On the Zp-extensions of a totally p-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent (original) (raw)

Let k be an imaginary quadratic field, and let p be an odd prime split in k. We analyze some properties of arbitrary Zp-extensions K/k. These properties are governed by the norm residue symbols of the fundamental p-unit x of k, in terms of the p-valuation, δp(k), of a suitable Fermat quotient of x, which also determines the order of the logarithmic class group (Theorem 2.2, Appendix A) and leads to some generalizations of the Gold-Sands criterion characterizing λp(K/k) = 1 (Results 5.1, 5.3, 5.4 in general, 5.5 for the cyclotomic case). This uses the higher rank Chevalley-Herbrand formulas, for the filtrations of the p-class groups in K, without any argument of Iwasawa's theory. This study is in connection with articles of Gold, Sands, Dummit-Ford-Kisilevsky-Sands, Hubbard-Washington, Ozaki, Kataoka, Ray, Jaulent, Fujii and others, and provides a different and elementary perspective on these questions. Illustrations (with pari/gp programs) are given in Appendix B.