Height and Weber’s Class Number Problem

Class numbers in cyclotomic Zp-extensions

“…This conjecture is further supported by the work of Morisawa, who proved in his thesis [16] that all primes p less than 400,000 do not the class number of B 3,n for all n. Morisawa's result has recently been further improved by Fukuda, Komatsu and Morisawa [9], who proved that no prime less than 10 9 divides the class number of B 3,n .…”

Tate–Shafarevich groups in the cyclotomic Zˆ-extension and Weber's class number problem

“…), e ≥ 1, whose p-rank is a multiple of the residue degree ρ N of p in Q p (µ N )/Q p ; thus ρ N → ∞ as N → ∞, which is considered "incredible" for arithmetic invariants, as class groups, for totally real fields. Indeed, interesting examples occur more easily when p totally splits in Q(µ N ) (i.e., p ≡ 1 (mod N )) and this "explains" the result of [38] and [39] claiming that # C Q(ℓ n ) is odd in Q(ℓ ∞ ) for all ℓ < 500, that of [37,51,52] and explicit deep analytic computations in [5,10,11,14,36,37,38,39,48,49,51,52,59] (e.g., Washington's theorem [59] claiming that for ℓ and p fixed, # C K is constant for all n large enough, whence C * K = 1 for all n ≫ 0, then [14, Theorems 2, 3, 4, Corollary 1]); mention also the numerous pioneering Horie's papers proving results of the form: "let ℓ 0 be a small prime; then a prime p, totally inert in some Q(ℓ n 0 0 ), yields C Q(ℓ n 0 ) = 1 for all n". In [5], a conjecture (from "speculative extensions of the Cohen-Lenstra-Martinet heuristics") implies C * Q(ℓ n ) = 1 for finitely many layers (possibly none).…”

“…Indeed, one may ask if the arithmetic of these fields is as smooth as it is conjectured (for the class group C Q(N ) ) by many authors after many verifications and partial proofs [2,5,10,11,12,13,14,15,34,35,36,37,38,39,40,46,47,48,49,50,51,52,59]. The triviality of C Q(ℓ n ) has, so far, no counterexamples as ℓ, n, p vary, but that of the Tate-Shafarevich group T Q(ℓ n ) (or more generally T Q(N ) ) is, on the contrary, not true as we shall see numerically, and, for composite N , few C Q(N ) = 1 have been discovered.…”

Minimal relative units of the cyclotomic $\mathbb Z_2$-extension

“…, where h n denotes the class number of B n . Therefore, by studying RE + n , we may obtain many partial results supporting Weber's class number problem (e.g., [H1,H2,FK1,FK2,FK3,MO1,MO2]). More directly, the second author proved the following.…”

“…This problem can be converted into a problem of studying the unit groups O × Bn of B n by the index formula (for example, see [Ha,§9], [H2, (1)], [Si], [Wa,Theorem 8.2]). Horie [H1, H2], FK2,FK3], and Morisawa-Okazaki [MO1,MO2] actually obtained many partial results. For example, Fukuda and Komatsu [FK3,Theorem 1.3] showed that l ∤ h n for all n and for all primes l with l ≡ 1 mod 32…”