On the Zp-extensions of a totally p-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent
Résumé
Let k be an imaginary quadratic field, and let p be an odd prime
number split in k. We analyze some properties of arbitrary Zp-extensions K/k.
These properties are governed by the Hase norm residue symbol of the fundamental
p-unit of k, in terms of the valuation δp(k) of a Fermat quotient, which determines
the order of the logarithmic class group Clogk (Theorem 2.2) and leads, under some
conditions, to generalizations of Gold's criterion characterizing λp(K/k) = 1
(Theorems 3.3, 5.1, 5.3). This uses the higher rank Chevalley--Herbrand
formulas, for the filtrations of the p-class groups in K, that
we gave in the 1994's, and the theorem of λ-stability (2022).
This study is in connection with articles of Gold, Sands,
Dummit--Ford--Kisilevsky--Sands, Ozaki, Jaulent, Fujii. In Appendix A, is given
a general proof, by Jaulent, of the link between Clogk and δp(k) in a
broader context. Numerical illustrations (with pari/gp programs) are given.
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