Weber's class number problem and $p$-rationality in the cyclotomic $\widehat{\mathbb{Z}}$-extension of $\mathbb{Q}$

Abstract

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, 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). Contents 1. Introduction 1.1. Class groups and torsion groups of abelian p-ramification, in Q(ℓ ∞ ) 1.2. The p-torsion groups T K in number theory 1.3. The logarithmic class group and Greenberg's conjecture 2. Abelian p-ramification theory for totally real fields 2.1. Main definitions and notations -The p-invariants of K 2.2. The case of the fields K = Q(ℓ n ) 2.3. General computation of the structure of T Q(ℓ n ) 2.4. Algebraic and analytic aspects 3. Definition of p-adic measures 3.1. General definition of the Stickelberger elements 3.2. Multipliers of Stickelberger elements 3.3. Spiegel involution 4. Annihilation theorem of TTest on the normalized p-adic regulator 4.5. Conjecture about the p-torsion groups T Q(ℓ n ) 5. Reflection theorem for p-class groups and p-torsion groups 5.1. Case p = 2 5.2. Case p = 2 5.3. Illustration of formula (10) of Theorem 5.3 5.4. Probabilistic analysis from the reflection theorem 6. The p-torsion groups in Q