, and give a method to compute #K_2Z_k. We utilize the regular kernel R_2Z_k of index 2^[k:\Q] in K_2Z_k, in relationship with the torsion group T_k of abelian p-ramification theory, whose order is given by the residue of the p-adic ζ-function ζ_p(s) of k at s=1. We intend to compute ζ_p(s), using Stickelberger's p-adic pseudo-measures and to compare #R_2Z_k and #T_k, by means of the powerful ``genus theory'' of these p-adic pseudo-measures, which was inaugurated in the 1970/80's, when k is cyclic of degree d p^e, e>0.
As application, we prove a conjecture of Deng--Li dealing with the 2-part of K_2Z_k of an interesting family of real quadratic fields. PARI programs and various tables are given for quadratic and cyclic cubic fields, with p=2,3.
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