Click here to flash read.
arXiv:2403.16603v3 Announce Type: replace
Abstract: 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 $\delta$p(k) of a Fermat quotient, which determines the order of the logarithmic class group Clogk (Theorem 2.2, Appendix 1) and leads, under some conditions, to generalizations of Gold's criterion characterizing $\lambda$p(K/k) = 1 (Theorems 5.1, 5.3, 5.5). 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 $\lambda$-stability (2022). This study is in connection with articles of Gold, Sands, Dummit--Ford--Kisilevsky--Sands, Ozaki, Hubbard--Washington,Kataoka, Ray, Jaulent, Fujii. In Appendix A, is given a general proof, by Jaulent, of the link between Clogk and $\delta$p(k) in a broader context. Numerical illustrations (with pari/gp programs) are given.
No creative common's license