Tate–Shafarevich groups in the cyclotomic Z^-extension and Weber’s class number problem

Final publication in: Journal of Number Theory 228 (2021), 219-252.https://doi.org/10.1016/j.jnt.2021.04.019Let K=Q(N) be the Nth layer in the cyclotomic Z^-extension. Many authors (Aoki, Fukuda, Horie, Ichimura, Inatomi, Komatsu, Miller, Morisawa, Nakajima, Okazaki, Washington) prove results on the p-class groups C_K. We enlarge ``Weber's problem'' to the Tate--Shafarevich groups Cha^1_K=C_K^S_p (S_p-class group) and Cha^2_K having same p-rank as the more easily computabletorsion group T_K, of the Galois group of the maximal abelian p-ramified pro-p-extension of K; but T_K is often non-trivial, which raises questions for class groups since #T_K = # C_K #R_K, where R_K is the normalized p-adic regulator.We give a new method testing T_K ne 1 (Theorem 4.6, Table 6.2) and characterize the p-extensions K_1=KQ(p) with C_K_1≠1 (Main Theorem 1.1 affirming, for short, that C_K_1≠1 if and only if p totally splits in K and T_K ≠1); this highlights the analytical results and justifies the 8 known examples. All PARI programs are given for further investigations