Tate-Shafarevich groups in the cyclotomic (Z)over-cap-extension and Weber's class number problem

[摘要] Let K = Q(N) be the Nth layer in the cyclotomic (Z) over cap -extension. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington, ...) prove results on the p-class groups l(K). We enlarge Weber's problem to the Tate-Shafarevich groups IIIK1 similar or equal to l(K)(Sp) [p] and IIIK2 having same p-rank as the more easily computable torsion group, J(K), of the Galois group of the maximal abelian p-ramified pro-p-extension of K; but J(K) is often non-trivial, which raises questions for class groups since #J(K) = #l(K) = #l(K) #R-K, where R-K is the normalized p-adic regulator. We give a new method testing J(K) not equal 1 (Theorem 4.6, Table in Appendix A.7) and characterize the fields K-1 = KQ(p) with l(K1 )not equal 1 (Main Theorem 1.1 affirming, for short, that l(K1) not equal 1 if and only if p totally splits in K and J(K) not equal 1); this highlights the analytical results and justifies the eight known examples. All PARI/GP programs are given for further investigations. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-01

[关键词] p-class groups;Cyclotomic Z(l)-extensions;Class field theory;p-adic regulators;p-ramification theory;PARI/GP programs