Weber's class number problem and p--rationality in the cyclotomic Z^-extension of Q
Résumé
Let K:=Q(ℓ^n) be the nth layer of the cyclotomic Z_ℓ-extension. It is conjectured that K is principal (Weber's conjecture for ℓ=2). Many studies (Ichimura--Miller--Morisawa--Nakajima--Okazaki) go in this direction. Nevertheless, we examine if a counterexample may be possible. For this, computations show that the p-torsion group T_K of the Galois group of the maximal abelian p-ramified pro-p-extension of K is not always trivial; whence the relevance of the conjecture since #T_K = #C_K # R_K (up to a canonical 2-power if p=2), where C_K is the p-class group and R_K the normalized p-adic regulator. We give a new method (Theorem 4.6 testing #T_K≠1), allowing larger values of ℓ^n than those of the literature. Finally, we search in the cyclotomic Z^--extension, cases of non-trivial class groups using genus theory related to a deep property of R_K (Theorem 6.3); we only find again the three known cases (Fukuda--Komatsu--Horie).
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